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([1,1]),K([0,-1]),K([1,0]),K([-32518,-5944]),K([-3247789,-592970])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([1,0]),Polrev([-32518,-5944]),Polrev([-3247789,-592970])], K);
magma: E := EllipticCurve([K![1,1],K![0,-1],K![1,0],K![-32518,-5944],K![-3247789,-592970]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((15,a)\) | = | \((3,a)\cdot(a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 15 \) | = | \(3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((164025)\) | = | \((3,a)^{16}\cdot(a+5)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 26904200625 \) | = | \(3^{16}\cdot5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{272223782641}{164025} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(0\) | |
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);
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Torsion generators: | $\left(-\frac{27}{2} a - \frac{311}{4} : \frac{365}{8} a + \frac{1927}{8} : 1\right)$ | $\left(-13 a - 75 : 44 a + 232 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 1.9616888821913405749936556941151444563 \) | ||
Tamagawa product: | \( 64 \) = \(2^{4}\cdot2^{2}\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 2.8652300042221108571245929026794987960 \) | ||
Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((3,a)\) | \(3\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
\((a+5)\) | \(5\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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, 4 and 8.
Its isogeny class
15.1-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 75.b2 |
\(\Q\) | 2880.bc2 |