Elliptic curve 49.2-b4 over number field 3.3.361.1

• Label: 3.3.361.1-49.2-b4
• Base field: 3.3.361.1
• Conductor norm: $49$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $4$
• Rank: $1$

Base field 3.3.361.1

Generator $a$, with minimal polynomial $x^{3} - x^{2} - 6 x + 7$; class number $1$.

Weierstrass equation

${y}^2+a{x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-4a+4\right){x}-a^{2}-3a+7$

This is a global minimal model.

Mordell-Weil group structure

$\Z \oplus \Z/{4}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(\frac{4}{121} a^{2} + \frac{17}{121} a - \frac{86}{121} : -\frac{1266}{1331} a^{2} - \frac{117}{1331} a + \frac{4229}{1331} : 1\right)$	$1.5670980788414340118090143703354116309$	$\infty$
$\left(0 : -a : 1\right)$	$0$	$4$

Invariants

Conductor:	$\frak{N}$	=	$(-2a^2-a+7)$	=	$(-a)\cdot(a^2+a-5)$
Conductor norm:	$N(\frak{N})$	=	$49$	=	$7\cdot7$
Discriminant:	$\Delta$	=	$-2a^2-a+7$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-2a^2-a+7)$	=	$(-a)\cdot(a^2+a-5)$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$49$	=	$7\cdot7$
j-invariant:	$j$	=	$-\frac{4733}{7} a^{2} - 42 a + \frac{24063}{7}$
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:	$\mathrm{Reg}(E/K)$	≈	$1.5670980788414340118090143703354116309$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$4.7012942365243020354270431110062348927$
Global period:	$\Omega(E/K)$	≈	$129.84057265560590393929750920154488319$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$1$  =  $1\cdot1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$4$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$2.0079563680684701506636056762284511218$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\begin{aligned}2.007956368 \approx L'(E/K,1) & \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \\ & \approx \frac{ 1 \cdot 129.840573 \cdot 4.701294 \cdot 1 } { {4^2 \cdot 19.000000} } \\ & \approx 2.007956368 \end{aligned}$

Local data at primes of bad reduction

This elliptic curve is semistable. There are 2 primes $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-a)$	$7$	$1$	$I_{1}$	Non-split multiplicative	$1$	$1$	$1$	$1$
$(a^2+a-5)$	$7$	$1$	$I_{1}$	Non-split multiplicative	$1$	$1$	$1$	$1$

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 and 4.
Its isogeny class 49.2-b consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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