Base field \(\Q(\sqrt{-1}) \)
Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))
gp: K = nfinit(Pol(Vecrev([1, 0, 1])));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([624,0]),K([0,9486])])
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([624,0])),Pol(Vecrev([0,9486]))], K);
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![624,0],K![0,9486]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((312)\) | = | \((i+1)^{6}\cdot(3)\cdot(-3i-2)\cdot(2i+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 97344 \) | = | \(2^{6}\cdot9\cdot13\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((25022177856)\) | = | \((i+1)^{12}\cdot(3)^{4}\cdot(-3i-2)^{6}\cdot(2i+3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 626109384657296756736 \) | = | \(2^{12}\cdot9^{4}\cdot13^{6}\cdot13^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{420526439488}{390971529} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(-15 i - 9 : 0 : 1\right)$ | $\left(31 i : 0 : 1\right)$ |
| 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: | \( 0.250676465104088 \) | ||
| Tamagawa product: | \( 64 \) = \(2^{2}\cdot2^{2}\cdot2\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 4.01082344166541 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((i+1)\) | \(2\) | \(4\) | \(I_{2}^{*}\) | Additive | \(1\) | \(6\) | \(12\) | \(0\) |
| \((3)\) | \(9\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((-3i-2)\) | \(13\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
| \((2i+3)\) | \(13\) | \(2\) | \(I_{6}\) | 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 |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
97344.2-e
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This curve is the base change of 1248.j2, 1248.e2, defined over \(\Q\), so it is also a \(\Q\)-curve.