Base field \(\Q(\sqrt{37}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 9 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-9, -1, 1]))
gp: K = nfinit(Polrev([-9, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([0,0]),K([0,0]),K([-3788,907]),K([-109976,29235])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([0,0]),Polrev([-3788,907]),Polrev([-109976,29235])], K);
magma: E := EllipticCurve([K![1,1],K![0,0],K![0,0],K![-3788,907],K![-109976,29235]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((14)\) | = | \((2)\cdot(-a-1)\cdot(a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 196 \) | = | \(4\cdot7\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-100156a+243530)\) | = | \((2)\cdot(-a-1)^{3}\cdot(a-2)^{9}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -55365148804 \) | = | \(-4\cdot7^{3}\cdot7^{9}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{195023039462640289}{40353607} a - \frac{963487859994700925}{80707214} \) | ||
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 | $\left(\frac{139}{9} a - 6 : \frac{1022}{27} a + \frac{182}{3} : 1\right)$ |
| Height | \(1.2758952112761822280201392951096659486\) |
| 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: | \( 1.2758952112761822280201392951096659486 \) | ||
| Period: | \( 0.26230195845171596054902038265320575832 \) | ||
| Tamagawa product: | \( 27 \) = \(1\cdot3\cdot3^{2}\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.9710464276084496258542969930697530838 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \((-a-1)\) | \(7\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
| \((a-2)\) | \(7\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
196.1-d
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.