Base field 6.6.1241125.1
Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 11, -2, -7, 0, 1]))
gp: K = nfinit(Polrev([1, 7, 11, -2, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-12,-41,-4,27,2,-4]),K([-2,5,1,-1,0,0]),K([-13,-42,-4,27,2,-4]),K([-35,-51,38,54,-5,-9]),K([-155,-598,-4,417,16,-63])])
gp: E = ellinit([Polrev([-12,-41,-4,27,2,-4]),Polrev([-2,5,1,-1,0,0]),Polrev([-13,-42,-4,27,2,-4]),Polrev([-35,-51,38,54,-5,-9]),Polrev([-155,-598,-4,417,16,-63])], K);
magma: E := EllipticCurve([K![-12,-41,-4,27,2,-4],K![-2,5,1,-1,0,0],K![-13,-42,-4,27,2,-4],K![-35,-51,38,54,-5,-9],K![-155,-598,-4,417,16,-63]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-7a^3-2a^2+10a+7)\) | = | \((-2a^5+a^4+13a^3-2a^2-19a-5)\cdot(2a^5-a^4-14a^3+2a^2+23a+6)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 405 \) | = | \(5\cdot9^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-31a^5+23a^4+215a^3-68a^2-339a-91)\) | = | \((-2a^5+a^4+13a^3-2a^2-19a-5)^{5}\cdot(2a^5-a^4-14a^3+2a^2+23a+6)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1660753125 \) | = | \(5^{5}\cdot9^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{13681459876}{125} a^{5} - \frac{36187576902}{125} a^{4} + \frac{25331233773}{125} a^{3} + \frac{99876956003}{125} a^{2} + \frac{10695502109}{25} a + \frac{7259236048}{125} \) | ||
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: | $\left(3 a^{5} - a^{4} - 21 a^{3} + 34 a + 14 : 20 a^{5} - 5 a^{4} - 135 a^{3} - 3 a^{2} + 204 a + 71 : 1\right)$ |
| 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: | \( 8255.6893739069702198802325059773302325 \) | ||
| Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 0.823385 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^5+a^4+13a^3-2a^2-19a-5)\) | \(5\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
| \((2a^5-a^4-14a^3+2a^2+23a+6)\) | \(9\) | \(1\) | \(I_0^{*}\) | Additive | \(1\) | \(2\) | \(6\) | \(0\) |
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 |
| \(5\) | 5B.4.1[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
405.2-m
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.