Base field \(\Q(\sqrt{5}, \sqrt{7})\)
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 15 x^{2} + 16 x + 29 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([29, 16, -15, -2, 1]))
gp: K = nfinit(Polrev([29, 16, -15, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 16, -15, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-125/23,30/23,16/23,-3/23]),K([36/23,76/23,-7/23,-3/23]),K([-125/23,30/23,16/23,-3/23]),K([-78/23,4/23,19/23,-5/23]),K([-291/23,-392/23,-22/23,30/23])])
gp: E = ellinit([Polrev([-125/23,30/23,16/23,-3/23]),Polrev([36/23,76/23,-7/23,-3/23]),Polrev([-125/23,30/23,16/23,-3/23]),Polrev([-78/23,4/23,19/23,-5/23]),Polrev([-291/23,-392/23,-22/23,30/23])], K);
magma: E := EllipticCurve([K![-125/23,30/23,16/23,-3/23],K![36/23,76/23,-7/23,-3/23],K![-125/23,30/23,16/23,-3/23],K![-78/23,4/23,19/23,-5/23],K![-291/23,-392/23,-22/23,30/23]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((3/23a^3+7/23a^2-53/23a-128/23)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(4^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-4)\) | = | \((3/23a^3+7/23a^2-53/23a-128/23)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 256 \) | = | \(4^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{6528}{23} a^{3} - \frac{9792}{23} a^{2} - \frac{140352}{23} a - \frac{87536}{23} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(0\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-\frac{4}{23} a^{3} + \frac{6}{23} a^{2} + \frac{40}{23} a - \frac{21}{23} : -\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a - \frac{45}{23} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 419.49484915123775133494304704779432958 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.24729383473877 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3/23a^3+7/23a^2-53/23a-128/23)\) | \(4\) | \(3\) | \(IV\) | Additive | \(1\) | \(2\) | \(4\) | \(0\) |
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 |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
16.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.