Base field 6.6.703493.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,14,5,-12,-1,2]),K([0,-4,0,1,0,0]),K([-1,14,5,-12,-1,2]),K([29,87,-11,-87,1,15]),K([226,192,-195,-142,32,22])])
gp: E = ellinit([Polrev([0,14,5,-12,-1,2]),Polrev([0,-4,0,1,0,0]),Polrev([-1,14,5,-12,-1,2]),Polrev([29,87,-11,-87,1,15]),Polrev([226,192,-195,-142,32,22])], K);
magma: E := EllipticCurve([K![0,14,5,-12,-1,2],K![0,-4,0,1,0,0],K![-1,14,5,-12,-1,2],K![29,87,-11,-87,1,15],K![226,192,-195,-142,32,22]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-3a^5+2a^4+17a^3-9a^2-15a)\) | = | \((-3a^5+2a^4+17a^3-9a^2-15a)\) |
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: | \((-a^5+5a^3+2a^2-3a-3)\) | = | \((-3a^5+2a^4+17a^3-9a^2-15a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -5041 \) | = | \(-71^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{808462120758107459268}{5041} a^{5} - \frac{2389999464232167402569}{5041} a^{4} - \frac{1756922926803356952415}{5041} a^{3} + \frac{148916966593396023766}{71} a^{2} - \frac{8493388913885282041853}{5041} a + \frac{845468825674899153375}{5041} \) | ||
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: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{11}{4} a^{5} - 2 a^{4} - 16 a^{3} + \frac{41}{4} a^{2} + \frac{63}{4} a - \frac{25}{4} : -\frac{31}{8} a^{5} + \frac{23}{8} a^{4} + \frac{45}{2} a^{3} - \frac{113}{8} a^{2} - \frac{89}{4} a + 6 : 1\right)$ |
| 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: | \( 940.58997874263557640866850262685358054 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.24285 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-3a^5+2a^4+17a^3-9a^2-15a)\) | \(71\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
71.2-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.