Base field 4.4.4525.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 3 x + 9 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, 3, -7, -1, 1]))
gp: K = nfinit(Polrev([9, 3, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 3, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1,0]),K([3,-1/3,-4/3,1/3]),K([0,1,0,0]),K([-24,-73/3,8/3,10/3]),K([60,53,-6,-8])])
gp: E = ellinit([Polrev([-3,0,1,0]),Polrev([3,-1/3,-4/3,1/3]),Polrev([0,1,0,0]),Polrev([-24,-73/3,8/3,10/3]),Polrev([60,53,-6,-8])], K);
magma: E := EllipticCurve([K![-3,0,1,0],K![3,-1/3,-4/3,1/3],K![0,1,0,0],K![-24,-73/3,8/3,10/3],K![60,53,-6,-8]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/3a^3+5/3a^2-4/3a-4)\) | = | \((-a^2+a+5)\cdot(1/3a^3-1/3a^2-7/3a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 45 \) | = | \(5\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1/3a^3+1/3a^2-2/3a-1)\) | = | \((-a^2+a+5)\cdot(1/3a^3-1/3a^2-7/3a+1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 45 \) | = | \(5\cdot9\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1037039198606}{15} a^{3} - 118962068606 a^{2} + \frac{2404382003012}{15} a + \frac{1143498460323}{5} \) | ||
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 | $\left(\frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{10}{3} a - 3 : \frac{7}{3} a^{3} - \frac{4}{3} a^{2} - \frac{37}{3} a - 4 : 1\right)$ |
| Height | \(0.021028658745267958607663639649332958194\) |
| 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: | \( 0.021028658745267958607663639649332958194 \) | ||
| Period: | \( 1579.5321674845683160232659827920998355 \) | ||
| Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.97510754342395 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^2+a+5)\) | \(5\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((1/3a^3-1/3a^2-7/3a+1)\) | \(9\) | \(1\) | \(I_{1}\) | 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
45.4-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.