Base field \(\Q(\sqrt{14}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 14 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 0, 1]))
gp: K = nfinit(Polrev([-14, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([29,5]),K([81,20])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([0,0]),Polrev([29,5]),Polrev([81,20])], K);
magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![29,5],K![81,20]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a+2)\) | = | \((-a+4)\cdot(-a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 10 \) | = | \(2\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((928a-2976)\) | = | \((-a+4)^{10}\cdot(-a-3)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -3200000 \) | = | \(-2^{10}\cdot5^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{73603923}{100000} a + \frac{358833109}{100000} \) | ||
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 | $\left(-2 : -3 : 1\right)$ |
| Height | \(0.054400494674780956742708434690663425336\) |
| 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: | \( 0.054400494674780956742708434690663425336 \) | ||
| Period: | \( 14.163134738548721329178436662472233718 \) | ||
| Tamagawa product: | \( 10 \) = \(( 2 \cdot 5 )\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.0591985216127295013337256651752767828 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+4)\) | \(2\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
| \((-a-3)\) | \(5\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B.4.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
10.2-c
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.