Elliptic curve 27648.2-t2 over number field Q(−2)\Q(\sqrt{-2})

• Label: 2.0.8.1-27648.2-t2
• Base field: $\Q(\sqrt{-2})$
• Conductor norm: $27648$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $2$
• Rank: $1$

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

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

Weierstrass equation

${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(2a-10\right){x}+8a-2$

This is a global minimal model.

Mordell-Weil group structure

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

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-2 : a + 4 : 1\right)$	$0.86236082633705213641333750053892097940$	$\infty$
$\left(-a + 3 : 0 : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(-96a-96)$	=	$(a)^{10}\cdot(-a-1)^{2}\cdot(a-1)$
Conductor norm:	$N(\frak{N})$	=	$27648$	=	$2^{10}\cdot3^{2}\cdot3$
Discriminant:	$\Delta$	=	$-44928a+148608$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-44928a+148608)$	=	$(a)^{14}\cdot(-a-1)^{3}\cdot(a-1)^{10}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$26121388032$	=	$2^{14}\cdot3^{3}\cdot3^{10}$
j-invariant:	$j$	=	$-\frac{4722784}{59049} a + \frac{38018528}{59049}$
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:	$\mathrm{Reg}(E/K)$	≈	$0.86236082633705213641333750053892097940$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1.72472165267410427282667500107784195880$
Global period:	$\Omega(E/K)$	≈	$3.7565270915393453828782249433865001800$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$8$  =  $2\cdot2\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$4.5813191062620558291783576925218088759$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}4.581319106 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 3.756527 \cdot 1.724722 \cdot 8 } { {2^2 \cdot 2.828427} } \\ & \approx 4.581319106 \end{aligned}$

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 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	$1$	$10$	$14$	$0$
$(-a-1)$	$3$	$2$	$III$	Additive	$1$	$2$	$3$	$0$
$(a-1)$	$3$	$2$	$I_{10}$	Non-split multiplicative	$1$	$1$	$10$	$10$

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.
Its isogeny class 27648.2-t consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is not a $\Q$-curve.

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