Base field \(\Q(\sqrt{5}, \sqrt{13})\)
Generator \(a\), with minimal polynomial \( x^{4} - 9 x^{2} + 4 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([4, 0, -9, 0, 1]))
gp: K = nfinit(Polrev([4, 0, -9, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, -9, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1/2,-7/4,0,1/4]),K([-1/2,5/4,1/2,-1/4]),K([0,1,0,0]),K([13,-19,-1,2]),K([-131,191,31/2,-45/2])])
gp: E = ellinit([Polrev([1/2,-7/4,0,1/4]),Polrev([-1/2,5/4,1/2,-1/4]),Polrev([0,1,0,0]),Polrev([13,-19,-1,2]),Polrev([-131,191,31/2,-45/2])], K);
magma: E := EllipticCurve([K![1/2,-7/4,0,1/4],K![-1/2,5/4,1/2,-1/4],K![0,1,0,0],K![13,-19,-1,2],K![-131,191,31/2,-45/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/4a^3-11/4a-1/2)\) | = | \((1/4a^3-11/4a-1/2)\) |
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: | \((1/4a^3-11/4a-7/2)\) | = | \((1/4a^3-11/4a-1/2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -81 \) | = | \(-9^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{22003355573863}{36} a^{3} - \frac{3766653496885}{9} a^{2} + \frac{20857051342593}{4} a + \frac{64267612784665}{18} \) | ||
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: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{9}{16} a^{3} - \frac{5}{8} a^{2} - \frac{69}{16} a + \frac{21}{8} : -\frac{5}{16} a^{3} + \frac{1}{4} a^{2} + \frac{41}{16} a - \frac{9}{4} : 1\right)$ |
| 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: | \( 182.57806002030811312104733717245444550 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.40444661554083 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/4a^3-11/4a-1/2)\) | \(9\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 8 and 16.
Its isogeny class
9.1-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.