Base field 4.4.11324.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 4 x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([2, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([-3,1,1,0]),K([1,-4,0,1]),K([1540,3071,-3922,-1149]),K([117393,264677,-216742,-99521])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([-3,1,1,0]),Polrev([1,-4,0,1]),Polrev([1540,3071,-3922,-1149]),Polrev([117393,264677,-216742,-99521])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![-3,1,1,0],K![1,-4,0,1],K![1540,3071,-3922,-1149],K![117393,264677,-216742,-99521]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-3a)\) | = | \((a^3-5a)\cdot(-a^2+a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 8 \) | = | \(2\cdot4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-5a^3-15a^2+19a+22)\) | = | \((a^3-5a)\cdot(-a^2+a+3)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 524288 \) | = | \(2\cdot4^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{9135977966106098429399}{512} a^{3} + \frac{867628940947068681587}{256} a^{2} - \frac{21807541432668180905747}{256} a - \frac{959710996587311585475}{32} \) | ||
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: | \(0\) |
| 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: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 0.027495042004777190487634686949203228441 \) | ||
| Tamagawa product: | \( 9 \) = \(1\cdot3^{2}\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.69521386163539 \) | ||
| Analytic order of Ш: | \( 729 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-5a)\) | \(2\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((-a^2+a+3)\) | \(4\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
8.1-d
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.