Elliptic curve 37632.2-g6 over number field Q(−3)\Q(\sqrt{-3})

• Label: 2.0.3.1-37632.2-g6
• Base field: $\Q(\sqrt{-3})$
• Conductor norm: $37632$
• CM: no
• Base change: yes
• Q-curve: yes
• Torsion order: $2$
• Rank: $1$

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

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

Weierstrass equation

${y}^2={x}^{3}-{x}^{2}-152{x}-672$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-6 : -4 a + 2 : 1\right)$	$0.68815432164241278023671161099664017193$	$\infty$
$\left(-7 : 0 : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(-224a+112)$	=	$(-2a+1)\cdot(2)^{4}\cdot(-3a+1)\cdot(3a-2)$
Conductor norm:	$N(\frak{N})$	=	$37632$	=	$3\cdot4^{4}\cdot7\cdot7$
Discriminant:	$\Delta$	=	$580608$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(580608)$	=	$(-2a+1)^{8}\cdot(2)^{10}\cdot(-3a+1)\cdot(3a-2)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$337105649664$	=	$3^{8}\cdot4^{10}\cdot7\cdot7$
j-invariant:	$j$	=	$\frac{381775972}{567}$
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:	$\mathrm{Reg}(E/K)$	≈	$0.68815432164241278023671161099664017193$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1.37630864328482556047342322199328034386$
Global period:	$\Omega(E/K)$	≈	$1.97235683460469410177459099558257460262$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$8$  =  $2\cdot2^{2}\cdot1\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$3.1345174717115116515199208353111505321$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 3.134517472 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 1.972357 \cdot 1.376309 \cdot 8 } { {2^2 \cdot 1.732051} } \approx 3.134517472$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-2a+1)$	$3$	$2$	$I_{8}$	Non-split multiplicative	$1$	$1$	$8$	$8$
$(2)$	$4$	$4$	$I_{2}^{*}$	Additive	$1$	$4$	$10$	$0$
$(-3a+1)$	$7$	$1$	$I_{1}$	Split multiplicative	$-1$	$1$	$1$	$1$
$(3a-2)$	$7$	$1$	$I_{1}$	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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 37632.2-g consists of curves linked by isogenies of degrees dividing 8.

Base change

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

Base field	Curve
$\Q$	336.c1
$\Q$	1008.e1