Elliptic curve with Cremona label 1309b1 (LMFDB label 1309.a1)

• Label: 1309b1
• Conductor: $1309$
• Discriminant: $-155771$
• j-invariant: $-\frac{1231925248}{155771}$
• CM: no
• Rank: $2$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2+y=x^3-x^2-22x+52$

(, )

Mordell-Weil group structure

$\Z \oplus \Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(-1, 8)$	$0.34930091685630860491820314459$	$\infty$
$(2, 3)$	$0.37810932623703224943979180788$	$\infty$

Integral points

$\left(-5, 3\right)$, $\left(-5, -4\right)$, $\left(-1, 8\right)$, $\left(-1, -9\right)$, $\left(1, 5\right)$, $\left(1, -6\right)$, $\left(2, 3\right)$, $\left(2, -4\right)$, $\left(4, 3\right)$, $\left(4, -4\right)$, $\left(5, 6\right)$, $\left(5, -7\right)$, $\left(16, 59\right)$, $\left(16, -60\right)$, $\left(50, 348\right)$, $\left(50, -349\right)$, $\left(79, 696\right)$, $\left(79, -697\right)$, $\left(92, 876\right)$, $\left(92, -877\right)$, $\left(14471, 1740735\right)$, $\left(14471, -1740736\right)$

Invariants

Conductor:	$N$	=	$1309$	=	$7 \cdot 11 \cdot 17$
Discriminant:	$\Delta$	=	$-155771$	=	$-1 \cdot 7^{2} \cdot 11 \cdot 17^{2}$
j-invariant:	$j$	=	$-\frac{1231925248}{155771}$	=	$-1 \cdot 2^{12} \cdot 7^{-2} \cdot 11^{-1} \cdot 17^{-2} \cdot 67^{3}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$    (no potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$-0.26967502590559402584263931121$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.26967502590559402584263931121$
$abc$ quality:	$Q$	≈	$0.7835434205577116$
Szpiro ratio:	$\sigma_{m}$	≈	$2.944044464329521$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$2$
Mordell-Weil rank:	$r$	=	$2$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$0.10619668700399062132334130963$
Real period:	$\Omega$	≈	$3.1455350713332992722247507180$
Tamagawa product:	$\prod_{p}c_p$	=	$4$  = $2\cdot1\cdot2$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(2)}(E,1)/2!$	≈	$1.3361816137218307792750409487$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\begin{aligned} 1.336181614 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 3.145535 \cdot 0.106197 \cdot 4}{1^2} \\ & \approx 1.336181614\end{aligned}$

Modular invariants

Modular form   1309.2.a.a

$q - 2 q^{2} - q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} - q^{7} - 2 q^{9} + 6 q^{10} - q^{11} - 2 q^{12} - 6 q^{13} + 2 q^{14} + 3 q^{15} - 4 q^{16} + q^{17} + 4 q^{18} - 4 q^{19} + O(q^{20})$

For more coefficients, see the Downloads section to the right.

Modular degree:	256
$\Gamma_0(N)$-optimal:	yes
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is semistable. There are 3 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$7$	$2$	$I_{2}$	nonsplit multiplicative	1	1	2	2
$11$	$1$	$I_{1}$	nonsplit multiplicative	1	1	1	1
$17$	$2$	$I_{2}$	split multiplicative	-1	1	2	2

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$.

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 22.2.0.a.1, level $22 = 2 \cdot 11$, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 1 \\ 21 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 2 \\ 20 & 3 \end{array}\right),\left(\begin{array}{rr} 13 & 2 \\ 13 & 3 \end{array}\right)$.

The torsion field $K:=\Q(E[22])$ is a degree-$39600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$	Reduction type	Serre weight	Serre conductor
$2$	good	$2$	$11$
$7$	nonsplit multiplicative	$8$	$187 = 11 \cdot 17$
$11$	nonsplit multiplicative	$12$	$119 = 7 \cdot 17$
$17$	split multiplicative	$18$	$77 = 7 \cdot 11$

Isogenies

This curve has no rational isogenies. Its isogeny class 1309b consists of this curve only.

Twists

This elliptic curve is its own minimal quadratic twist.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.44.1	$\Z/2\Z$	not in database
$6$	6.0.21296.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$8$	deg 8	$\Z/3\Z$	not in database
$12$	deg 12	$\Z/4\Z$	not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	ss	ord	ord	nonsplit	nonsplit	ord	split	ord	ord	ord	ord	ord	ord	ord	ord
$\lambda$-invariant(s)	6,7	2	6	2	2	2	3	2	2	2	2	2	2	2	2
$\mu$-invariant(s)	0,0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

$p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.