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([3/2,11/4,0,-1/4]),K([0,4,1/2,-1/2]),K([1/2,11/4,0,-1/4]),K([-11/2,-27/4,3/2,3/4]),K([-51/2,-145/4,7/2,17/4])])
gp: E = ellinit([Polrev([3/2,11/4,0,-1/4]),Polrev([0,4,1/2,-1/2]),Polrev([1/2,11/4,0,-1/4]),Polrev([-11/2,-27/4,3/2,3/4]),Polrev([-51/2,-145/4,7/2,17/4])], K);
magma: E := EllipticCurve([K![3/2,11/4,0,-1/4],K![0,4,1/2,-1/2],K![1/2,11/4,0,-1/4],K![-11/2,-27/4,3/2,3/4],K![-51/2,-145/4,7/2,17/4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-1/4a^3+1/2a^2+9/4a-1/2)\) | = | \((-1/4a^3+1/2a^2+9/4a-1/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 29 \) | = | \(29\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((1/4a^3-1/2a^2-9/4a+1/2)\) | = | \((-1/4a^3+1/2a^2+9/4a-1/2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -29 \) | = | \(-29\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{42577135}{116} a^{3} + \frac{61063303}{58} a^{2} + \frac{23791499}{116} a - \frac{25310505}{58} \) | ||
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: | $\left(\frac{5}{16} a^{3} - \frac{1}{8} a^{2} - \frac{49}{16} a - \frac{15}{8} : -\frac{5}{16} a^{3} + \frac{57}{16} a + 3 : 1\right)$ |
| 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: | \( 497.36870993014090070663903210224994942 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.91295657665439 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-1/4a^3+1/2a^2+9/4a-1/2)\) | \(29\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
29.1-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.