Base field 4.4.9909.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -3, -6, 0, 1]))
gp: K = nfinit(Polrev([3, -3, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1,0]),K([2,2,-1,0]),K([2,-3,-1,1]),K([85,-124,-98,51]),K([467,-720,-560,296])])
gp: E = ellinit([Polrev([-3,0,1,0]),Polrev([2,2,-1,0]),Polrev([2,-3,-1,1]),Polrev([85,-124,-98,51]),Polrev([467,-720,-560,296])], K);
magma: E := EllipticCurve([K![-3,0,1,0],K![2,2,-1,0],K![2,-3,-1,1],K![85,-124,-98,51],K![467,-720,-560,296]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-5a)\) | = | \((-a^3+a^2+4a)\cdot(-a^3+2a^2+3a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 15 \) | = | \(3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-17a^3-23a^2+150a+105)\) | = | \((-a^3+a^2+4a)^{2}\cdot(-a^3+2a^2+3a-4)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 87890625 \) | = | \(3^{2}\cdot5^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{422494591886387}{29296875} a^{3} + \frac{1087832991903677}{29296875} a^{2} + \frac{19383695526463}{1953125} a - \frac{160867452574272}{9765625} \) | ||
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 | $\left(2 a^{3} - 2 a^{2} - 6 a : -3 a^{2} - 4 a - 1 : 1\right)$ | |
| Height | \(0.30064573076647616729489308124548338046\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(2 a^{3} - 3 a^{2} - \frac{27}{4} a + \frac{1}{2} : -\frac{1}{8} a^{3} + \frac{7}{4} a^{2} - \frac{9}{8} a - \frac{19}{4} : 1\right)$ | $\left(a^{3} - 2 a^{2} - 2 a + 2 : -a^{3} + a^{2} + 3 a - 1 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.30064573076647616729489308124548338046 \) | ||
| Period: | \( 168.35735220100399423838613234850383842 \) | ||
| Tamagawa product: | \( 20 \) = \(2\cdot( 2 \cdot 5 )\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 2.54239027220219 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^3+a^2+4a)\) | \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((-a^3+2a^2+3a-4)\) | \(5\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
15.1-b
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.