The separable degree of an isogeny φ : E₁ → E₂, i.e., the degree of the separable part of
the induced extension of function fields.
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The separable degree of an isogeny is strictly positive.
Separable degree is multiplicative under composition of isogenies.
The multiplication-by-p map [p] : E → E viewed as an isogeny, where p is the
characteristic of the base field.
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The multiplication-by-p isogeny acts on points by the usual ℤ-scalar multiplication by
p.
For any isogeny φ : E₁ → E₂, pre- and post-composition with the multiplication-by-p maps
yield isogenies with the same separable degree.
The p-torsion subgroup of E: the set of points P killed by multiplication by p.
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An elliptic curve E over a field of characteristic p is supersingular if its p-torsion
subgroup is trivial.
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An elliptic curve E over a field of characteristic p is ordinary if it admits a nonzero
p-torsion point.
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E is ordinary iff it is not supersingular.
Supersingularity is equivalent to the multiplication-by-p isogeny having separable degree
1 (i.e., being purely inseparable).
Every elliptic curve over a field of characteristic p is either ordinary or supersingular.
Isogenous elliptic curves have the same separable degree for their multiplication-by-p
isogenies; in particular, this is an isogeny invariant.
Supersingularity is an isogeny invariant: if E₁ and E₂ are isogenous, then E₁ is
supersingular iff E₂ is.
Scalar multiplication by an integer m (acting on E.Point via m • id) gives a well-defined
algebraic endomorphism of E.
The endomorphism of E given by multiplication by the integer m, packaged as an element of
the endomorphism ring of E.
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The Frobenius endomorphism of E over a finite field F of characteristic p, as an element
of the endomorphism ring.
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The dual (Verschiebung) of the Frobenius endomorphism, as an element of the endomorphism ring.
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The Frobenius endomorphism is inseparable.
The defining identity π + π̂ = [t] in the endomorphism ring: Frobenius plus its dual equals
multiplication by the trace of Frobenius.
Multiplication-by-m is inseparable iff p ∣ m.
Supersingularity of E/F_q is equivalent to inseparability of the dual Frobenius.
Supersingularity of E/F_q is equivalent to p dividing the trace of Frobenius. The proof
uses Lemma 7.1: if α is an inseparable isogeny, then α + β is inseparable iff β is.
If E : y² = x³ + Ax + B is supersingular, then E is isomorphic to its p²-Frobenius
twist: there exists μ ≠ 0 with A^(p²) = μ⁴ A and B^(p²) = μ⁶ B.
A supersingular curve has the same j-invariant as its p²-Frobenius twist.
The j-invariant of a supersingular elliptic curve lies in F_{p²}: it is fixed by the
p²-power Frobenius.
The geometric endomorphism algebra of E: End(E_{\bar F}) ⊗_ℤ ℚ, the endomorphism algebra
of E after base-change to the algebraic closure.
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Ring structure on the geometric endomorphism algebra.
ℚ-algebra structure on the geometric endomorphism algebra.
The geometric endomorphism algebra of E is finite-dimensional over ℚ.
Positive-definite involution algebra structure (Rosati involution) on the geometric
endomorphism algebra of E.
The geometric endomorphism algebra of E is nontrivial.
The geometric endomorphism algebra of any elliptic curve falls into the three-way Albert classification (rational / imaginary quadratic / quaternion).
The predicate that the geometric endomorphism algebra of E is a quaternion algebra: it has
ℚ-dimension 4 with two anticommuting elements squaring to negative rationals.
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For a supersingular elliptic curve, the geometric endomorphism algebra has ℚ-dimension
strictly greater than 1, i.e., it is not just ℚ.
For a supersingular elliptic curve, the geometric endomorphism algebra is not an imaginary
quadratic field, i.e., its ℚ-dimension is not 2.
The geometric endomorphism algebra of a supersingular elliptic curve is a quaternion algebra
(over ℚ).
If the geometric endomorphism algebra of E is a quaternion algebra, there exist two nonzero
anticommuting endomorphisms α, β in the endomorphism ring of E.
For an ordinary elliptic curve, two nonzero anticommuting endomorphisms cannot exist: the
existence of a nontrivial p-torsion point contradicts the anticommutation relation.
The geometric endomorphism algebra of an ordinary elliptic curve is not a quaternion algebra.
If the geometric endomorphism algebra of E is a quaternion algebra, then E is
supersingular.
The predicate that the geometric endomorphism algebra of E is an imaginary quadratic field:
it has ℚ-dimension 2 with a non-rational element squaring to a negative rational.
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The geometric endomorphism algebra of an ordinary elliptic curve over a finite field is an imaginary quadratic field.
The supersingular/ordinary dichotomy over a finite field: every elliptic curve E/F_q is
either supersingular (in which case p divides the trace of Frobenius and the geometric endomorphism
algebra is a quaternion algebra) or ordinary (in which case p does not divide the trace and the
algebra is an imaginary quadratic field).
If the conductor-f₁ order is contained in the conductor-f₂ order, then f₂ ∣ f₁. This is
the standard correspondence between containment of orders and divisibility of conductors.
The Frobenius discriminant of E/F_q: t² - 4q, where t = tr π_E.
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The predicate that the Frobenius endomorphism is not an integer (i.e., t² ≠ 4q), so that
ℤ[π_E] is a non-trivial quadratic order.
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The conductor of the order ℤ[π_E] inside the maximal quadratic order with discriminant
frobDiscriminant E.
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The conductor of the endomorphism ring of E inside the maximal quadratic order with
discriminant frobDiscriminant E.
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The Frobenius order ℤ[π_E] is contained in the endomorphism ring End(E), expressed as a
containment of the corresponding conductor orders.
The conductor of End(E) divides the conductor of ℤ[π_E], as a consequence of the order
containment ℤ[π_E] ⊆ End(E).
The endomorphism ring is contained in the maximal order O_K (the conductor-1 order), since
its conductor is a divisor of 1.
Theorem 13.8 of Sutherland: for an elliptic curve E/F_q whose geometric endomorphism algebra
is an imaginary quadratic field K, the inclusions ℤ[π_E] ⊆ End(E) ⊆ O_K hold, and the
conductor of End(E) divides [O_K : ℤ[π_E]].