Elliptic curve 7056.2-e4 over number field Q(−2)\Q(\sqrt{-2})

• Label: 2.0.8.1-7056.2-e4
• Base field: $\Q(\sqrt{-2})$
• Conductor norm: $7056$
• CM: no
• Base change: yes
• Q-curve: yes
• Torsion order: $2$
• Rank: $1$

Base field $\Q(\sqrt{-2})$

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

Weierstrass equation

${y}^2+a{x}{y}+a{y}={x}^{3}-1007{x}+12488$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(\frac{75}{4} : -\frac{79}{8} a - \frac{15}{8} : 1\right)$	$1.1608336823679598003488745302960338324$	$\infty$
$\left(\frac{37}{2} : -\frac{39}{4} a : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(84)$	=	$(a)^{4}\cdot(-a-1)\cdot(a-1)\cdot(7)$
Conductor norm:	$N(\frak{N})$	=	$7056$	=	$2^{4}\cdot3\cdot3\cdot49$
Discriminant:	$\Delta$	=	$3024$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(3024)$	=	$(a)^{8}\cdot(-a-1)^{3}\cdot(a-1)^{3}\cdot(7)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$9144576$	=	$2^{8}\cdot3^{3}\cdot3^{3}\cdot49$
j-invariant:	$j$	=	$\frac{7080974546692}{189}$
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:	$\mathrm{Reg}(E/K)$	≈	$1.1608336823679598003488745302960338324$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$2.3216673647359196006977490605920676648$
Global period:	$\Omega(E/K)$	≈	$1.39085186580099922122501587665379307130$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$18$  =  $2\cdot3\cdot3\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$5.1374593002317605989361706706006737096$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 5.137459300 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 1.390852 \cdot 2.321667 \cdot 18 } { {2^2 \cdot 2.828427} } \approx 5.137459300$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(a)$	$2$	$2$	$I_0^{*}$	Additive	$-1$	$4$	$8$	$0$
$(-a-1)$	$3$	$3$	$I_{3}$	Split multiplicative	$-1$	$1$	$3$	$3$
$(a-1)$	$3$	$3$	$I_{3}$	Split multiplicative	$-1$	$1$	$3$	$3$
$(7)$	$49$	$1$	$I_{1}$	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 and 4.
Its isogeny class 7056.2-e 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
$\Q$	336.e1
$\Q$	1344.m1