Base field \(\Q(\sqrt{-1}) \)
Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))
gp: K = nfinit(Polrev([1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,1]),K([375,0]),K([12344,0])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,1]),Polrev([375,0]),Polrev([12344,0])], K);
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,1],K![375,0],K![12344,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((225)\) | = | \((-i-2)^{2}\cdot(2i+1)^{2}\cdot(3)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 50625 \) | = | \(5^{2}\cdot5^{2}\cdot9^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-69198046875)\) | = | \((-i-2)^{8}\cdot(2i+1)^{8}\cdot(3)^{11}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 4788369691314697265625 \) | = | \(5^{8}\cdot5^{8}\cdot9^{11}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{20480}{243} \) | ||
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 | $\left(-100 : 1012 i : 1\right)$ |
| Height | \(0.12794045225631289494197893161604903247\) |
| 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: | \( 0.12794045225631289494197893161604903247 \) | ||
| Period: | \( 0.21830687274085464277266753163384842235 \) | ||
| Tamagawa product: | \( 36 \) = \(3\cdot3\cdot2^{2}\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.0109801620970927648412025410244519244 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-i-2)\) | \(5\) | \(3\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
| \((2i+1)\) | \(5\) | \(3\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
| \((3)\) | \(9\) | \(4\) | \(I_{5}^{*}\) | Additive | \(1\) | \(2\) | \(11\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B.4.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
50625.3-a
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 225.a2 |
| \(\Q\) | 3600.bk2 |