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([-21/23,63/23,6/23,-4/23]),K([104/23,33/23,-10/23,-1/23]),K([-103/23,-13/23,13/23,-1/23]),K([292/23,228/23,25/23,-9/23]),K([743/23,554/23,-186/23,-83/23])])
gp: E = ellinit([Polrev([-21/23,63/23,6/23,-4/23]),Polrev([104/23,33/23,-10/23,-1/23]),Polrev([-103/23,-13/23,13/23,-1/23]),Polrev([292/23,228/23,25/23,-9/23]),Polrev([743/23,554/23,-186/23,-83/23])], K);
magma: E := EllipticCurve([K![-21/23,63/23,6/23,-4/23],K![104/23,33/23,-10/23,-1/23],K![-103/23,-13/23,13/23,-1/23],K![292/23,228/23,25/23,-9/23],K![743/23,554/23,-186/23,-83/23]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a-1)\) | = | \((a-1)\) |
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: | \((a-1)\) | = | \((a-1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 29 \) | = | \(29\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{5568759973}{667} a^{3} + \frac{154187636}{667} a^{2} - \frac{83217892687}{667} a - \frac{2746289531}{23} \) | ||
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: | \(1\) |
| Generator | $\left(\frac{3}{23} a^{3} + \frac{7}{23} a^{2} - \frac{30}{23} a - \frac{36}{23} : \frac{1}{23} a^{3} + \frac{10}{23} a^{2} + \frac{36}{23} a + \frac{11}{23} : 1\right)$ |
| Height | \(0.18325482942207945916954487270676713506\) |
| 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: | \( 0.18325482942207945916954487270676713506 \) | ||
| Period: | \( 501.70341450493858853440672853273254364 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.62684496130221 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a-1)\) | \(29\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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
29.4-a
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.