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,-7,-1,1]),K([7,-7,-2,1]),K([-6,-1,1,0]),K([-4126,-2446,444,316]),K([-73463,-45676,7714,5945])])
gp: E = ellinit([Polrev([0,-7,-1,1]),Polrev([7,-7,-2,1]),Polrev([-6,-1,1,0]),Polrev([-4126,-2446,444,316]),Polrev([-73463,-45676,7714,5945])], K);
magma: E := EllipticCurve([K![0,-7,-1,1],K![7,-7,-2,1],K![-6,-1,1,0],K![-4126,-2446,444,316],K![-73463,-45676,7714,5945]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^3+a^2-8a-13)\) | = | \((a^3+a^2-8a-13)\) |
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: | \((-35848a^3+29661a^2+265176a-46410)\) | = | \((a^3+a^2-8a-13)^{12}\) |
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: | \( -\frac{221757824275299}{13841287201} a^{3} + \frac{172791318451968}{1977326743} a^{2} - \frac{1101656472345774}{13841287201} a - \frac{221447638110366}{1977326743} \) | ||
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 | $\left(\frac{74}{9} a^{3} - \frac{35}{3} a^{2} - \frac{454}{9} a + \frac{428}{9} : -\frac{784}{27} a^{3} - \frac{4646}{27} a^{2} + \frac{899}{3} a + \frac{33775}{27} : 1\right)$ |
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: | $\left(-\frac{11}{4} a^{3} + 3 a^{2} + \frac{71}{4} a - \frac{37}{4} : \frac{25}{8} a^{3} - \frac{27}{4} a^{2} - 17 a + \frac{135}{4} : 1\right)$ |
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^3+a^2-8a-13)\) | \(49\) | \(12\) | \(I_{12}\) | 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
49.2-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.