Base field 5.5.36497.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 1, 5, -3, -2, 1]))
gp: K = nfinit(Polrev([-1, 1, 5, -3, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 5, -3, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0,0]),K([1,0,-3,-1,1]),K([3,7,-7,-3,2]),K([26,-23,-134,37,9]),K([-38,-258,868,-62,-112])])
gp: E = ellinit([Polrev([1,1,0,0,0]),Polrev([1,0,-3,-1,1]),Polrev([3,7,-7,-3,2]),Polrev([26,-23,-134,37,9]),Polrev([-38,-258,868,-62,-112])], K);
magma: E := EllipticCurve([K![1,1,0,0,0],K![1,0,-3,-1,1],K![3,7,-7,-3,2],K![26,-23,-134,37,9],K![-38,-258,868,-62,-112]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^4-2a^3-4a^2+6a+2)\) | = | \((a^4-2a^3-4a^2+6a+2)\) |
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: | \((-a^4+2a^3+4a^2-6a-2)\) | = | \((a^4-2a^3-4a^2+6a+2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -49 \) | = | \(-49\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{620948251468503566}{7} a^{4} + 28210247096660731 a^{3} - \frac{1405103809967670307}{7} a^{2} - \frac{152315338627341705}{7} a + \frac{267878898946671192}{7} \) | ||
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 | $\left(-7 a^{4} + 11 a^{3} + 21 a^{2} - 22 a + 5 : -25 a^{4} + 49 a^{3} + 59 a^{2} - 102 a + 27 : 1\right)$ |
Height | \(0.065885382282140315158463570073523639338\) |
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: | $\left(-a^{4} + a^{3} + \frac{3}{4} a^{2} + \frac{15}{2} a - \frac{13}{4} : \frac{17}{8} a^{3} - \frac{25}{8} a^{2} - \frac{49}{8} a + \frac{5}{8} : 1\right)$ |
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: | \( 0.065885382282140315158463570073523639338 \) | ||
Period: | \( 4763.6892269615729898711964846425785883 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.05359085 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((a^4-2a^3-4a^2+6a+2)\) | \(49\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
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.