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([0,0,0,0]),K([0,0,0,0]),K([-22/23,43/23,3/23,-2/23]),K([-1865/23,-2064/23,-144/23,96/23]),K([853,946,66,-44])])
gp: E = ellinit([Polrev([0,0,0,0]),Polrev([0,0,0,0]),Polrev([-22/23,43/23,3/23,-2/23]),Polrev([-1865/23,-2064/23,-144/23,96/23]),Polrev([853,946,66,-44])], K);
magma: E := EllipticCurve([K![0,0,0,0],K![0,0,0,0],K![-22/23,43/23,3/23,-2/23],K![-1865/23,-2064/23,-144/23,96/23],K![853,946,66,-44]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((4/23a^3-6/23a^2-40/23a+21/23)\) | = | \((4/23a^3-6/23a^2-40/23a+21/23)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-125)\) | = | \((4/23a^3-6/23a^2-40/23a+21/23)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 244140625 \) | = | \(25^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{110592}{125} \) | ||
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: | 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: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 148.30156367778230507597189046333948622 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.11859376682546 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((4/23a^3-6/23a^2-40/23a+21/23)\) | \(25\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Nn |
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 25.1-d consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q(\sqrt{7}) \) | 2.2.28.1-25.1-b1 |
| \(\Q(\sqrt{7}) \) | 2.2.28.1-625.1-b1 |