Base field \(\Q(\zeta_{15})^+\)
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + 4 x + 1 \); class number \(1\).
Weierstrass equation
This is a global minimal model.
Invariants
| Conductor: | \((a^2-8)\) | = | \((a^3+2a^2-3a-3)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 961 \) | = | \(31^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-92a^3+a^2+417a-19)\) | = | \((a^3+2a^2-3a-3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 887503681 \) | = | \(31^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -16554983445 a^{3} + 49664950335 a + 10231546590 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z[\sqrt{-15}]\) | (potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $N(\mathrm{U}(1))$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(763 a^{3} + 632 a^{2} - 1897 a - 417 : 32564 a^{3} + 26936 a^{2} - 81041 a - 17823 : 1\right)$ |
| Height | \(0.33365178768598527317687008905484660855\) |
| 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: | $\left(\frac{663}{4} a^{3} + 137 a^{2} - \frac{1653}{4} a - \frac{183}{2} : -\frac{533}{4} a^{3} - \frac{221}{2} a^{2} + \frac{1325}{4} a + \frac{583}{8} : 1\right)$ |
| 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: | \( 0.33365178768598527317687008905484660855 \) | ||
| Period: | \( 129.41812778404945617525530334770416574 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.57479290305553 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+2a^2-3a-3)\) | \(31\) | \(2\) | \(I_0^{*}\) | 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 |
|---|---|
| \(5\) | 5B.4.1[2] |
For all other primes \(p\), the image is a Borel subgroup if \(p\in \{ 2, 3\}\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -15 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -15 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 5, 6, 10, 15 and 30.
Its isogeny class
961.7-b
consists of curves linked by isogenies of
degrees dividing 30.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.