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([0,-1]),K([0,0]),K([7,3]),K([-680,-204])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([0,0]),Polrev([7,3]),Polrev([-680,-204])], K);
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![7,3],K![-680,-204]]);
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: | \((-3072a+2048)\) | = | \((a-4)^{17}\cdot(a+3)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -134217728 \) | = | \(-2^{17}\cdot2^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{925430099}{32} a + \frac{3956137073}{32} \) | ||
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 | $\left(7 a + 25 : -52 a - 168 : 1\right)$ |
Height | \(1.1749805434692234780837502465135373062\) |
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: | $\left(2 a + 4 : 0 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 1.1749805434692234780837502465135373062 \) | ||
Period: | \( 2.1703760933475409720267540680132828281 \) | ||
Tamagawa product: | \( 16 \) = \(2^{2}\cdot2^{2}\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.7022046150103699995894226917452920965 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a-4)\) | \(2\) | \(4\) | \(I_{9}^{*}\) | Additive | \(-1\) | \(4\) | \(17\) | \(5\) |
\((a+3)\) | \(2\) | \(4\) | \(I_{2}^{*}\) | Additive | \(1\) | \(4\) | \(10\) | \(0\) |
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
256.1-j
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.