Base field \(\Q(\sqrt{249}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 62 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-62, -1, 1]))
gp: K = nfinit(Polrev([-62, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-62, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([1,1]),K([-864079084,102990805]),K([13425981367840,-1600261541361])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([1,1]),Polrev([-864079084,102990805]),Polrev([13425981367840,-1600261541361])], K);
magma: E := EllipticCurve([K![0,1],K![0,0],K![1,1],K![-864079084,102990805],K![13425981367840,-1600261541361]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((41a+303)\) | = | \((59a-495)\cdot(18a-151)\) |
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: | \((11589a-97223)\) | = | \((59a-495)^{18}\cdot(18a-151)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -1310720 \) | = | \(-2^{18}\cdot5\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{8025743347527}{1310720} a + \frac{29314806044891}{655360} \) | ||
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 | $\left(-1908 a + 16002 : -127613 a + 1070676 : 1\right)$ |
| Height | \(0.64562407970798553177336575333972510789\) |
| Torsion structure: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-\frac{3790}{3} a + \frac{31780}{3} : \frac{463079}{9} a - \frac{3884992}{9} : 1\right)$ |
| 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: | \( 0.64562407970798553177336575333972510789 \) | ||
| Period: | \( 21.596732818265221371066476457139048348 \) | ||
| Tamagawa product: | \( 18 \) = \(( 2 \cdot 3^{2} )\cdot1\) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 3.5345008714561200709276311210798559367 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((59a-495)\) | \(2\) | \(18\) | \(I_{18}\) | Split multiplicative | \(-1\) | \(1\) | \(18\) | \(18\) |
| \((18a-151)\) | \(5\) | \(1\) | \(I_{1}\) | Non-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 |
|---|---|
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
10.2-a
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.