Elliptic curve with LMFDB label 26569.a2 (Cremona label 26569a1)

• Label: 26569.a2
• Conductor: $26569$
• Discriminant: $-4330747$
• j-invariant: $-262537412640768000$
• CM: yes ($D=-163$)
• Rank: $1$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2+y=x^3-2174420x+1234136692$

(, )

Mordell-Weil group structure

$\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(850, 68)$	$2.2912801323260268704544630700$	$\infty$

Integral points

$\left(850, 68\right)$, $\left(850, -69\right)$

Invariants

Conductor:	$N$	=	$26569$	=	$163^{2}$
Discriminant:	$\Delta$	=	$-4330747$	=	$-1 \cdot 163^{3}$
j-invariant:	$j$	=	$-262537412640768000$	=	$-1 \cdot 2^{18} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3} \cdot 29^{3}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z[(1+\sqrt{-163})/2]$    (potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$N(\mathrm{U}(1))$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$1.8511273564650038098091683491$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$0.57768980626331322629293525757$
$abc$ quality:	$Q$	≈	$1.2036213229011967$
Szpiro ratio:	$\sigma_{m}$	≈	$5.43709628563842$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$1$
Mordell-Weil rank:	$r$	=	$1$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$2.2912801323260268704544630700$
Real period:	$\Omega$	≈	$0.79364722618265765456256125300$
Tamagawa product:	$\prod_{p}c_p$	=	$2$  = $2$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L'(E,1)$	≈	$3.6369362428559680165082260567$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\displaystyle 3.636936243 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.793647 \cdot 2.291280 \cdot 2}{1^2} \approx 3.636936243$

Modular invariants

Modular form   26569.2.a.a

$q - 2 q^{4} - 3 q^{9} + 4 q^{16} + O(q^{20})$

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

Modular degree:	59368
$\Gamma_0(N)$-optimal:	yes
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is not semistable. There is only one prime $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$163$	$2$	$III$	additive	1	2	3	0

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
$163$	163B.1.145	163.26568.1027.57

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
$163$	additive	$6806$	$1$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 163.
Its isogeny class 26569.a consists of 2 curves linked by isogenies of degree 163.

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
$3$	3.1.652.1	$\Z/2\Z$	not in database
$6$	6.0.69291952.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$8$	8.2.41017993267105683.1	$\Z/3\Z$	not in database
$12$	deg 12	$\Z/4\Z$	not in database
$16$	deg 16	$\Z/3\Z \oplus \Z/3\Z$	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	163
Reduction type	ss	ss	ss	ss	ss	ss	ss	ss	ss	ss	ss	ss	ord	ord	ord	add
$\lambda$-invariant(s)	?	1,1	1,1	1,1	1,1	1,1	1,1	1,1	1,1	1,1	1,1	1,1	1	1	1	-
$\mu$-invariant(s)	?	0,0	0,0	0,0	0,0	0,0	0,0	0,0	0,0	0,0	0,0	0,0	0	0	0	-

An entry ? indicates that the invariants have not yet been computed.

An entry - indicates that the invariants are not computed because the reduction is additive.

$p$-adic regulators

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