Base field \(\Q(\sqrt{205}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 51 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-51, -1, 1]))
gp: K = nfinit(Polrev([-51, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-51, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([0,-1]),K([0,1]),K([-19,-2]),K([-174,11])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,-1]),Polrev([0,1]),Polrev([-19,-2]),Polrev([-174,11])], K);
magma: E := EllipticCurve([K![0,1],K![0,-1],K![0,1],K![-19,-2],K![-174,11]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((14,2a+10)\) | = | \((2)\cdot(7,a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 28 \) | = | \(4\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-1284a+13936)\) | = | \((2)^{2}\cdot(7,a+5)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 92236816 \) | = | \(4^{2}\cdot7^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{780348831047775}{23059204} a + \frac{1299078889648623}{5764801} \) | ||
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 | $\left(\frac{13}{9} a - \frac{16}{3} : \frac{50}{27} a - \frac{200}{9} : 1\right)$ | $\left(\frac{1}{3} a - \frac{68}{9} : \frac{40}{9} a - \frac{505}{27} : 1\right)$ |
| Heights | \(2.1402472986621658288241245179362099579\) | \(2.6553291318083252600060544984930074555\) |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{3}{4} a - \frac{43}{4} : \frac{9}{2} a - \frac{153}{8} : 1\right)$ | $\left(-5 : 2 a : 1\right)$ |
| 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: | \( 5.4587435659360312087544588253619862003 \) | ||
| Period: | \( 3.9432131433469299769855676139040918823 \) | ||
| Tamagawa product: | \( 16 \) = \(2\cdot2^{3}\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 6.0134818783477308128885275361820399861 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((7,a+5)\) | \(7\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
28.2-b
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.