Base field 3.3.1620.1
Generator \(a\), with minimal polynomial \( x^{3} - 12 x - 14 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -12, 0, 1]))
gp: K = nfinit(Polrev([-14, -12, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -12, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-7,-1,1]),K([0,-1,0]),K([1,1,0]),K([-17363891,-2399476,1725112]),K([-14855556426422,-2052847260013,1475906218126])])
gp: E = ellinit([Polrev([-7,-1,1]),Polrev([0,-1,0]),Polrev([1,1,0]),Polrev([-17363891,-2399476,1725112]),Polrev([-14855556426422,-2052847260013,1475906218126])], K);
magma: E := EllipticCurve([K![-7,-1,1],K![0,-1,0],K![1,1,0],K![-17363891,-2399476,1725112],K![-14855556426422,-2052847260013,1475906218126]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2+3a+4)\) | = | \((a+2)\cdot(a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 6 \) | = | \(2\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((432a^2-648a-4320)\) | = | \((a+2)^{10}\cdot(a+1)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -60466176 \) | = | \(-2^{10}\cdot3^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{11038321}{1296} a^{2} - \frac{22299361}{648} a - \frac{5084711}{162} \) | ||
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 | $\left(-4589 a^{2} + 6382 a + 46193 : -602602 a^{2} + 838160 a + 6065428 : 1\right)$ |
| Height | \(0.12091350567705845082777508021559359756\) |
| 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: | \( 0.12091350567705845082777508021559359756 \) | ||
| Period: | \( 46.212281324297175678538093735020035970 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.6659269743379270174493606873323946670 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+2)\) | \(2\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
| \((a+1)\) | \(3\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
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
6.1-a
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.