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([-14624,0]),K([669300,0])])
gp: E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([0,0]),Polrev([-14624,0]),Polrev([669300,0])], K);
magma: E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![-14624,0],K![669300,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-96a+48)\) | = | \((a)^{4}\cdot(-a+1)^{4}\cdot(-2a+1)\cdot(3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 16128 \) | = | \(2^{4}\cdot2^{4}\cdot7\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((3825245233152)\) | = | \((a)^{13}\cdot(-a+1)^{13}\cdot(-2a+1)^{16}\cdot(3)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 14632501093752098839855104 \) | = | \(2^{13}\cdot2^{13}\cdot7^{16}\cdot9^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{84448510979617}{933897762} \) | ||
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: | \(1\) |
Generator | $\left(-63 a - 23 : -252 a - 1470 : 1\right)$ |
Height | \(2.3108118147069246137026070875318320986\) |
Torsion structure: | \(\Z/8\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(82 : 168 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 2.3108118147069246137026070875318320986 \) | ||
Period: | \( 0.085636479133251818449030599440513078347 \) | ||
Tamagawa product: | \( 1024 \) = \(2^{2}\cdot2^{2}\cdot2^{4}\cdot2^{2}\) | ||
Torsion order: | \(8\) | ||
Leading coefficient: | \( 4.7868997978382464906921047090734783638 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a)\) | \(2\) | \(4\) | \(I_{5}^{*}\) | Additive | \(-1\) | \(4\) | \(13\) | \(1\) |
\((-a+1)\) | \(2\) | \(4\) | \(I_{5}^{*}\) | Additive | \(-1\) | \(4\) | \(13\) | \(1\) |
\((-2a+1)\) | \(7\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
\((3)\) | \(9\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 8 and 16.
Its isogeny class
16128.5-n
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 336.d2 |
\(\Q\) | 2352.l2 |