Base field \(\Q(\sqrt{-7}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
gp: K = nfinit(Polrev([2, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([0,0]),K([-5,0]),K([25,0])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([0,0]),Polrev([-5,0]),Polrev([25,0])], K);
magma: E := EllipticCurve([K![0,0],K![-1,0],K![0,0],K![-5,0],K![25,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-160a+80)\) | = | \((a)^{4}\cdot(-a+1)^{4}\cdot(-2a+1)\cdot(5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 44800 \) | = | \(2^{4}\cdot2^{4}\cdot7\cdot25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-224000)\) | = | \((a)^{8}\cdot(-a+1)^{8}\cdot(-2a+1)^{2}\cdot(5)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 50176000000 \) | = | \(2^{8}\cdot2^{8}\cdot7^{2}\cdot25^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{65536}{875} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(2\) | |
Generators | $\left(-\frac{5}{4} a - \frac{5}{4} : \frac{5}{8} a - \frac{55}{8} : 1\right)$ | $\left(5 : -10 : 1\right)$ |
Heights | \(0.50194825281248066244089243243125884895\) | \(0.11244053540398151283635559917433576368\) |
Torsion structure: | trivial | |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
|
BSD invariants
Analytic rank: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 0.053278611790844889891244912010057090569 \) | ||
Period: | \( 1.7923320732805115299266341159066338146 \) | ||
Tamagawa product: | \( 24 \) = \(2\cdot2\cdot2\cdot3\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 6.9298460335186668219744587662797586944 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(4\) | \(8\) | \(0\) |
\((-a+1)\) | \(2\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(4\) | \(8\) | \(0\) |
\((-2a+1)\) | \(7\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
\((5)\) | \(25\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
44800.5-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 560.c2 |
\(\Q\) | 3920.u2 |