Elliptic curve 18000.2-g2 over number field Q(−1)\Q(\sqrt{-1})

• Label: 2.0.4.1-18000.2-g2
• Base field: $\Q(\sqrt{-1})$
• Conductor norm: $18000$
• CM: no
• Base change: no
• Q-curve: yes
• Torsion order: $4$
• Rank: $1$

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

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

Weierstrass equation

${y}^2+\left(i+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-16i-12\right){x}-22i-4$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(4 i - 7 : 15 i + 25 : 1\right)$	$0.56105509073466330829093502803492620949$	$\infty$
$\left(-2 i - 4 : 3 i + 1 : 1\right)$	$0$	$2$
$\left(-\frac{1}{2} i - 1 : \frac{3}{4} i + \frac{1}{4} : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(-60i-120)$	=	$(i+1)^{4}\cdot(-i-2)^{2}\cdot(2i+1)\cdot(3)$
Conductor norm:	$N(\frak{N})$	=	$18000$	=	$2^{4}\cdot5^{2}\cdot5\cdot9$
Discriminant:	$\Delta$	=	$158400i-421200$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(158400i-421200)$	=	$(i+1)^{8}\cdot(-i-2)^{8}\cdot(2i+1)^{2}\cdot(3)^{2}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$202500000000$	=	$2^{8}\cdot5^{8}\cdot5^{2}\cdot9^{2}$
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:	$r_{\mathrm{an}}$	=	$1$
Mordell-Weil rank:	$r$	=	$1$
Regulator:	$\mathrm{Reg}(E/K)$	≈	$0.56105509073466330829093502803492620949$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1.12211018146932661658187005606985241898$
Global period:	$\Omega(E/K)$	≈	$3.1140259784385669952365615480237270560$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$32$  =  $2\cdot2^{2}\cdot2\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$4$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$3.4942802557658977848141442989112341274$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}3.494280256 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.114026 \cdot 1.122110 \cdot 32 } { {4^2 \cdot 2.000000} } \\ & \approx 3.494280256 \end{aligned}$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(i+1)$	$2$	$2$	$I_0^{*}$	Additive	$-1$	$4$	$8$	$0$
$(-i-2)$	$5$	$4$	$I_{2}^{*}$	Additive	$1$	$2$	$8$	$2$
$(2i+1)$	$5$	$2$	$I_{2}$	Split multiplicative	$-1$	$1$	$2$	$2$
$(3)$	$9$	$2$	$I_{2}$	Non-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 18000.2-g consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a $\Q$-curve.

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