Elliptic curve 8064.7-b4 over number field Q(−7)\Q(\sqrt{-7})

• Label: 2.0.7.1-8064.7-b4
• Base field: $\Q(\sqrt{-7})$
• Conductor norm: $8064$
• CM: no
• Base change: no
• Q-curve: yes
• Torsion order: $2$
• Rank: $0$

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

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

Weierstrass equation

${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1928a+2700\right){x}-19010a+15206$

This is a global minimal model.

Mordell-Weil group structure

$\Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(\frac{9}{4} a - \frac{35}{4} : \frac{17}{8} a + \frac{53}{8} : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(66a-54)$	=	$(a)\cdot(-a+1)^{6}\cdot(-2a+1)\cdot(3)$
Conductor norm:	$N(\frak{N})$	=	$8064$	=	$2\cdot2^{6}\cdot7\cdot9$
Discriminant:	$\Delta$	=	$358579185930a+1927890446706$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(358579185930a+1927890446706)$	=	$(a)\cdot(-a+1)^{19}\cdot(-2a+1)^{4}\cdot(3)^{16}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$4665221026606764776226816$	=	$2\cdot2^{19}\cdot7^{4}\cdot9^{16}$
j-invariant:	$j$	=	$\frac{6359387729183}{4218578658}$
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:	$\Omega(E/K)$	≈	$0.242216540448250546346412529568477292240$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$16$  =  $1\cdot2\cdot2^{2}\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$1.4647879530342590172098244799766989577$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$4$ (rounded)

BSD formula

$\displaystyle 1.464787953 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 4 \cdot 0.242217 \cdot 1 \cdot 16 } { {2^2 \cdot 2.645751} } \approx 1.464787953$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(a)$	$2$	$1$	$I_{1}$	Non-split multiplicative	$1$	$1$	$1$	$1$
$(-a+1)$	$2$	$2$	$I_{9}^{*}$	Additive	$1$	$6$	$19$	$1$
$(-2a+1)$	$7$	$4$	$I_{4}$	Split multiplicative	$-1$	$1$	$4$	$4$
$(3)$	$9$	$2$	$I_{16}$	Non-split multiplicative	$1$	$1$	$16$	$16$

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, 8 and 16.
Its isogeny class 8064.7-b 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.