Base field \(\Q(\sqrt{5}, \sqrt{17})\)
Generator \(a\), with minimal polynomial \( x^{4} - 11 x^{2} + 9 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, 0, -11, 0, 1]))
gp: K = nfinit(Polrev([9, 0, -11, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, -11, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,11/6,1/2,-1/6]),K([3,-11/6,-1/2,1/6]),K([3/2,7/3,0,-1/6]),K([8,-22/3,-1,2/3]),K([9,-6,-6,-1])])
gp: E = ellinit([Polrev([-2,11/6,1/2,-1/6]),Polrev([3,-11/6,-1/2,1/6]),Polrev([3/2,7/3,0,-1/6]),Polrev([8,-22/3,-1,2/3]),Polrev([9,-6,-6,-1])], K);
magma: E := EllipticCurve([K![-2,11/6,1/2,-1/6],K![3,-11/6,-1/2,1/6],K![3/2,7/3,0,-1/6],K![8,-22/3,-1,2/3],K![9,-6,-6,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/2a^3-3a+1/2)\) | = | \((1/6a^3-1/2a^2-5/6a+2)\cdot(1/2a^2+1/2a-11/2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 236 \) | = | \(4\cdot59\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((28/3a^3-1/2a^2-715/6a-63/2)\) | = | \((1/6a^3-1/2a^2-5/6a+2)\cdot(1/2a^2+1/2a-11/2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 48469444 \) | = | \(4\cdot59^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{35867214803}{48469444} a^{3} - \frac{15888032143}{48469444} a^{2} - \frac{455784216367}{48469444} a + \frac{114223634126}{12117361} \) | ||
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: | $\left(-\frac{1}{24} a^{3} + \frac{3}{8} a^{2} + \frac{29}{24} a - 2 : -\frac{5}{16} a^{3} - \frac{1}{8} a^{2} + \frac{3}{2} a - \frac{45}{16} : 1\right)$ |
| 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: | \( 271.37308184670033938491768186645411095 \) | ||
| Tamagawa product: | \( 4 \) = \(1\cdot2^{2}\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.19262449231412 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/6a^3-1/2a^2-5/6a+2)\) | \(4\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((1/2a^2+1/2a-11/2)\) | \(59\) | \(4\) | \(I_{4}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
236.3-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.