Base field 4.4.6125.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 9 x^{2} + 9 x + 11 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 9, -9, -1, 1]))
gp: K = nfinit(Polrev([11, 9, -9, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 9, -9, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0]),K([-1,-1,0,0]),K([-11,-12,3,2]),K([68,86,-20,-15]),K([117,145,-34,-25])])
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([-1,-1,0,0]),Polrev([-11,-12,3,2]),Polrev([68,86,-20,-15]),Polrev([117,145,-34,-25])], K);
magma: E := EllipticCurve([K![1,1,0,0],K![-1,-1,0,0],K![-11,-12,3,2],K![68,86,-20,-15],K![117,145,-34,-25]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((7a^3+9a^2-41a-31)\) | = | \((7a^3+9a^2-41a-31)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 71 \) | = | \(71\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((38a^3+46a^2-227a-164)\) | = | \((7a^3+9a^2-41a-31)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 357911 \) | = | \(71^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{6737231751}{357911} a^{3} + \frac{39748625183}{357911} a^{2} - \frac{18717825915}{357911} a - \frac{86019846650}{357911} \) | ||
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 | $\left(9 a^{3} + 12 a^{2} - 52 a - 41 : 22 a^{3} + 30 a^{2} - 126 a - 101 : 1\right)$ |
| Height | \(0.012087042536891051962558966425690842916\) |
| 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: | \( 0.012087042536891051962558966425690842916 \) | ||
| Period: | \( 1375.4792214357158959221372569506678997 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.54919045785147 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((7a^3+9a^2-41a-31)\) | \(71\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 71.2-e consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.