Elliptic curve 11.1-a1 over number field 3.3.361.1

• Label: 3.3.361.1-11.1-a1
• Base field: 3.3.361.1
• Conductor norm: $11$
• CM: no
• Base change: no
• Q-curve: no
• Torsion order: $3$
• 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

${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{2}+a-5\right){x}^{2}+\left(5a^{2}+3a-24\right){x}-6a^{2}+a+40$

This is a global minimal model.

Mordell-Weil group structure

$\Z \oplus \Z/{3}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$\left(-2 a^{2} - a + 11 : -3 a^{2} + 18 : 1\right)$	$0.53308313887267125662016467355710358125$	$\infty$
$\left(-a^{2} - a + 5 : a^{2} + a - 4 : 1\right)$	$0$	$3$

Invariants

Conductor:	$\frak{N}$	=	$(a+1)$	=	$(a+1)$
Conductor norm:	$N(\frak{N})$	=	$11$	=	$11$
Discriminant:	$\Delta$	=	$-5a^2+16$
Discriminant ideal:	$\frak{D}_{\mathrm{min}} = (\Delta)$	=	$(-5a^2+16)$	=	$(a+1)^{3}$
Discriminant norm:	$N(\frak{D}_{\mathrm{min}}) = N(\Delta)$	=	$-1331$	=	$-11^{3}$
j-invariant:	$j$	=	$\frac{15157070}{1331} a^{2} + \frac{3336499}{1331} a - \frac{87193995}{1331}$
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:	$\mathrm{Reg}(E/K)$	≈	$0.53308313887267125662016467355710358125$
Néron-Tate Regulator:	$\mathrm{Reg}_{\mathrm{NT}}(E/K)$	≈	$1.59924941661801376986049402067131074375$
Global period:	$\Omega(E/K)$	≈	$135.12600381213910995001336254578457720$
Tamagawa product:	$\prod_{\frak{p}}c_{\frak{p}}$	=	$1$
Torsion order:	$\#E(K)_{\mathrm{tor}}$	=	$3$
Special value:	$L^{(r)}(E/K,1)/r!$	≈	$1.2637437588683448907628200985591220104$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$ (rounded)

BSD formula

$\displaystyle 1.263743759 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 135.126004 \cdot 1.599249 \cdot 1 } { {3^2 \cdot 19.000000} } \approx 1.263743759$

Local data at primes of bad reduction

This elliptic curve is semistable. There is only one prime $\frak{p}$ of bad reduction.

$\mathfrak{p}$	$N(\mathfrak{p})$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}$)	$\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))$
$(a+1)$	$11$	$1$	$I_{3}$	Non-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
$3$	3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 11.1-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.