Base field 6.6.1387029.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 2 x^{4} + 9 x^{3} - x^{2} - 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))
gp: K = nfinit(Polrev([1, -4, -1, 9, -2, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,-3,-13,6,5,-2]),K([-3,5,13,-6,-5,2]),K([0,-3,0,1,0,0]),K([788,-2073,-3636,2097,1301,-573]),K([-4380,11495,20192,-11619,-7234,3182])])
gp: E = ellinit([Polrev([3,-3,-13,6,5,-2]),Polrev([-3,5,13,-6,-5,2]),Polrev([0,-3,0,1,0,0]),Polrev([788,-2073,-3636,2097,1301,-573]),Polrev([-4380,11495,20192,-11619,-7234,3182])], K);
magma: E := EllipticCurve([K![3,-3,-13,6,5,-2],K![-3,5,13,-6,-5,2],K![0,-3,0,1,0,0],K![788,-2073,-3636,2097,1301,-573],K![-4380,11495,20192,-11619,-7234,3182]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-3a)\) | = | \((a^3-a^2-3a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-14a^4+28a^3+150a^2-164a+459)\) | = | \((a^3-a^2-3a)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 59604644775390625 \) | = | \(25^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{72134063979419}{244140625} a^{4} - \frac{144268127958838}{244140625} a^{3} - \frac{4976237481849}{244140625} a^{2} + \frac{77110301461268}{244140625} a - \frac{17613430193543}{244140625} \) | ||
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 | $\left(-a^{5} + 4 a^{4} - 2 a^{3} - 5 a^{2} + 2 a : -36 a^{5} + 76 a^{4} + 151 a^{3} - 239 a^{2} - 141 a + 54 : 1\right)$ |
| Height | \(0.15447431300235532200623930770860795219\) |
| 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: | $\left(\frac{23}{2} a^{5} - \frac{103}{4} a^{4} - 43 a^{3} + \frac{285}{4} a^{2} + \frac{177}{4} a - 15 : \frac{67}{4} a^{5} - \frac{151}{4} a^{4} - 62 a^{3} + \frac{213}{2} a^{2} + \frac{253}{4} a - \frac{195}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.15447431300235532200623930770860795219 \) | ||
| Period: | \( 1109.1079557560972380386102210625184978 \) | ||
| Tamagawa product: | \( 12 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.61854 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-3a)\) | \(25\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
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 |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
25.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| 3.3.257.1 | a curve with conductor norm 245 (not in the database) |
| 3.3.257.1 | 3.3.257.1-45.2-a3 |