Base field \(\Q(\sqrt{15}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 15 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-15, 0, 1]))
gp: K = nfinit(Pol(Vecrev([-15, 0, 1])));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([1,1]),K([-16744,-4323]),K([-1189678,-307168])])
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,-1])),Pol(Vecrev([1,1])),Pol(Vecrev([-16744,-4323])),Pol(Vecrev([-1189678,-307168]))], K);
magma: E := EllipticCurve([K![1,1],K![0,-1],K![1,1],K![-16744,-4323],K![-1189678,-307168]]);
This is not a global minimal model: it is minimal at all primes except \((2,a+1)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a)\) | = | \((3,a)\cdot(5,a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 15 \) | = | \(3\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((10497600)\) | = | \((2,a+1)^{12}\cdot(3,a)^{16}\cdot(5,a)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 110199605760000 \) | = | \(2^{12}\cdot3^{16}\cdot5^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| Minimal discriminant: | \((164025)\) | = | \((3,a)^{16}\cdot(5,a)^{4}\) |
| Minimal discriminant norm: | \( 26904200625 \) | = | \(3^{16}\cdot5^{4}\) |
| j-invariant: | \( \frac{272223782641}{164025} \) | ||
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 | $\left(108 a + 430 : -3388 a - 13054 : 1\right)$ | |
| Height | \(2.60062568798625\) | |
| Torsion structure: | \(\Z/2\Z\times\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(-\frac{27}{2} a - 56 : \frac{137}{4} a + \frac{515}{4} : 1\right)$ | $\left(-13 a - 54 : 33 a + 124 : 1\right)$ |
| 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: | \( 2.60062568798625 \) | ||
| Period: | \( 1.96168888219134 \) | ||
| Tamagawa product: | \( 32 \) = \(1\cdot2^{4}\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 2.63446446464044 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a+1)\) | \(2\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
| \((3,a)\) | \(3\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
| \((5,a)\) | \(5\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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, 4, 8 and 16.
Its isogeny class
15.1-d
consists of curves linked by isogenies of
degrees dividing 32.
Base change
This curve is the base change of 75.b2, 720.c2, defined over \(\Q\), so it is also a \(\Q\)-curve.