Elliptic curve 45000.3-o2 over number field Q(−1)\Q(\sqrt{-1})

• Label: 2.0.4.1-45000.3-o2
• Base field: $\Q(\sqrt{-1})$
• Conductor norm: $45000$
• CM: no
• Base change: yes
• Q-curve: yes
• Torsion order: $4$
• Rank: $0$

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

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

Weierstrass equation

${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}-i{x}^{2}+\left(-i+102\right){x}+148i$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-9 i : 4 i - 5 : 1\right)$	$0$	$2$
$\left(-\frac{3}{2} i : \frac{1}{4} i - \frac{5}{4} : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(150i+150)$	=	$(i+1)^{3}\cdot(-i-2)^{2}\cdot(2i+1)^{2}\cdot(3)$
Conductor norm:	$N(\frak{N})$	=	$45000$	=	$2^{3}\cdot5^{2}\cdot5^{2}\cdot9$
Discriminant:	$\Delta$	=	$-56250000$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-56250000)$	=	$(i+1)^{8}\cdot(-i-2)^{8}\cdot(2i+1)^{8}\cdot(3)^{2}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$3164062500000000$	=	$2^{8}\cdot5^{8}\cdot5^{8}\cdot9^{2}$
j-invariant:	$j$	=	$\frac{470596}{225}$
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:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/K)$	=	$1$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	=	$1$
Global period:	$\Omega(E/K)$	≈	$1.39263475429778604363833564089450576300$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$64$  =  $2\cdot2^{2}\cdot2^{2}\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$4$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$2.7852695085955720872766712817890115260$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}2.785269509 \approx L(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.392635 \cdot 1 \cdot 64 } { {4^2 \cdot 2.000000} } \\ & \approx 2.785269509 \end{aligned}$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 primes $\frak{p}$ of bad reduction.

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

Galois Representations

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

prime	Image of Galois Representation
$2$	2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 45000.3-o consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a $\Q$-curve. It is the base change of the following 2 elliptic curves:

Base field	Curve
$\Q$	600.a3
$\Q$	1200.r3