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([-7,-8,0,1]),K([-1,0,0,0]),K([-6,-1,1,0]),K([-5812,609,1392,-316]),K([-105480,12413,25549,-5945])])
gp: E = ellinit([Polrev([-7,-8,0,1]),Polrev([-1,0,0,0]),Polrev([-6,-1,1,0]),Polrev([-5812,609,1392,-316]),Polrev([-105480,12413,25549,-5945])], K);
magma: E := EllipticCurve([K![-7,-8,0,1],K![-1,0,0,0],K![-6,-1,1,0],K![-5812,609,1392,-316],K![-105480,12413,25549,-5945]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^2-10)\) | = | \((a^2-10)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 49 \) | = | \(49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((8053a^3+73613a^2-165832a-447961)\) | = | \((a^2-10)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 191581231380566414401 \) | = | \(49^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{221757824275299}{13841287201} a^{3} + \frac{544265756337879}{13841287201} a^{2} - \frac{93164073307983}{1977326743} a - \frac{1664008534229859}{13841287201} \) | ||
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 | $\left(-\frac{74}{9} a^{3} + 13 a^{2} + \frac{442}{9} a - \frac{19}{3} : -\frac{907}{27} a^{3} + \frac{6491}{27} a^{2} - \frac{1840}{27} a - \frac{32182}{27} : 1\right)$ |
Height | \(1.0779552952515073919864183042768190239\) |
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: | $\left(\frac{11}{4} a^{3} - \frac{21}{4} a^{2} - \frac{31}{2} a + \frac{35}{4} : \frac{37}{8} a^{3} + \frac{9}{8} a^{2} - \frac{317}{8} a - \frac{387}{8} : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 1.0779552952515073919864183042768190239 \) | ||
Period: | \( 19.605972930637165480756659791336255381 \) | ||
Tamagawa product: | \( 12 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.90492233135250 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^2-10)\) | \(49\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
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 and 4.
Its isogeny class
49.1-a
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.