Elliptic curve with LMFDB label 26.b2 (Cremona label 26b1)

• Label: 26.b2
• Conductor: $26$
• Discriminant: $-1664$
• j-invariant: $-\frac{2146689}{1664}$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{7}\Z$

Minimal Weierstrass equation

$y^2+xy+y=x^3-x^2-3x+3$

(, )

Mordell-Weil group structure

$\Z/{7}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(1, 0)$	$0$	$7$

Integral points

$\left(-1, 2\right)$, $\left(-1, -2\right)$, $\left(1, 0\right)$, $\left(1, -2\right)$, $\left(3, 2\right)$, $\left(3, -6\right)$

Invariants

Conductor:	$N$	=	$26$	=	$2 \cdot 13$
Discriminant:	$\Delta$	=	$-1664$	=	$-1 \cdot 2^{7} \cdot 13$
j-invariant:	$j$	=	$-\frac{2146689}{1664}$	=	$-1 \cdot 2^{-7} \cdot 3^{3} \cdot 13^{-1} \cdot 43^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$-0.68316054842427208735666930170$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.68316054842427208735666930170$
$abc$ quality:	$Q$	≈	$0.96783604338842$
Szpiro ratio:	$\sigma_{m}$	≈	$4.7356960628832425$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$4.3467574468433882646310387089$
Tamagawa product:	$\prod_{p}c_p$	=	$7$  = $7\cdot1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$7$
Special value:	$L(E,1)$	≈	$0.62096534954905546637586267270$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\begin{aligned} 0.620965350 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 4.346757 \cdot 1.000000 \cdot 7}{7^2} \\ & \approx 0.620965350\end{aligned}$

Modular invariants

Modular form   26.2.a.b

$q + q^{2} - 3 q^{3} + q^{4} - q^{5} - 3 q^{6} + q^{7} + q^{8} + 6 q^{9} - q^{10} - 2 q^{11} - 3 q^{12} - q^{13} + q^{14} + 3 q^{15} + q^{16} - 3 q^{17} + 6 q^{18} + 6 q^{19} + O(q^{20})$

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

Modular degree:	2
$\Gamma_0(N)$-optimal:	yes
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is semistable. There are 2 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$7$	$I_{7}$	split multiplicative	-1	1	7	7
$13$	$1$	$I_{1}$	nonsplit multiplicative	1	1	1	1

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$	mod-$\ell$ image	$\ell$-adic image
$7$	7B.1.1	7.48.0.1

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $728 = 2^{3} \cdot 7 \cdot 13$, index $96$, genus $2$, and generators

$\left(\begin{array}{rr} 183 & 14 \\ 553 & 99 \end{array}\right),\left(\begin{array}{rr} 365 & 14 \\ 371 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 561 & 14 \\ 287 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 715 & 14 \\ 714 & 15 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 547 & 378 \\ 0 & 339 \end{array}\right)$.

The torsion field $K:=\Q(E[728])$ is a degree-$845365248$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/728\Z)$.

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
$2$	split multiplicative	$4$	$13$
$7$	good	$2$	$13$
$13$	nonsplit multiplicative	$14$	$2$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 26.b consists of 2 curves linked by isogenies of degree 7.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.104.1	$\Z/14\Z$	not in database
$6$	6.0.1124864.1	$\Z/2\Z \oplus \Z/14\Z$	not in database
$8$	8.2.999406512.1	$\Z/21\Z$	not in database
$12$	12.2.8421963387109376.7	$\Z/28\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$	2	3	5	7	13
Reduction type	split	ss	ord	ord	nonsplit
$\lambda$-invariant(s)	1	0,0	0	4	0
$\mu$-invariant(s)	0	0,0	0	0	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.

$p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.