Base field \(\Q(\sqrt{14}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 14 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 0, 1]))
gp: K = nfinit(Polrev([-14, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([-635,-150]),K([-9041,-2318])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,1]),Polrev([-635,-150]),Polrev([-9041,-2318])], K);
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,1],K![-635,-150],K![-9041,-2318]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2a)\) | = | \((-a+4)^{3}\cdot(-2a+7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 56 \) | = | \(2^{3}\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-32a)\) | = | \((-a+4)^{11}\cdot(-2a+7)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -14336 \) | = | \(-2^{11}\cdot7\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{39775849362076815}{7} a + 21261085797246696 \) | ||
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: | \(2\) | |
Generators | $\left(-\frac{156337}{8450} a + \frac{807133}{16900} : -\frac{53898228}{274625} a + \frac{1697471733}{2197000} : 1\right)$ | $\left(-\frac{19598}{4225} a - \frac{6947}{1690} : \frac{8047431}{1098500} a + \frac{5975163}{549250} : 1\right)$ |
Heights | \(5.9221351593970926978791183875431340406\) | \(5.9221351593970926978791183875431340153\) |
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);
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Torsion generator: | $\left(-4 a - \frac{13}{2} : \frac{11}{4} a + 28 : 1\right)$ | |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 2 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(2\) | ||
Regulator: | \( 32.363353551208464212080134907405718051 \) | ||
Period: | \( 0.65907338218172640606237416363638825668 \) | ||
Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.8503177441011686065450864444986300260 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a+4)\) | \(2\) | \(1\) | \(II^{*}\) | Additive | \(-1\) | \(3\) | \(11\) | \(0\) |
\((-2a+7)\) | \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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, 4 and 8.
Its isogeny class
56.1-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.