Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
gp: K = nfinit(Polrev([1, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,1]),K([0,1]),K([132,-235]),K([667,-757])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,1]),Polrev([0,1]),Polrev([132,-235]),Polrev([667,-757])], K);
magma: E := EllipticCurve([K![0,1],K![1,1],K![0,1],K![132,-235],K![667,-757]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((322a-308)\) | = | \((-2a+1)\cdot(2)\cdot(-3a+1)\cdot(3a-2)\cdot(-4a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 99372 \) | = | \(3\cdot4\cdot7\cdot7\cdot13^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-501308514a+482662152)\) | = | \((-2a+1)^{14}\cdot(2)\cdot(-3a+1)^{6}\cdot(3a-2)^{2}\cdot(-4a+1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 242310332998997172 \) | = | \(3^{14}\cdot4\cdot7^{6}\cdot7^{2}\cdot13^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{321426784225}{514596726} a + \frac{370102537351}{171532242} \) | ||
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 | $\left(-5 a - 5 : -27 a + 8 : 1\right)$ |
Height | \(0.051874010343000356378475577923278939091\) |
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: | $\left(-\frac{5}{4} a - \frac{11}{4} : \frac{3}{2} a - \frac{5}{8} : 1\right)$ |
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: | \( 0.051874010343000356378475577923278939091 \) | ||
Period: | \( 0.48539907291875729029627604003779238070 \) | ||
Tamagawa product: | \( 336 \) = \(( 2 \cdot 7 )\cdot1\cdot( 2 \cdot 3 )\cdot2\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 4.8845821362727072056999383998933863924 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-2a+1)\) | \(3\) | \(14\) | \(I_{14}\) | Split multiplicative | \(-1\) | \(1\) | \(14\) | \(14\) |
\((2)\) | \(4\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
\((-3a+1)\) | \(7\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
\((3a-2)\) | \(7\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((-4a+1)\) | \(13\) | \(2\) | \(III\) | Additive | \(-1\) | \(2\) | \(3\) | \(0\) |
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
99372.4-k
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.