Base field 4.4.19225.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 15 x^{2} + 2 x + 44 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([44, 2, -15, -1, 1]))
gp: K = nfinit(Polrev([44, 2, -15, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44, 2, -15, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([9,-5/2,-3/2,1/2]),K([8,-5/2,-3/2,1/2]),K([-22,6,4,-1]),K([-1238,464,220,-64]),K([16959,-13031/2,-6007/2,1809/2])])
gp: E = ellinit([Polrev([9,-5/2,-3/2,1/2]),Polrev([8,-5/2,-3/2,1/2]),Polrev([-22,6,4,-1]),Polrev([-1238,464,220,-64]),Polrev([16959,-13031/2,-6007/2,1809/2])], K);
magma: E := EllipticCurve([K![9,-5/2,-3/2,1/2],K![8,-5/2,-3/2,1/2],K![-22,6,4,-1],K![-1238,464,220,-64],K![16959,-13031/2,-6007/2,1809/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((3/2a^3-11/2a^2-19/2a+31)\) | = | \((3/2a^3-11/2a^2-19/2a+31)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((9/2a^3-31/2a^2-59/2a+89)\) | = | \((3/2a^3-11/2a^2-19/2a+31)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -6561 \) | = | \(-9^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{148735529594573}{162} a^{3} - \frac{44389227275927}{54} a^{2} + \frac{1978654649544127}{162} a + \frac{575471171779328}{27} \) | ||
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 | $\left(\frac{17}{4} a^{3} - \frac{59}{4} a^{2} - \frac{61}{2} a + 85 : \frac{145}{8} a^{3} - \frac{237}{4} a^{2} - 131 a + 332 : 1\right)$ |
| Height | \(2.3843167261047738450544367727045423333\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-\frac{35}{8} a^{3} + \frac{115}{8} a^{2} + \frac{249}{8} a - \frac{167}{2} : \frac{13}{8} a^{3} - \frac{37}{8} a^{2} - \frac{75}{8} a + \frac{223}{8} : 1\right)$ |
| 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: | \( 2.3843167261047738450544367727045423333 \) | ||
| Period: | \( 57.606375284767081023797471751060590644 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.96242734276914 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3/2a^3-11/2a^2-19/2a+31)\) | \(9\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
9.2-b
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.