Base field \(\Q(\sqrt{209}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 52 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-52, -1, 1]))
gp: K = nfinit(Polrev([-52, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-52, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,1]),K([-7197740,931337]),K([-10446391276,1351685914])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([0,1]),Polrev([-7197740,931337]),Polrev([-10446391276,1351685914])], K);
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,1],K![-7197740,931337],K![-10446391276,1351685914]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-15a+116)\) | = | \((11a+74)^{4}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(2^{4}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((5a-76)\) | = | \((11a+74)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 4096 \) | = | \(2^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -32768 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z[(1+\sqrt{-11})/2]\) | (potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $N(\mathrm{U}(1))$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(-209709 a + 1620721 : 365142927 a - 2821976497 : 1\right)$ |
| Height | \(4.4522854538708100592553980919276405245\) |
| 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: | \( 4.4522854538708100592553980919276405245 \) | ||
| Period: | \( 3.4769158424352161557941951589028318460 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.1415786686789418636283898123342378140 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((11a+74)\) | \(2\) | \(1\) | \(II^{*}\) | Additive | \(-1\) | \(4\) | \(12\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
The image is a Borel subgroup if \(p=11\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
11.
Its isogeny class
16.4-a
consists of curves linked by isogenies of
degree 11.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.