Base field \(\Q(\sqrt{11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 11 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-11, 0, 1]))
gp: K = nfinit(Polrev([-11, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,0]),K([-68965,-20794]),K([-6044337,-1822436])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,-1]),Polrev([1,0]),Polrev([-68965,-20794]),Polrev([-6044337,-1822436])], K);
magma: E := EllipticCurve([K![1,0],K![-1,-1],K![1,0],K![-68965,-20794],K![-6044337,-1822436]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((21)\) | = | \((a+2)\cdot(a-2)\cdot(3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 441 \) | = | \(7\cdot7\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((11573604a+43401015)\) | = | \((a+2)^{4}\cdot(a-2)^{2}\cdot(3)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 410216697993249 \) | = | \(7^{4}\cdot7^{2}\cdot9^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{303371613810203}{141776649} a + \frac{335523283547294}{47258883} \) | ||
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 | $\left(-47 a - \frac{301}{2} : \frac{715}{4} a + \frac{2297}{4} : 1\right)$ |
| Height | \(1.3594806153831828604558875122455124661\) |
| 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: | $\left(-47 a - \frac{629}{4} : \frac{47}{2} a + \frac{625}{8} : 1\right)$ |
| 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: | \( 1.3594806153831828604558875122455124661 \) | ||
| Period: | \( 1.2871724670729138829984482872717775686 \) | ||
| Tamagawa product: | \( 40 \) = \(2\cdot2\cdot( 2 \cdot 5 )\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 5.2761048603663785441492521881218973599 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+2)\) | \(7\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((a-2)\) | \(7\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((3)\) | \(9\) | \(10\) | \(I_{10}\) | Split multiplicative | \(-1\) | \(1\) | \(10\) | \(10\) |
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
441.1-j
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.