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([-1,1]),K([0,0]),K([-447,-701]),K([-76839,61718])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,1]),Polrev([0,0]),Polrev([-447,-701]),Polrev([-76839,61718])], K);
magma: E := EllipticCurve([K![1,1],K![-1,1],K![0,0],K![-447,-701],K![-76839,61718]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((100i-240)\) | = | \((i+1)^{4}\cdot(-i-2)\cdot(2i+1)\cdot(2i+3)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 67600 \) | = | \(2^{4}\cdot5\cdot5\cdot13^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4128181204560i-975184625920)\) | = | \((i+1)^{8}\cdot(-i-2)\cdot(2i+1)^{3}\cdot(2i+3)^{18}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 17992865112313182900640000 \) | = | \(2^{8}\cdot5\cdot5^{3}\cdot13^{18}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{40605232846917732}{2912260640310125} i + \frac{15507117639303424}{2912260640310125} \) | ||
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 | $\left(-\frac{718313}{16854} i + \frac{3792755}{101124} : \frac{2105886370}{4019679} i - \frac{896000587}{10719144} : 1\right)$ |
| Height | \(6.9993748452367707654490313501593112517\) |
| 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: | $\left(-\frac{71}{2} i - 37 : \frac{145}{4} i + \frac{3}{4} : 1\right)$ |
| 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: | \( 6.9993748452367707654490313501593112517 \) | ||
| Period: | \( 0.11021910992577144587358562037137029732 \) | ||
| Tamagawa product: | \( 12 \) = \(1\cdot1\cdot3\cdot2^{2}\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 4.6287891928729868309465404522743628462 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((i+1)\) | \(2\) | \(1\) | \(I_0^{*}\) | Additive | \(1\) | \(4\) | \(8\) | \(0\) |
| \((-i-2)\) | \(5\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((2i+1)\) | \(5\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
| \((2i+3)\) | \(13\) | \(4\) | \(I_{12}^{*}\) | Additive | \(1\) | \(2\) | \(18\) | \(12\) |
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 |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
67600.6-i
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.