Base field 4.4.18625.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 14 x^{2} + 9 x + 41 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, 9, -14, -1, 1]))
gp: K = nfinit(Polrev([41, 9, -14, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 9, -14, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-35/6,-1/3,1,1/6]),K([-1,-1,0,0]),K([-6,1,1,0]),K([-2023/6,-524/3,34,107/6]),K([5222/3,3587/3,-142,-337/3])])
gp: E = ellinit([Polrev([-35/6,-1/3,1,1/6]),Polrev([-1,-1,0,0]),Polrev([-6,1,1,0]),Polrev([-2023/6,-524/3,34,107/6]),Polrev([5222/3,3587/3,-142,-337/3])], K);
magma: E := EllipticCurve([K![-35/6,-1/3,1,1/6],K![-1,-1,0,0],K![-6,1,1,0],K![-2023/6,-524/3,34,107/6],K![5222/3,3587/3,-142,-337/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((1/6a^3-7/3a-11/6)\) | = | \((1/6a^3-7/3a-11/6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 11 \) | = | \(11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-1/6a^3+10/3a-25/6)\) | = | \((1/6a^3-7/3a-11/6)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 121 \) | = | \(11^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{4823165761}{242} a^{3} - \frac{6386985120}{121} a^{2} - \frac{23238969544}{121} a + \frac{120028019549}{242} \) | ||
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 | $\left(\frac{1}{8} a^{3} - \frac{15}{4} a^{2} - \frac{3}{2} a + \frac{273}{8} : \frac{185}{24} a^{3} - \frac{109}{8} a^{2} - \frac{1711}{24} a + \frac{1633}{12} : 1\right)$ | |
Height | \(1.3050711245166679721433609244299533266\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
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 generators: | $\left(\frac{5}{8} a^{3} + \frac{5}{4} a^{2} - \frac{15}{2} a - \frac{139}{8} : \frac{1}{24} a^{3} + \frac{13}{8} a^{2} + \frac{91}{24} a - \frac{17}{6} : 1\right)$ | $\left(-\frac{2}{3} a^{3} - 2 a^{2} + \frac{19}{3} a + \frac{55}{3} : 2 a^{3} + 2 a^{2} - 17 a - 13 : 1\right)$ |
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: | \( 1.3050711245166679721433609244299533266 \) | ||
Period: | \( 756.70631293927004821000076957810414603 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(4\) | ||
Leading coefficient: | \( 3.61812362218058 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((1/6a^3-7/3a-11/6)\) | \(11\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
11.1-b
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.