Base field \(\Q(\sqrt{-163}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 41 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, -1, 1]))
gp: K = nfinit(Polrev([41, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-135,0]),K([-660,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([1,0]),Polrev([-135,0]),Polrev([-660,0])], K);
magma: E := EllipticCurve([K![1,0],K![1,0],K![1,0],K![-135,0],K![-660,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((15)\) | = | \((3)\cdot(5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 225 \) | = | \(9\cdot25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((164025)\) | = | \((3)^{8}\cdot(5)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 26904200625 \) | = | \(9^{8}\cdot25^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| 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: | \(0 \le r \le 1\) | |
| 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: | $\left(-7 : 3 : 1\right)$ | $\left(-\frac{29}{4} : \frac{25}{8} : 1\right)$ |
| 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: | \(0 \le r \le 1\) | ||
| Regulator: | not available | ||
| Period: | \( 1.1178508563087645508833713910561208612 \) | ||
| Tamagawa product: | \( 16 \) = \(2^{3}\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 4.3134439442326855993429445266892711426 \) | ||
| Analytic order of Ш: | not available | ||
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3)\) | \(9\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
| \((5)\) | \(25\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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 and 8.
Its isogeny class
225.1-a
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 15.a2 |
| \(\Q\) | 398535.i2 |