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([1,1]),K([0,-1]),K([1,1]),K([52,-1]),K([0,-177])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([1,1]),Polrev([52,-1]),Polrev([0,-177])], K);
magma: E := EllipticCurve([K![1,1],K![0,-1],K![1,1],K![52,-1],K![0,-177]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((200)\) | = | \((i+1)^{6}\cdot(-i-2)^{2}\cdot(2i+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 40000 \) | = | \(2^{6}\cdot5^{2}\cdot5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((3125000)\) | = | \((i+1)^{6}\cdot(-i-2)^{8}\cdot(2i+1)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 9765625000000 \) | = | \(2^{6}\cdot5^{8}\cdot5^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -5000 \) | ||
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 | $\left(-4 i : 14 i - 15 : 1\right)$ |
| Height | \(0.19637779133168667685371410477367146271\) |
| 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: | \( 0.19637779133168667685371410477367146271 \) | ||
| Period: | \( 1.0639399808954382081491386800466204358 \) | ||
| Tamagawa product: | \( 9 \) = \(1\cdot3\cdot3\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.7608153040390153307100844263779010253 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((i+1)\) | \(2\) | \(1\) | \(II\) | Additive | \(1\) | \(6\) | \(6\) | \(0\) |
| \((-i-2)\) | \(5\) | \(3\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
| \((2i+1)\) | \(5\) | \(3\) | \(IV^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5Ns.2.1 |
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 40000.3-b consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 800.h1 |
| \(\Q\) | 800.b1 |