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([0,0]),K([0,1]),K([-630,1502]),K([4419,25525])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([0,1]),Polrev([-630,1502]),Polrev([4419,25525])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,1],K![-630,1502],K![4419,25525]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((195i+195)\) | = | \((i+1)\cdot(-i-2)\cdot(2i+1)\cdot(3)\cdot(-3i-2)\cdot(2i+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 76050 \) | = | \(2\cdot5\cdot5\cdot9\cdot13\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((4453031700i+12995576100)\) | = | \((i+1)^{5}\cdot(-i-2)^{2}\cdot(2i+1)^{3}\cdot(3)^{2}\cdot(-3i-2)^{10}\cdot(2i+3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 188714489492096100000 \) | = | \(2^{5}\cdot5^{2}\cdot5^{3}\cdot9^{2}\cdot13^{10}\cdot13^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( \frac{3127785311994266947}{137858491849000} i - \frac{32765280950314910759}{1240726426641000} \) | ||
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 | $\left(-20 i - 50 : 307 i - 275 : 1\right)$ |
Height | \(0.14454663684423494886925545819006993082\) |
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: | $\left(10 i - \frac{81}{4} : -\frac{11}{2} i + \frac{81}{8} : 1\right)$ |
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: | \( 0.14454663684423494886925545819006993082 \) | ||
Period: | \( 0.23471832605691715532736695606101230702 \) | ||
Tamagawa product: | \( 240 \) = \(1\cdot2\cdot3\cdot2\cdot( 2 \cdot 5 )\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 4.0713293564683119996718869559798148140 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((i+1)\) | \(2\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
\((-i-2)\) | \(5\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((2i+1)\) | \(5\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
\((3)\) | \(9\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((-3i-2)\) | \(13\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
\((2i+3)\) | \(13\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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
76050.5-k
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.