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([9,3,-1]),K([1,1,0]),K([91,-383,-316]),K([7885,15090,6778])])
gp: E = ellinit([Polrev([-7,-1,1]),Polrev([9,3,-1]),Polrev([1,1,0]),Polrev([91,-383,-316]),Polrev([7885,15090,6778])], K);
magma: E := EllipticCurve([K![-7,-1,1],K![9,3,-1],K![1,1,0],K![91,-383,-316],K![7885,15090,6778]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-a-10)\) | = | \((a+2)\cdot(2a^2-3a-19)\) |
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: | \((2560a^2-4096a-24064)\) | = | \((a+2)^{27}\cdot(2a^2-3a-19)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 16777216000 \) | = | \(2^{27}\cdot5^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{58374455049}{64000} a^{2} - \frac{216373317399}{64000} a - \frac{187800406911}{64000} \) | ||
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: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{11}{3} a^{2} - \frac{4}{3} a - 11 : -\frac{5}{3} a^{2} - 17 a - 24 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 83.063438389301239741102128041884740346 \) | ||
| Tamagawa product: | \( 27 \) = \(3^{3}\cdot1\) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 6.1911831561114404082256402428941817547 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+2)\) | \(2\) | \(27\) | \(I_{27}\) | Split multiplicative | \(-1\) | \(1\) | \(27\) | \(27\) |
| \((2a^2-3a-19)\) | \(5\) | \(1\) | \(I_{3}\) | Non-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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
10.1-d
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.