Base field 4.4.11661.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([3, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,-1,1,0]),K([0,-5,-1,1]),K([-3,0,1,0]),K([-105,68,46,-22]),K([732,626,-64,-106])])
gp: E = ellinit([Polrev([-2,-1,1,0]),Polrev([0,-5,-1,1]),Polrev([-3,0,1,0]),Polrev([-105,68,46,-22]),Polrev([732,626,-64,-106])], K);
magma: E := EllipticCurve([K![-2,-1,1,0],K![0,-5,-1,1],K![-3,0,1,0],K![-105,68,46,-22],K![732,626,-64,-106]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((a^3-a^2-5a)\) | = | \((-a)\cdot(a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 27 \) | = | \(3\cdot3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((10a^3-49a^2+13a+69)\) | = | \((-a)^{5}\cdot(a+1)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 4782969 \) | = | \(3^{5}\cdot3^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( -\frac{43579931283769}{243} a^{3} + \frac{107777583890159}{243} a^{2} + \frac{123309264089110}{243} a - \frac{276240756518333}{243} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(1\) |
Generator | $\left(-7 a^{3} + 2 a^{2} + 35 a + 21 : -59 a^{3} + 30 a^{2} + 283 a + 126 : 1\right)$ |
Height | \(0.021370245877703063977483871739899342419\) |
Torsion structure: | trivial |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
|||
Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.021370245877703063977483871739899342419 \) | ||
Period: | \( 408.46516301754559657854036357247184624 \) | ||
Tamagawa product: | \( 10 \) = \(5\cdot2\) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 3.23337916276245 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a)\) | \(3\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
\((a+1)\) | \(3\) | \(2\) | \(III^{*}\) | Additive | \(1\) | \(2\) | \(9\) | \(0\) |
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
27.1-e
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.