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(Polrev([1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([1,1]),K([3999,1332]),K([-49560,95335])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,1]),Polrev([1,1]),Polrev([3999,1332]),Polrev([-49560,95335])], K);
magma: E := EllipticCurve([K![1,1],K![0,1],K![1,1],K![3999,1332],K![-49560,95335]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-20i-40)\) | = | \((i+1)^{4}\cdot(-i-2)^{2}\cdot(2i+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 2000 \) | = | \(2^{4}\cdot5^{2}\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-260000i+180000)\) | = | \((i+1)^{11}\cdot(-i-2)^{7}\cdot(2i+1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 100000000000 \) | = | \(2^{11}\cdot5^{7}\cdot5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{15332659200009}{625} i + \frac{5763174879987}{625} \) | ||
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: | $\left(-\frac{75}{2} i + 6 : \frac{61}{4} i - \frac{89}{4} : 1\right)$ |
| 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: | \( 0.33506237422083295535926473386331894523 \) | ||
| Tamagawa product: | \( 16 \) = \(2\cdot2^{2}\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.3402494968833318214370589354532757809 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((i+1)\) | \(2\) | \(2\) | \(I_{3}^{*}\) | Additive | \(-1\) | \(4\) | \(11\) | \(0\) |
| \((-i-2)\) | \(5\) | \(4\) | \(I_{1}^{*}\) | Additive | \(1\) | \(2\) | \(7\) | \(1\) |
| \((2i+1)\) | \(5\) | \(2\) | \(I_{4}\) | 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, 4, 8 and 16.
Its isogeny class
2000.2-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.