Base field 4.4.17725.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 12 x^{2} + 13 x + 41 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, 13, -12, -2, 1]))
gp: K = nfinit(Polrev([41, 13, -12, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 13, -12, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-6,-1,1]),K([-7,7,2,-1]),K([0,-6,-1,1]),K([33,-1,-5,1]),K([-11,1,2,0])])
gp: E = ellinit([Polrev([0,-6,-1,1]),Polrev([-7,7,2,-1]),Polrev([0,-6,-1,1]),Polrev([33,-1,-5,1]),Polrev([-11,1,2,0])], K);
magma: E := EllipticCurve([K![0,-6,-1,1],K![-7,7,2,-1],K![0,-6,-1,1],K![33,-1,-5,1],K![-11,1,2,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a-1)\) | = | \((-a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 19 \) | = | \(19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((a^3-5a^2-5a+20)\) | = | \((-a-1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 6859 \) | = | \(19^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{2454829497}{6859} a^{3} - \frac{3223188207}{6859} a^{2} + \frac{18885670830}{6859} a + \frac{30593217186}{6859} \) | ||
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: | \(0 \le r \le 1\) |
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: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(0 \le r \le 1\) | ||
Regulator: | not available | ||
Period: | \( 186.15935077318305010334480621957312951 \) | ||
Tamagawa product: | \( 3 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.43072884180034 \) | ||
Analytic order of Ш: | not available |
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a-1)\) | \(19\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 19.1-b consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.