Base field 4.4.16609.1
Generator \(a\), with minimal polynomial \( x^{4} - 7 x^{2} - x + 9 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, -1, -7, 0, 1]))
gp: K = nfinit(Polrev([9, -1, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -1, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-4,0,1]),K([5,0,-1,0]),K([1,0,0,0]),K([-1110,-823,172,147]),K([9470,6962,-1468,-1242])])
gp: E = ellinit([Polrev([0,-4,0,1]),Polrev([5,0,-1,0]),Polrev([1,0,0,0]),Polrev([-1110,-823,172,147]),Polrev([9470,6962,-1468,-1242])], K);
magma: E := EllipticCurve([K![0,-4,0,1],K![5,0,-1,0],K![1,0,0,0],K![-1110,-823,172,147],K![9470,6962,-1468,-1242]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2+3)\) | = | \((-a^2-a+2)\cdot(a^2-a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 6 \) | = | \(2\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^3+a^2-9a)\) | = | \((-a^2-a+2)^{2}\cdot(a^2-a-3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 2916 \) | = | \(2^{2}\cdot3^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1106561590963}{2916} a^{3} + \frac{262445678183}{324} a^{2} + \frac{1352062034885}{1458} a - \frac{4665604095371}{2916} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(4 a^{3} + 5 a^{2} - 23 a - 33 : 7 a^{3} + 8 a^{2} - 38 a - 50 : 1\right)$ | $\left(-\frac{5}{4} a^{3} - \frac{3}{4} a^{2} + \frac{13}{2} a + \frac{19}{4} : -\frac{5}{8} a^{3} - \frac{15}{8} a^{2} + \frac{19}{4} a + \frac{95}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 486.15937402994335077496077347213717014 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 0.943076515696087 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^2-a+2)\) | \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((a^2-a-3)\) | \(3\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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
6.1-a
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.