Properties

Label 2.0.4.1-40000.3-CMc2
Base field \(\Q(\sqrt{-1}) \)
Conductor \((200)\)
Conductor norm \( 40000 \)
CM yes (\(-16\))
Base change yes: 800.d1,800.d2
Q-curve yes
Torsion order \( 2 \)
Rank \( 2 \)

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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(Pol(Vecrev([1, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+68{x}+253i\)
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([0,0]),K([68,0]),K([0,253])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([68,0])),Pol(Vecrev([0,253]))], K);
 
magma: E := EllipticCurve([K![1,1],K![0,1],K![0,0],K![68,0],K![0,253]]);
 

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: \((-125000)\) = \((i+1)^{6}\cdot(-i-2)^{6}\cdot(2i+1)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 15625000000 \) = \(2^{6}\cdot5^{6}\cdot5^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 287496 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z[\sqrt{-4}]\) (complex multiplication)
Geometric endomorphism ring: \(\Z[\sqrt{-4}]\)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{U}(1)$

Mordell-Weil group

Rank: \(2\)
Generators $\left(-7 i - 2 : 6 i - 13 : 1\right)$ $\left(-\frac{9}{2} i - \frac{3}{4} : \frac{53}{8} i - \frac{7}{4} : 1\right)$
Heights \(0.949741086265898\) \(1.89948217253180\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{11}{2} i : \frac{11}{4} i - \frac{11}{4} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 0.902008130941528 \)
Period: \( 1.37503716360407 \)
Tamagawa product: \( 4 \)  =  \(1\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 4.96117880767060 \)
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 \(-1\) \(6\) \(6\) \(0\)
\((-i-2)\) \(5\) \(2\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)
\((2i+1)\) \(5\) \(2\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

The image is a Borel subgroup if \(p=2\), a split Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies (excluding endomorphisms) of degree \(d\) for \(d=\) 2.
Its isogeny class 40000.3-CMc consists of curves linked by isogenies of degree 2.

Base change

This curve is the base change of elliptic curves 800.d1, 800.d2, defined over \(\Q\), so it is also a \(\Q\)-curve.