Elliptic curve 52650.4-a8 over number field Q(−1)\Q(\sqrt{-1})

• Label: 2.0.4.1-52650.4-a8
• Base field: $\Q(\sqrt{-1})$
• Conductor norm: $52650$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $2$
• Rank: $1$

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

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

Weierstrass equation

${y}^2+i{x}{y}+\left(i+1\right){y}={x}^{3}+{x}^{2}+\left(-5477i-94\right){x}-114521i+107406$

This is a global minimal model.

Mordell-Weil group structure

$\Z \oplus \Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-21 i - 24 : -20 i + 106 : 1\right)$	$0.83600035890869194296645343085430366642$	$\infty$
$\left(-30 i - \frac{123}{4} : \frac{119}{8} i - \frac{31}{2} : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(45i-225)$	=	$(i+1)\cdot(-i-2)\cdot(2i+1)\cdot(3)^{2}\cdot(2i+3)$
Conductor norm:	$N(\frak{N})$	=	$52650$	=	$2\cdot5\cdot5\cdot9^{2}\cdot13$
Discriminant:	$\Delta$	=	$-44286750i+27155250$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-44286750i+27155250)$	=	$(i+1)^{3}\cdot(-i-2)^{8}\cdot(2i+1)^{3}\cdot(3)^{6}\cdot(2i+3)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$2698723828125000$	=	$2^{3}\cdot5^{8}\cdot5^{3}\cdot9^{6}\cdot13$
j-invariant:	$j$	=	$-\frac{4023422266102893}{20312500} i + \frac{5856979210600901}{20312500}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$    (no potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$1$
Mordell-Weil rank:	$r$	=	$1$
Regulator:	$\mathrm{Reg}(E/K)$	≈	$0.83600035890869194296645343085430366642$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1.67200071781738388593290686170860733284$
Global period:	$\Omega(E/K)$	≈	$0.482141548074313006598481621892285673100$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$32$  =  $1\cdot2^{3}\cdot1\cdot2^{2}\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$3.2245640578793441937991055818722362542$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 3.224564058 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 0.482142 \cdot 1.672001 \cdot 32 } { {2^2 \cdot 2.000000} } \approx 3.224564058$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 5 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(i+1)$	$2$	$1$	$I_{3}$	Non-split multiplicative	$1$	$1$	$3$	$3$
$(-i-2)$	$5$	$8$	$I_{8}$	Split multiplicative	$-1$	$1$	$8$	$8$
$(2i+1)$	$5$	$1$	$I_{3}$	Non-split multiplicative	$1$	$1$	$3$	$3$
$(3)$	$9$	$4$	$I_0^{*}$	Additive	$1$	$2$	$6$	$0$
$(2i+3)$	$13$	$1$	$I_{1}$	Split multiplicative	$-1$	$1$	$1$	$1$

Galois Representations

The mod $p$ Galois Representation has maximal image for all primes $p < 1000$ except those listed.

prime	Image of Galois Representation
$2$	2B
$3$	3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3, 4, 6 and 12.
Its isogeny class 52650.4-a consists of curves linked by isogenies of degrees dividing 12.

Base change

This elliptic curve is not a $\Q$-curve.

It is not the base change of an elliptic curve defined over any subfield.