Base field \(\Q(\sqrt{30}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 30 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-30, 0, 1]))
gp: K = nfinit(Polrev([-30, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-30, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,0]),K([-1432,262]),K([22370,-4084])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([0,0]),Polrev([-1432,262]),Polrev([22370,-4084])], K);
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,0],K![-1432,262],K![22370,-4084]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a-12)\) | = | \((2,a)^{3}\cdot(3,a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 24 \) | = | \(2^{3}\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((1296)\) | = | \((2,a)^{8}\cdot(3,a)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1679616 \) | = | \(2^{8}\cdot3^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1556068}{81} \) | ||
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 | $\left(3 a - \frac{35}{2} : -\frac{91}{4} a + \frac{255}{2} : 1\right)$ | |
| Height | \(1.0685417217729154588958216948483363469\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(6 a - 34 : 17 a - 90 : 1\right)$ | $\left(-2 a + 10 : -5 a + 30 : 1\right)$ |
| 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: | \( 1.0685417217729154588958216948483363469 \) | ||
| Period: | \( 22.734034070006347604148590511773734758 \) | ||
| Tamagawa product: | \( 16 \) = \(2\cdot2^{3}\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 4.4351403050949894729315324693857331733 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(2\) | \(I_{1}^{*}\) | Additive | \(1\) | \(3\) | \(8\) | \(0\) |
| \((3,a)\) | \(3\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
24.1-c
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 72.a3 |
| \(\Q\) | 4800.cc3 |