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([1,1]),K([26,-15]),K([61,54])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([1,1]),Polrev([26,-15]),Polrev([61,54])], K);
magma: E := EllipticCurve([K![1,1],K![1,1],K![1,1],K![26,-15],K![61,54]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((84i-276)\) | = | \((i+1)^{5}\cdot(3)\cdot(i+4)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 83232 \) | = | \(2^{5}\cdot9\cdot17^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-720i+14256)\) | = | \((i+1)^{9}\cdot(3)^{2}\cdot(i+4)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 203751936 \) | = | \(2^{9}\cdot9^{2}\cdot17^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1949750}{9} i - 15622 \) | ||
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 | $\left(-5 i - 3 : i + 6 : 1\right)$ |
| Height | \(1.5269546363420535445786074792852389224\) |
| 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: | $\left(-\frac{7}{2} i - 1 : \frac{7}{4} i - \frac{7}{4} : 1\right)$ |
| 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: | \( 1.5269546363420535445786074792852389224 \) | ||
| Period: | \( 2.0219501215021977485769468264764806925 \) | ||
| Tamagawa product: | \( 4 \) = \(1\cdot2\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 6.1748522249603186847986083322923691262 \) | ||
| 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 | \(-1\) | \(5\) | \(9\) | \(0\) |
| \((3)\) | \(9\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((i+4)\) | \(17\) | \(2\) | \(III\) | Additive | \(1\) | \(2\) | \(3\) | \(0\) |
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.
Its isogeny class
83232.1-e
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.