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([0,0]),K([-1,1]),K([0,0]),K([10242,-2395]),K([-103332,24172])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,1]),Polrev([0,0]),Polrev([10242,-2395]),Polrev([-103332,24172])], K);
magma: E := EllipticCurve([K![0,0],K![-1,1],K![0,0],K![10242,-2395],K![-103332,24172]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((16)\) | = | \((a-4)^{4}\cdot(a+3)^{4}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 256 \) | = | \(2^{4}\cdot2^{4}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((6144a-26624)\) | = | \((a-4)^{11}\cdot(a+3)^{13}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 16777216 \) | = | \(2^{11}\cdot2^{13}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{4171}{2} a - 5149 \) | ||
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 | $\left(-33 a + 140 : 582 a - 2488 : 1\right)$ |
Height | \(0.15665631725403995911354127642548143957\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(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: | \( 0.15665631725403995911354127642548143957 \) | ||
Period: | \( 11.068680763719633563085051782961640006 \) | ||
Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.6747375697761644341777567910909756779 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a-4)\) | \(2\) | \(4\) | \(I_{3}^{*}\) | Additive | \(1\) | \(4\) | \(11\) | \(0\) |
\((a+3)\) | \(2\) | \(2\) | \(I_{5}^{*}\) | Additive | \(-1\) | \(4\) | \(13\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 256.1-p consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.