Elliptic curve 8.1-a2 over number field Q(241)\Q(\sqrt{241})

• Label: 2.2.241.1-8.1-a2
• Base field: $\Q(\sqrt{241})$
• Conductor norm: $8$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $1$
• Rank: $0$

Base field $\Q(\sqrt{241})$

Generator $a$, with minimal polynomial $x^{2} - x - 60$; class number $1$.

Weierstrass equation

${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2879460343066a+23790352869908\right){x}-22072265823414034036a+182362988204191549224$

This is a global minimal model.

Mordell-Weil group structure

trivial

Invariants

Conductor:	$\frak{N}$	=	$(-786a-5708)$	=	$(-393a-2854)^{2}\cdot(393a-3247)$
Conductor norm:	$N(\frak{N})$	=	$8$	=	$2^{2}\cdot2$
Discriminant:	$\Delta$	=	$-64a+512$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-64a+512)$	=	$(-393a-2854)^{8}\cdot(393a-3247)^{6}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$16384$	=	$2^{8}\cdot2^{6}$
j-invariant:	$j$	=	$\frac{615}{64} a + \frac{3917}{16}$
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:	$\Omega(E/K)$	≈	$17.508712912343282093537368966918063611$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$6$  =  $1\cdot( 2 \cdot 3 )$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$6.7670120653414288820198977929875419481$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}6.767012065 \approx L(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 17.508713 \cdot 1 \cdot 6 } { {1^2 \cdot 15.524175} } \\ & \approx 6.767012065 \end{aligned}$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-393a-2854)$	$2$	$1$	$IV^{*}$	Additive	$-1$	$2$	$8$	$0$
$(393a-3247)$	$2$	$6$	$I_{6}$	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
$3$	3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 8.1-a consists of curves linked by isogenies of degree 3.

Base change

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

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