Elliptic curve 784.3-a1 over number field Q(−7)\Q(\sqrt{-7})

• Label: 2.0.7.1-784.3-a1
• Base field: $\Q(\sqrt{-7})$
• Conductor norm: $784$
• CM: no
• Base change: yes
• Q-curve: yes
• Torsion order: $3$
• Rank: $1$

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

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

Weierstrass equation

${y}^2={x}^{3}-{x}^{2}-2{x}+1$

This is a global minimal model.

Mordell-Weil group structure

$\Z \oplus \Z/{3}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-a + 2 : 2 a - 1 : 1\right)$	$0.086035450966578930195721660820160569546$	$\infty$
$\left(-2 : -2 a + 1 : 1\right)$	$0$	$3$

Invariants

Conductor:	$\frak{N}$	=	$(28)$	=	$(a)^{2}\cdot(-a+1)^{2}\cdot(-2a+1)^{2}$
Conductor norm:	$N(\frak{N})$	=	$784$	=	$2^{2}\cdot2^{2}\cdot7^{2}$
Discriminant:	$\Delta$	=	$784$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(784)$	=	$(a)^{4}\cdot(-a+1)^{4}\cdot(-2a+1)^{4}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$614656$	=	$2^{4}\cdot2^{4}\cdot7^{4}$
j-invariant:	$j$	=	$1792$
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:	$\mathrm{Reg}(E/K)$	≈	$0.086035450966578930195721660820160569546$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$0.1720709019331578603914433216403211390920$
Global period:	$\Omega(E/K)$	≈	$9.0415355856050104008888145051174614866$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$27$  =  $3\cdot3\cdot3$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$3$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$1.7640945805104228875420790787208623829$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}1.764094581 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 9.041536 \cdot 0.172071 \cdot 27 } { {3^2 \cdot 2.645751} } \\ & \approx 1.764094581 \end{aligned}$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(a)$	$2$	$3$	$IV$	Additive	$-1$	$2$	$4$	$0$
$(-a+1)$	$2$	$3$	$IV$	Additive	$-1$	$2$	$4$	$0$
$(-2a+1)$	$7$	$3$	$IV$	Additive	$1$	$2$	$4$	$0$

Galois Representations

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

prime	Image of Galois Representation
$2$	2Cn
$3$	3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 784.3-a consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is a $\Q$-curve. It is the base change of the following 2 elliptic curves:

Base field	Curve
$\Q$	196.b2
$\Q$	196.a2