Elliptic curve 86700.1-g2 over number field Q(−3)\Q(\sqrt{-3})

• Label: 2.0.3.1-86700.1-g2
• Base field: $\Q(\sqrt{-3})$
• Conductor norm: $86700$
• CM: no
• Base change: yes
• Q-curve: yes
• Torsion order: $6$
• Rank: $1$

Base field $\Q(\sqrt{-3})$

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

Weierstrass equation

${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(25a-25\right){x}-375$

This is a global minimal model.

Mordell-Weil group structure

$\Z \oplus \Z/{6}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(2 a : 22 a - 11 : 1\right)$	$0.60834903062564688853376260745246899160$	$\infty$
$\left(-40 a : 40 a + 235 : 1\right)$	$0$	$6$

Invariants

Conductor:	$\frak{N}$	=	$(-340a+170)$	=	$(-2a+1)\cdot(2)\cdot(5)\cdot(17)$
Conductor norm:	$N(\frak{N})$	=	$86700$	=	$3\cdot4\cdot25\cdot289$
Discriminant:	$\Delta$	=	$-62424000$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-62424000)$	=	$(-2a+1)^{6}\cdot(2)^{6}\cdot(5)^{3}\cdot(17)^{2}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$3896755776000000$	=	$3^{6}\cdot4^{6}\cdot25^{3}\cdot289^{2}$
j-invariant:	$j$	=	$\frac{1723683599}{62424000}$
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:	$\mathrm{Reg}(E/K)$	≈	$0.60834903062564688853376260745246899160$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1.21669806125129377706752521490493798320$
Global period:	$\Omega(E/K)$	≈	$1.40740452488398312870526605720065948450$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$216$  =  $( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot3\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$6$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$5.9318803444091643748060409033807391042$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}5.931880344 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 1.407405 \cdot 1.216698 \cdot 216 } { {6^2 \cdot 1.732051} } \\ & \approx 5.931880344 \end{aligned}$

Local data at primes of bad reduction

This elliptic curve is semistable. There are 4 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-2a+1)$	$3$	$6$	$I_{6}$	Split multiplicative	$-1$	$1$	$6$	$6$
$(2)$	$4$	$6$	$I_{6}$	Split multiplicative	$-1$	$1$	$6$	$6$
$(5)$	$25$	$3$	$I_{3}$	Split multiplicative	$-1$	$1$	$3$	$3$
$(17)$	$289$	$2$	$I_{2}$	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
$2$	2B
$3$	3B.1.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 86700.1-g consists of curves linked by isogenies of degrees dividing 6.

Base change

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

Base field	Curve
$\Q$	510.g4
$\Q$	1530.c4