Base field 6.6.1241125.1
Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 11, -2, -7, 0, 1]))
gp: K = nfinit(Polrev([1, 7, 11, -2, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-5,-11,2,7,0,-1]),K([15,41,3,-27,-2,4]),K([-3,-8,-3,6,1,-1]),K([43,97,-50,-105,6,18]),K([22,-56,-338,-218,62,47])])
gp: E = ellinit([Polrev([-5,-11,2,7,0,-1]),Polrev([15,41,3,-27,-2,4]),Polrev([-3,-8,-3,6,1,-1]),Polrev([43,97,-50,-105,6,18]),Polrev([22,-56,-338,-218,62,47])], K);
magma: E := EllipticCurve([K![-5,-11,2,7,0,-1],K![15,41,3,-27,-2,4],K![-3,-8,-3,6,1,-1],K![43,97,-50,-105,6,18],K![22,-56,-338,-218,62,47]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2a^5-a^4-14a^3+2a^2+23a+6)\) | = | \((2a^5-a^4-14a^3+2a^2+23a+6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 9 \) | = | \(9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((3a^5+a^4-19a^3-8a^2+23a+9)\) | = | \((2a^5-a^4-14a^3+2a^2+23a+6)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -729 \) | = | \(-9^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{79572442844}{27} a^{5} + \frac{179028656408}{27} a^{4} - \frac{154197900412}{27} a^{3} - \frac{506013769217}{27} a^{2} - \frac{263202231980}{27} a - \frac{35350337779}{27} \) | ||
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: | 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: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 1226.5360945033932874353896896020530865 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 1.10096 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2a^5-a^4-14a^3+2a^2+23a+6)\) | \(9\) | \(1\) | \(I_{3}\) | Non-split multiplicative | \(1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(5\) | 5B.4.1[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
9.1-a
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.