Elliptic curve 49.2-a2 over number field 3.3.361.1

• Label: 3.3.361.1-49.2-a2
• Base field: 3.3.361.1
• Conductor norm: $49$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $2$
• Rank: $0$

Base field 3.3.361.1

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

Weierstrass equation

${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-6a^{2}-4a+29\right){x}+18a^{2}-6a-128$

This is a global minimal model.

Mordell-Weil group structure

$\Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-2 a^{2} - \frac{5}{4} a + 9 : \frac{5}{4} a^{2} + \frac{15}{8} a - \frac{27}{8} : 1\right)$	$0$	$2$

Invariants

Conductor:	$\frak{N}$	=	$(-2a^2-a+7)$	=	$(-a)\cdot(a^2+a-5)$
Conductor norm:	$N(\frak{N})$	=	$49$	=	$7\cdot7$
Discriminant:	$\Delta$	=	$22a^2-3a-70$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(22a^2-3a-70)$	=	$(-a)^{4}\cdot(a^2+a-5)^{2}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$-117649$	=	$-7^{4}\cdot7^{2}$
j-invariant:	$j$	=	$\frac{386002347110}{2401} a^{2} + \frac{496019123941}{2401} a - \frac{1182368108178}{2401}$
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:	$\Omega(E/K)$	≈	$32.407923948165358780876303276271972295$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$4$  =  $2\cdot2$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$2$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$1.7056802077981767779408580671722090682$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 1.705680208 \approx L(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 32.407924 \cdot 1 \cdot 4 } { {2^2 \cdot 19.000000} } \approx 1.705680208$

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	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(-a)$	$7$	$2$	$I_{4}$	Non-split multiplicative	$1$	$1$	$4$	$4$
$(a^2+a-5)$	$7$	$2$	$I_{2}$	Split multiplicative	$-1$	$1$	$2$	$2$

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 49.2-a 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.