Base field 4.4.2777.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 1, -4, -1, 1]))
gp: K = nfinit(Polrev([2, 1, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 1, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,-3,-1,1]),K([-2,0,1,0]),K([-2,-1,1,0]),K([-58,53,23,-15]),K([90,-85,-34,21])])
gp: E = ellinit([Polrev([2,-3,-1,1]),Polrev([-2,0,1,0]),Polrev([-2,-1,1,0]),Polrev([-58,53,23,-15]),Polrev([90,-85,-34,21])], K);
magma: E := EllipticCurve([K![2,-3,-1,1],K![-2,0,1,0],K![-2,-1,1,0],K![-58,53,23,-15],K![90,-85,-34,21]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-2a^2+8)\) | = | \((-a)^{6}\cdot(a^3-a^2-4a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 512 \) | = | \(2^{6}\cdot8\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-62a^3-86a^2+140a+352)\) | = | \((-a)^{30}\cdot(a^3-a^2-4a+1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -8589934592 \) | = | \(-2^{30}\cdot8\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( -\frac{14270157697541427}{4096} a^{3} + \frac{35796422109839569}{4096} a^{2} + \frac{1541200524057185}{2048} a - \frac{18919901006348423}{4096} \) | ||
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 | $\left(a^{2} : -4 a^{3} + a^{2} + 11 a - 4 : 1\right)$ |
Height | \(0.34438803798673382856260663694771192546\) |
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: | $\left(\frac{5}{4} a^{3} - 2 a^{2} - 4 a + 6 : -\frac{13}{4} a^{3} + \frac{17}{4} a^{2} + \frac{81}{8} a - \frac{31}{4} : 1\right)$ |
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: | \( 0.34438803798673382856260663694771192546 \) | ||
Period: | \( 138.45959113046384611660493512357507602 \) | ||
Tamagawa product: | \( 4 \) = \(2^{2}\cdot1\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 3.61945477478369 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(2\) | \(4\) | \(I_{20}^{*}\) | Additive | \(1\) | \(6\) | \(30\) | \(12\) |
\((a^3-a^2-4a+1)\) | \(8\) | \(1\) | \(I_{1}\) | 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 |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
512.3-n
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.