Elliptic curve 45056.14-y1 over number field Q(−7)\Q(\sqrt{-7})

• Label: 2.0.7.1-45056.14-y1
• Base field: $\Q(\sqrt{-7})$
• Conductor norm: $45056$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $2$
• Rank: $0$

Base field $\Q(\sqrt{-7})$

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

Weierstrass equation

${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-27a+61\right){x}-73a-77$

This is a global minimal model.

Mordell-Weil group structure

$\Z/{2}\Z$

Mordell-Weil generators

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

Invariants

Conductor:	$\frak{N}$	=	$(128a+64)$	=	$(a)^{6}\cdot(-a+1)^{6}\cdot(2a+1)$
Conductor norm:	$N(\frak{N})$	=	$45056$	=	$2^{6}\cdot2^{6}\cdot11$
Discriminant:	$\Delta$	=	$-345088a-54272$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-345088a-54272)$	=	$(a)^{10}\cdot(-a+1)^{21}\cdot(2a+1)^{2}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$259845521408$	=	$2^{10}\cdot2^{21}\cdot11^{2}$
j-invariant:	$j$	=	$-\frac{1374871}{968} a - \frac{22437003}{484}$
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:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/K)$	=	$1$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	=	$1$
Global period:	$\Omega(E/K)$	≈	$2.5147445761182747099021912061892684456$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$16$  =  $2\cdot2^{2}\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$3.8019364338614248239509039054870860288$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}3.801936434 \approx L(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 2.514745 \cdot 1 \cdot 16 } { {2^2 \cdot 2.645751} } \\ & \approx 3.801936434 \end{aligned}$

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	$1$	$6$	$10$	$0$
$(-a+1)$	$2$	$4$	$I_{11}^{*}$	Additive	$1$	$6$	$21$	$3$
$(2a+1)$	$11$	$2$	$I_{2}$	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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 45056.14-y consists of curves linked by isogenies of degrees dividing 6.

Base change

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

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