Elliptic curve 1568.2-i1 over number field Q(2)\Q(\sqrt{2})

• Label: 2.2.8.1-1568.2-i1
• Base field: $\Q(\sqrt{2})$
• Conductor norm: $1568$
• CM: yes ($-64$)
• Base change: no
• Q-curve: yes
• Torsion order: $2$
• Rank: $0$

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

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

Weierstrass equation

${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-55a-97\right){x}+1397a+1934$

This is a global minimal model.

Mordell-Weil group structure

$\Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-6 a - \frac{23}{2} : \frac{21}{4} a + 6 : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(4a+40)$	=	$(a)^{5}\cdot(-2a+1)^{2}$
Conductor norm:	$N(\frak{N})$	=	$1568$	=	$2^{5}\cdot7^{2}$
Discriminant:	$\Delta$	=	$-2416a-8480$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-2416a-8480)$	=	$(a)^{9}\cdot(-2a+1)^{6}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$-60236288$	=	$-2^{9}\cdot7^{6}$
j-invariant:	$j$	=	$-29071392966 a + 41113158120$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z[\sqrt{-16}]$    (potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$N(\mathrm{U}(1))$

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:	$\Omega(E/K)$	≈	$2.5985759845485830206907096824875924345$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$2$  =  $1\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$1.8374707001028121947568890733149830343$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$4$ (rounded)

BSD formula

$\begin{aligned}1.837470700 \approx L(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 4 \cdot 2.598576 \cdot 1 \cdot 2 } { {2^2 \cdot 2.828427} } \\ & \approx 1.837470700 \end{aligned}$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(a)$	$2$	$1$	$I_0^{*}$	Additive	$-1$	$5$	$9$	$0$
$(-2a+1)$	$7$	$2$	$I_0^{*}$	Additive	$-1$	$2$	$6$	$0$

Galois Representations

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

The image is a Borel subgroup if $p=2$, the normalizer of a split Cartan subgroup if $\left(\frac{ -1 }{p}\right)=+1$ or the normalizer of a nonsplit Cartan subgroup if $\left(\frac{ -1 }{p}\right)=-1$.

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4, 8 and 16.
Its isogeny class 1568.2-i consists of curves linked by isogenies of degrees dividing 16.

Base change

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

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