Base field \(\Q(\sqrt{-11}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
gp: K = nfinit(Polrev([3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,1]),K([138,-163]),K([-354,-715])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([1,1]),Polrev([138,-163]),Polrev([-354,-715])], K);
magma: E := EllipticCurve([K![1,0],K![1,0],K![1,1],K![138,-163],K![-354,-715]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-84a+42)\) | = | \((-a)\cdot(a-1)\cdot(2)\cdot(-2a+1)\cdot(7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 19404 \) | = | \(3\cdot3\cdot4\cdot11\cdot49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1065577632a+1575064512)\) | = | \((-a)^{26}\cdot(a-1)\cdot(2)^{5}\cdot(-2a+1)\cdot(7)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 4208841773473784832 \) | = | \(3^{26}\cdot3\cdot4^{5}\cdot11\cdot49\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{380239251435504673}{6263157401002656} a + \frac{6832513042456643645}{6263157401002656} \) | ||
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 | $\left(-\frac{3194}{529} a - \frac{6083}{529} : \frac{457211}{12167} a - \frac{443372}{12167} : 1\right)$ |
| Height | \(2.8539515719756900441132091741540964187\) |
| 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: | \( 2.8539515719756900441132091741540964187 \) | ||
| Period: | \( 0.38727080959224989109291682159847339531 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\cdot1\cdot1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.6659684605388579293021432058497570831 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(3\) | \(2\) | \(I_{26}\) | Non-split multiplicative | \(1\) | \(1\) | \(26\) | \(26\) |
| \((a-1)\) | \(3\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((2)\) | \(4\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
| \((-2a+1)\) | \(11\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((7)\) | \(49\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 19404.2-a consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.