Base field \(\Q(\sqrt{57}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 14 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -1, 1]))
gp: K = nfinit(Polrev([-14, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,1]),K([-605,-188]),K([-8777,-2683])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([0,1]),Polrev([-605,-188]),Polrev([-8777,-2683])], K);
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,1],K![-605,-188],K![-8777,-2683]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2a+4)\) | = | \((a-4)^{4}\cdot(a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 32 \) | = | \(2^{4}\cdot2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-6720a+29056)\) | = | \((a-4)^{18}\cdot(a+3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 16777216 \) | = | \(2^{18}\cdot2^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{2146689}{64} \) | ||
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: | \(1\) | |
Generator | $\left(\frac{23}{3} a + 23 : \frac{193}{9} a + \frac{616}{9} : 1\right)$ | |
Height | \(1.2113779133630752666297891567890812280\) | |
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: | $\left(-3 a - 12 : 7 a + 21 : 1\right)$ | $\left(-\frac{9}{4} a - \frac{19}{2} : \frac{43}{8} a + \frac{63}{4} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 1.2113779133630752666297891567890812280 \) | ||
Period: | \( 11.546389806438462227735118517231273373 \) | ||
Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 1.8526289166364359055876409393416604020 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a-4)\) | \(2\) | \(4\) | \(I_{10}^{*}\) | Additive | \(-1\) | \(4\) | \(18\) | \(6\) |
\((a+3)\) | \(2\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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 |
\(3\) | 3Ns |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
32.3-f
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.