Base field 5.5.70601.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, 3, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([5,0,-11,-1,2]),K([4,-2,-10,-1,2]),K([0,1,-4,-1,1]),K([19,-8,-44,-5,9]),K([-9,32,58,0,-11])])
gp: E = ellinit([Polrev([5,0,-11,-1,2]),Polrev([4,-2,-10,-1,2]),Polrev([0,1,-4,-1,1]),Polrev([19,-8,-44,-5,9]),Polrev([-9,32,58,0,-11])], K);
magma: E := EllipticCurve([K![5,0,-11,-1,2],K![4,-2,-10,-1,2],K![0,1,-4,-1,1],K![19,-8,-44,-5,9],K![-9,32,58,0,-11]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((2)\) | = | \((2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 32 \) | = | \(32\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-2)\) | = | \((2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 32 \) | = | \(32\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( 410269356 a^{4} - 278026353 a^{3} - 2140972273 a^{2} + 130455752 a + \frac{2545730995}{2} \) | ||
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 | $\left(-2 a^{4} + a^{3} + 10 a^{2} + 2 a - 4 : -a^{4} + 4 a^{2} + 5 a + 2 : 1\right)$ |
Height | \(0.0077324031395937786037311795173222637679\) |
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: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.0077324031395937786037311795173222637679 \) | ||
Period: | \( 16912.423030468034734678745091318148216 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.46084863 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2)\) | \(32\) | \(1\) | \(I_{1}\) | 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 |
---|---|
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
32.1-c
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.