Elliptic curve with LMFDB label 11.a2 (Cremona label 11a1)

• Label: 11.a2
• Conductor: $11$
• Discriminant: $-161051$
• j-invariant: $-\frac{122023936}{161051}$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{5}\Z$

This is a model for the modular curve $X_0(11)$.

Minimal Weierstrass equation

$y^2+y=x^3-x^2-10x-20$

(, )

Mordell-Weil group structure

$\Z/{5}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(5, 5)$	$0$	$5$

Integral points

$\left(5, 5\right)$, $\left(5, -6\right)$, $\left(16, 60\right)$, $\left(16, -61\right)$

Invariants

Conductor:	$N$	=	$11$	=	$11$
Discriminant:	$\Delta$	=	$-161051$	=	$-1 \cdot 11^{5}$
j-invariant:	$j$	=	$-\frac{122023936}{161051}$	=	$-1 \cdot 2^{12} \cdot 11^{-5} \cdot 31^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$-0.30800984111840306468901426146$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.30800984111840306468901426146$
$abc$ quality:	$Q$	≈	$1.012998630378065$
Szpiro ratio:	$\sigma_{m}$	≈	$8.26048375415454$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$1.2692093042795534216887946168$
Tamagawa product:	$\prod_{p}c_p$	=	$5$  = $5$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$5$
Special value:	$L(E,1)$	≈	$0.25384186085591068433775892335$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\begin{aligned} 0.253841861 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.269209 \cdot 1.000000 \cdot 5}{5^2} \\ & \approx 0.253841861\end{aligned}$

Modular invariants

Modular form   11.2.a.a

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

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

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

Local data at primes of bad reduction

This elliptic curve is semistable. There is only one prime $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$11$	$5$	$I_{5}$	split multiplicative	-1	1	5	5

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
$5$	5Cs.1.1	5.120.0.1

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $550 = 2 \cdot 5^{2} \cdot 11$, index $1200$, genus $37$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 501 & 50 \\ 500 & 51 \end{array}\right),\left(\begin{array}{rr} 16 & 35 \\ 115 & 11 \end{array}\right),\left(\begin{array}{rr} 101 & 50 \\ 204 & 277 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 10 & 501 \end{array}\right),\left(\begin{array}{rr} 381 & 50 \\ 235 & 21 \end{array}\right)$.

The torsion field $K:=\Q(E[550])$ is a degree-$19800000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/550\Z)$.

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
$5$	good	$2$	$1$
$11$	split multiplicative	$12$	$1$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 5.
Its isogeny class 11.a consists of 3 curves linked by isogenies of degrees dividing 25.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.44.1	$\Z/10\Z$	not in database
$4$	$\Q(\zeta_{5})$	$\Z/5\Z \oplus \Z/5\Z$	not in database
$6$	6.0.21296.1	$\Z/2\Z \oplus \Z/10\Z$	not in database
$8$	8.2.32019867.1	$\Z/15\Z$	not in database
$12$	12.2.20433779818496.3	$\Z/20\Z$	not in database
$12$	12.0.7320500000000.2	$\Z/5\Z \oplus \Z/10\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$	2	3	5	11
Reduction type	ss	ord	ord	split
$\lambda$-invariant(s)	0,1	0	0	1
$\mu$-invariant(s)	0,0	0	1	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

$p$-adic regulators

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