Elliptic curve 784.1-f4 over number field Q(−22)\Q(\sqrt{-22})

• Label: 2.0.88.1-784.1-f4
• Base field: $\Q(\sqrt{-22})$
• Conductor norm: $784$
• CM: no
• Base change: yes
• Q-curve: yes
• Torsion order: $2$
• Rank: $0$

Base field $\Q(\sqrt{-22})$

Generator $a$, with minimal polynomial $x^{2} + 22$; class number $2$.

Weierstrass equation

${y}^2+a{x}{y}={x}^3-{x}^2-128{x}+832$

This is a global minimal model.

Mordell-Weil group structure

$\Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(\frac{13}{2} : -\frac{13}{4} a : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(28)$	=	$(2,a)^{4}\cdot(7)$
Conductor norm:	$N(\frak{N})$	=	$784$	=	$2^{4}\cdot49$
Discriminant:	$\Delta$	=	$60236288$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(60236288)$	=	$(2,a)^{18}\cdot(7)^{6}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$3628410392018944$	=	$2^{18}\cdot49^{6}$
j-invariant:	$j$	=	$\frac{4956477625}{941192}$
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:	$\Omega(E/K)$	≈	$1.31312570279100809324892748443678470128$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$24$  =  $2^{2}\cdot( 2 \cdot 3 )$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$3.3595120860000879036050131487196610703$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$4$ (rounded)

BSD formula

$\displaystyle 3.359512086 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 4 \cdot 1.313126 \cdot 1 \cdot 24 } { {2^2 \cdot 9.380832} } \approx 3.359512086$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(2,a)$	$2$	$4$	$I_{10}^{*}$	Additive	$1$	$4$	$18$	$6$
$(7)$	$49$	$6$	$I_{6}$	Split multiplicative	$-1$	$1$	$6$	$6$

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 784.1-f consists of curves linked by isogenies of degrees dividing 18.

Base change

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

Base field	Curve
$\Q$	448.a3
$\Q$	13552.w3