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([0,1]),K([-1,-1]),K([1,0]),K([-78105,-23543]),K([11940924,3600318])])
gp: E = ellinit([Polrev([0,1]),Polrev([-1,-1]),Polrev([1,0]),Polrev([-78105,-23543]),Polrev([11940924,3600318])], K);
magma: E := EllipticCurve([K![0,1],K![-1,-1],K![1,0],K![-78105,-23543],K![11940924,3600318]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a-1)\) | = | \((a+3)\cdot(a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 10 \) | = | \(2\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a-1)\) | = | \((a+3)\cdot(a-4)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -10 \) | = | \(-2\cdot5\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{466209435421917067326607}{10} a + \frac{1546241771017925012924657}{10} \) | ||
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 | $\left(\frac{937}{25} a + \frac{3147}{25} : -\frac{19299}{125} a - \frac{63744}{125} : 1\right)$ |
| Height | \(0.41154070631662457691910948682021389272\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.41154070631662457691910948682021389272 \) | ||
| Period: | \( 8.6588982306646701996801077588831869643 \) | ||
| Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.0744323880512441395294914877717627447 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+3)\) | \(2\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((a-4)\) | \(5\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(7\) | 7B.6.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
10.1-c
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.