Base field 4.4.17725.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 12 x^{2} + 13 x + 41 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, 13, -12, -2, 1]))
gp: K = nfinit(Polrev([41, 13, -12, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 13, -12, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-6,-1,1]),K([8,-5,-2,1]),K([0,-7,-1,1]),K([12,-66,-15,12]),K([84,-113,-41,23])])
gp: E = ellinit([Polrev([1,-6,-1,1]),Polrev([8,-5,-2,1]),Polrev([0,-7,-1,1]),Polrev([12,-66,-15,12]),Polrev([84,-113,-41,23])], K);
magma: E := EllipticCurve([K![1,-6,-1,1],K![8,-5,-2,1],K![0,-7,-1,1],K![12,-66,-15,12],K![84,-113,-41,23]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a-1)\) | = | \((a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 41 \) | = | \(41\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((80a^3+37a^2-595a-834)\) | = | \((a-1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -4750104241 \) | = | \(-41^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1496441838018450}{4750104241} a^{3} + \frac{7679003528285540}{4750104241} a^{2} + \frac{3087746109933194}{4750104241} a - \frac{756285282553335}{115856201} \) | ||
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: | $\left(\frac{1}{4} a^{3} - \frac{5}{4} a^{2} - \frac{3}{4} a + 4 : \frac{7}{8} a^{3} - \frac{1}{2} a^{2} - \frac{37}{8} a - 2 : 1\right)$ |
| 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: | \( 59.586250184918610389639255640296011703 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.01402773828800 \) | ||
| Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a-1)\) | \(41\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
41.2-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.