Elliptic curve 12.1-c1 over number field Q(337)\Q(\sqrt{337})

• Label: 2.2.337.1-12.1-c1
• Base field: $\Q(\sqrt{337})$
• Conductor norm: $12$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $2$
• Rank: $1$

Base field $\Q(\sqrt{337})$

Generator $a$, with minimal polynomial $x^{2} - x - 84$; class number $1$.

Weierstrass equation

${y}^2+{x}{y}={x}^{3}+\left(7098a+61602\right){x}+230272a+1998480$

This is a global minimal model.

Mordell-Weil group structure

$\Z \oplus \Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(224 a + 1944 : -14532 a - 126120 : 1\right)$	$0.34462549774262994984268183861336367121$	$\infty$
$\left(-\frac{7}{4} a - \frac{61}{4} : \frac{7}{8} a + \frac{61}{8} : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(-56a-486)$	=	$(3a+26)\cdot(3a-29)\cdot(-28a-243)$
Conductor norm:	$N(\frak{N})$	=	$12$	=	$2\cdot2\cdot3$
Discriminant:	$\Delta$	=	$-4002284a-34734948$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-4002284a-34734948)$	=	$(3a+26)^{2}\cdot(3a-29)^{22}\cdot(-28a-243)^{3}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$-452984832$	=	$-2^{2}\cdot2^{22}\cdot3^{3}$
j-invariant:	$j$	=	$-\frac{3072738109}{113246208} a + \frac{53840373619}{28311552}$
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:	$\mathrm{Reg}(E/K)$	≈	$0.34462549774262994984268183861336367121$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$0.689250995485259899685363677226727342420$
Global period:	$\Omega(E/K)$	≈	$7.1285654371801663395914728888792810179$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$132$  =  $2\cdot( 2 \cdot 11 )\cdot3$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$8.8323959934034647463095094084120494900$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 8.832395993 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 7.128565 \cdot 0.689251 \cdot 132 } { {2^2 \cdot 18.357560} } \approx 8.832395993$

Local data at primes of bad reduction

This elliptic curve is semistable. There are 3 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(3a+26)$	$2$	$2$	$I_{2}$	Split multiplicative	$-1$	$1$	$2$	$2$
$(3a-29)$	$2$	$22$	$I_{22}$	Split multiplicative	$-1$	$1$	$22$	$22$
$(-28a-243)$	$3$	$3$	$I_{3}$	Split multiplicative	$-1$	$1$	$3$	$3$

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.
Its isogeny class 12.1-c consists of curves linked by isogenies of degree 2.

Base change

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

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