Properties

Label 2.0.4.1-50625.3-c1
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 50625 \)
CM yes (\(-3\))
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{-1}) \)

Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))
 
Copy content gp:K = nfinit(Polrev([1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\({y}^2+i{y}={x}^{3}+34\)
Copy content comment:Define the curve
 
Copy content sage:E = EllipticCurve([K([0,0]),K([0,0]),K([0,1]),K([0,0]),K([34,0])])
 
Copy content gp:E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,1]),Polrev([0,0]),Polrev([34,0])], K);
 
Copy content magma:E := EllipticCurve([K![0,0],K![0,0],K![0,1],K![0,0],K![34,0]]);
 

This is a global minimal model.

Copy content comment:Test whether it is a global minimal model
 
Copy content sage:E.is_global_minimal_model()
 

Mordell-Weil group structure

\(\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(-6 : -14 i : 1\right)$$0.46092010503988343429723097198891843500$$\infty$

Invariants

Conductor: $\frak{N}$ = \((225)\) = \((-i-2)^{2}\cdot(2i+1)^{2}\cdot(3)^{2}\)
Copy content comment:Compute the conductor
 
Copy content sage:E.conductor()
 
Copy content gp:ellglobalred(E)[1]
 
Copy content magma:Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 50625 \) = \(5^{2}\cdot5^{2}\cdot9^{2}\)
Copy content comment:Compute the norm of the conductor
 
Copy content sage:E.conductor().norm()
 
Copy content gp:idealnorm(ellglobalred(E)[1])
 
Copy content magma:Norm(Conductor(E));
 
Discriminant: $\Delta$ = $-492075$
Discriminant ideal: $\frak{D}_{\mathrm{min}} = (\Delta)$ = \((-492075)\) = \((-i-2)^{2}\cdot(2i+1)^{2}\cdot(3)^{9}\)
Copy content comment:Compute the discriminant
 
Copy content sage:E.discriminant()
 
Copy content gp:E.disc
 
Copy content magma:Discriminant(E);
 
Discriminant norm: $N(\frak{D}_{\mathrm{min}}) = N(\Delta)$ = \( 242137805625 \) = \(5^{2}\cdot5^{2}\cdot9^{9}\)
Copy content comment:Compute the norm of the discriminant
 
Copy content sage:E.discriminant().norm()
 
Copy content gp:norm(E.disc)
 
Copy content magma:Norm(Discriminant(E));
 
j-invariant: $j$ = \( 0 \)
Copy content comment:Compute the j-invariant
 
Copy content sage:E.j_invariant()
 
Copy content gp:E.j
 
Copy content magma:jInvariant(E);
 
Endomorphism ring: $\mathrm{End}(E)$ = \(\Z\)   
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$ = \(\Z[(1+\sqrt{-3})/2]\)    (potential complex multiplication)
Copy content comment:Test for Complex Multiplication
 
Copy content sage:E.has_cm(), E.cm_discriminant()
 
Copy content magma:HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $N(\mathrm{U}(1))$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$= \( 1 \)
Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 
Mordell-Weil rank: $r$ = \(1\)
Regulator: $\mathrm{Reg}(E/K)$ \( 0.46092010503988343429723097198891843500 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ \( 0.921840210079766868594461943977836870000 \)
Global period: $\Omega(E/K)$ \( 3.1613030502258025158853715196151958922 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 2 \)  =  \(1\cdot1\cdot2\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(1\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 2.9142162679459615836569325337890834657 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$$\begin{aligned}2.914216268 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# ะจ(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.161303 \cdot 0.921840 \cdot 2 } { {1^2 \cdot 2.000000} } \\ & \approx 2.914216268 \end{aligned}$$

Local data at primes of bad reduction

Copy content comment:Compute the local reduction data at primes of bad reduction
 
Copy content sage:E.local_data()
 
Copy content magma:LocalInformation(E);
 

This elliptic curve is not semistable. There are 3 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((-i-2)\) \(5\) \(1\) \(II\) Additive \(1\) \(2\) \(2\) \(0\)
\((2i+1)\) \(5\) \(1\) \(II\) Additive \(1\) \(2\) \(2\) \(0\)
\((3)\) \(9\) \(2\) \(III^{*}\) Additive \(1\) \(2\) \(9\) \(0\)

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

For all other primes \(p\), the image is a Borel subgroup if \(p=3\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 50625.3-c consists of curves linked by isogenies of degree 3.

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.c1
\(\Q\) 3600.br2