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([-103/23,-13/23,13/23,-1/23]),K([21/23,-63/23,-6/23,4/23]),K([1,0,0,0]),K([3527/23,-1473/23,-367/23,122/23]),K([-1061/23,423/23,129/23,-40/23])])
gp: E = ellinit([Polrev([-103/23,-13/23,13/23,-1/23]),Polrev([21/23,-63/23,-6/23,4/23]),Polrev([1,0,0,0]),Polrev([3527/23,-1473/23,-367/23,122/23]),Polrev([-1061/23,423/23,129/23,-40/23])], K);
magma: E := EllipticCurve([K![-103/23,-13/23,13/23,-1/23],K![21/23,-63/23,-6/23,4/23],K![1,0,0,0],K![3527/23,-1473/23,-367/23,122/23],K![-1061/23,423/23,129/23,-40/23]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-4/23a^3+6/23a^2+63/23a-44/23)\) | = | \((-4/23a^3+6/23a^2+63/23a-44/23)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 29 \) | = | \(29\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-29/23a^3+9/23a^2+336/23a+95/23)\) | = | \((-4/23a^3+6/23a^2+63/23a-44/23)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 24389 \) | = | \(29^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{26708755}{560947} a^{3} - \frac{78978730}{560947} a^{2} - \frac{312565105}{560947} a + \frac{20685109}{19343} \) | ||
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: | \(1\) |
| Generator | $\left(1 : \frac{7}{23} a^{3} - \frac{22}{23} a^{2} - \frac{93}{23} a + \frac{238}{23} : 1\right)$ |
| Height | \(0.17579277387571287464615004550657019811\) |
| 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: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.17579277387571287464615004550657019811 \) | ||
| Period: | \( 217.94907400688153885671810394993589335 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.28404619571252 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-4/23a^3+6/23a^2+63/23a-44/23)\) | \(29\) | \(3\) | \(I_{3}\) | 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
29.3-b
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.