Base field \(\Q(\sqrt{2}, \sqrt{7})\)
Generator \(a\), with minimal polynomial \( x^{4} - 8 x^{2} + 9 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([9, 0, -8, 0, 1]))
gp: K = nfinit(Polrev([9, 0, -8, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, -8, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0]),K([0,0,0,0]),K([1,0,0,0]),K([-11,0,0,0]),K([12,0,0,0])])
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([0,0,0,0]),Polrev([1,0,0,0]),Polrev([-11,0,0,0]),Polrev([12,0,0,0])], K);
magma: E := EllipticCurve([K![1,0,0,0],K![0,0,0,0],K![1,0,0,0],K![-11,0,0,0],K![12,0,0,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2/3a^3+2a^2-1/3a-1)\) | = | \((1/3a^3+a^2-2/3a-2)\cdot(1/3a^3+a^2-2/3a-3)\cdot(a+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 98 \) | = | \(2\cdot7\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((98)\) | = | \((1/3a^3+a^2-2/3a-2)^{4}\cdot(1/3a^3+a^2-2/3a-3)^{4}\cdot(a+2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 92236816 \) | = | \(2^{4}\cdot7^{4}\cdot7^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{128787625}{98} \) | ||
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: | \(2\) | |
| Generators | $\left(a^{2} - 4 : -a^{2} + 7 : 1\right)$ | $\left(\frac{4}{9} a^{2} - 1 : -\frac{17}{27} a^{3} - \frac{2}{9} a^{2} + \frac{13}{3} a : 1\right)$ |
| Heights | \(0.19484484261192328250586595732048315124\) | \(0.70169503751341272691958723147940411863\) |
| Torsion structure: | \(\Z/2\Z\oplus\Z/6\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{2}{3} a^{3} - \frac{10}{3} a - 1 : -\frac{1}{3} a^{3} + \frac{5}{3} a : 1\right)$ | $\left(0 : 3 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 0.13672165914586850633072038759878035679 \) | ||
| Period: | \( 1248.3109013368628226986843369146119165 \) | ||
| Tamagawa product: | \( 32 \) = \(2\cdot2^{2}\cdot2^{2}\) | ||
| Torsion order: | \(12\) | ||
| Leading coefficient: | \( 5.41813135113176 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/3a^3+a^2-2/3a-2)\) | \(2\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((1/3a^3+a^2-2/3a-3)\) | \(7\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((a+2)\) | \(7\) | \(4\) | \(I_{4}\) | 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 |
|---|---|
| \(2\) | 2Cs |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 6, 9 and 18.
Its isogeny class
98.1-n
consists of curves linked by isogenies of
degrees dividing 36.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 10 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 14.a4 |
| \(\Q\) | 448.g4 |
| \(\Q\) | 784.b4 |
| \(\Q\) | 3136.z4 |
| \(\Q(\sqrt{14}) \) | 2.2.56.1-14.1-a5 |
| \(\Q(\sqrt{14}) \) | 2.2.56.1-112.1-a5 |
| \(\Q(\sqrt{2}) \) | 2.2.8.1-98.1-a8 |
| \(\Q(\sqrt{2}) \) | a curve with conductor norm 38416 (not in the database) |
| \(\Q(\sqrt{7}) \) | 2.2.28.1-14.1-b5 |
| \(\Q(\sqrt{7}) \) | a curve with conductor norm 1792 (not in the database) |