Base field \(\Q(\zeta_{11})^+\)
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 3, -4, -1, 1]))
gp: K = nfinit(Polrev([-1, 3, 3, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,-3,-4,1,1]),K([0,2,-3,-1,1]),K([0,0,-3,0,1]),K([0,1,-1,-1,0]),K([0,4,-1,-2,0])])
gp: E = ellinit([Polrev([3,-3,-4,1,1]),Polrev([0,2,-3,-1,1]),Polrev([0,0,-3,0,1]),Polrev([0,1,-1,-1,0]),Polrev([0,4,-1,-2,0])], K);
magma: E := EllipticCurve([K![3,-3,-4,1,1],K![0,2,-3,-1,1],K![0,0,-3,0,1],K![0,1,-1,-1,0],K![0,4,-1,-2,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^4+3a^2+a-2)\) | = | \((-a^4+3a^2+a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 23 \) | = | \(23\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-3a^4+24a^3+9a^2-63a-2)\) | = | \((-a^4+3a^2+a-2)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -6436343 \) | = | \(-23^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{21045078235}{6436343} a^{4} - \frac{81762784}{6436343} a^{3} - \frac{37127244372}{6436343} a^{2} - \frac{539767896}{6436343} a + \frac{2876686312}{6436343} \) | ||
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: | \(0\) |
Torsion structure: | \(\Z/10\Z\) |
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: | $\left(-5 a^{4} + 10 a^{3} + 10 a^{2} - 28 a + 12 : 45 a^{4} - 80 a^{3} - 115 a^{2} + 221 a - 44 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 1986.2521339973071898146350140826083125 \) | ||
Tamagawa product: | \( 5 \) | ||
Torsion order: | \(10\) | ||
Leading coefficient: | \( 0.820765345 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^4+3a^2+a-2)\) | \(23\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
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 |
\(5\) | 5B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
23.2-a
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.