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([1,0]),K([0,-1]),K([0,0]),K([-70,-54]),K([-971,141])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,-1]),Polrev([0,0]),Polrev([-70,-54]),Polrev([-971,141])], K);
magma: E := EllipticCurve([K![1,0],K![0,-1],K![0,0],K![-70,-54],K![-971,141]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((51i+51)\) | = | \((i+1)\cdot(3)\cdot(i+4)\cdot(i-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 5202 \) | = | \(2\cdot9\cdot17\cdot17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((140240871i-398264865)\) | = | \((i+1)\cdot(3)\cdot(i+4)\cdot(i-4)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 178282404592306866 \) | = | \(2\cdot9\cdot17\cdot17^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{461275687224792005}{3495733423378566} i + \frac{140679328163848447}{1165244474459522} \) | ||
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 | $\left(\frac{166}{25} i - \frac{87}{25} : -\frac{1574}{125} i + \frac{1343}{125} : 1\right)$ |
| Height | \(0.57806728202429234250753181450867660161\) |
| 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: | $\left(-7 i - \frac{33}{4} : \frac{7}{2} i + \frac{33}{8} : 1\right)$ |
| 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: | \( 0.57806728202429234250753181450867660161 \) | ||
| Period: | \( 0.50918407614522534410697783614502854888 \) | ||
| Tamagawa product: | \( 12 \) = \(1\cdot1\cdot1\cdot( 2^{2} \cdot 3 )\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.7660559296839243555352182127137366381 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((i+1)\) | \(2\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((3)\) | \(9\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((i+4)\) | \(17\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((i-4)\) | \(17\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
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
5202.2-d
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.