Base field \(\Q(\sqrt{-11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
gp: K = nfinit(Polrev([3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([-1,-1]),K([0,1]),K([561,341]),K([-11032,3327])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,-1]),Polrev([0,1]),Polrev([561,341]),Polrev([-11032,3327])], K);
magma: E := EllipticCurve([K![1,1],K![-1,-1],K![0,1],K![561,341],K![-11032,3327]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((60a+135)\) | = | \((-a)^{2}\cdot(a-1)\cdot(-a-1)^{2}\cdot(a-2)\cdot(-2a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 37125 \) | = | \(3^{2}\cdot3\cdot5^{2}\cdot5\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((36073125a-120470625)\) | = | \((-a)^{7}\cdot(a-1)^{2}\cdot(-a-1)^{7}\cdot(a-2)^{4}\cdot(-2a+1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 14071230615234375 \) | = | \(3^{7}\cdot3^{2}\cdot5^{7}\cdot5^{4}\cdot11^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( \frac{16206760423}{61875} a - \frac{164293539484}{226875} \) | ||
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 | $\left(-11 a + 14 : -9 a - 11 : 1\right)$ |
Height | \(0.35300562880171292759958398959341723415\) |
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: | $\left(-\frac{39}{4} a + \frac{23}{2} : \frac{7}{2} a - \frac{163}{8} : 1\right)$ |
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: | \( 0.35300562880171292759958398959341723415 \) | ||
Period: | \( 0.40300188434959806375677472083046544715 \) | ||
Tamagawa product: | \( 128 \) = \(2\cdot2\cdot2^{2}\cdot2^{2}\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 5.4903791206258719818685936233559541460 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(3\) | \(2\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(2\) | \(7\) | \(1\) |
\((a-1)\) | \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
\((-a-1)\) | \(5\) | \(4\) | \(I_{1}^{*}\) | Additive | \(1\) | \(2\) | \(7\) | \(1\) |
\((a-2)\) | \(5\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
\((-2a+1)\) | \(11\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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 and 4.
Its isogeny class
37125.6-h
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.