Base field \(\Q(\sqrt{-11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
gp: K = nfinit(Polrev([3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([-9,0]),K([9,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-9,0]),Polrev([9,0])], K);
magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![-9,0],K![9,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((158)\) | = | \((2)\cdot(79)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 24964 \) | = | \(4\cdot6241\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((20224)\) | = | \((2)^{8}\cdot(79)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 409010176 \) | = | \(4^{8}\cdot6241\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{72511713}{20224} \) | ||
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: | \(3\) | ||
Generators | $\left(-2 a - 3 : 4 a - 12 : 1\right)$ | $\left(\frac{4}{9} a + \frac{11}{3} : -\frac{44}{27} a - \frac{58}{9} : 1\right)$ | $\left(-4 a - 1 : 4 a + 20 : 1\right)$ |
Heights | \(1.3583686801440819593247033724710284059\) | \(1.9707721046446770617621822836734928918\) | \(1.1634172785902033711405945156547541167\) |
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: | \( 3 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(3\) | ||
Regulator: | \( 0.039162963641991509572597277218434134462 \) | ||
Period: | \( 2.5425179645875854469509940832880648204 \) | ||
Tamagawa product: | \( 8 \) = \(2^{3}\cdot1\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.8428479997686886859404070185744245520 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(4\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
\((79)\) | \(6241\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
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 24964.1-a consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 158.c1 |
\(\Q\) | 19118.a1 |