Elliptic curve with LMFDB label 162.c2 (Cremona label 162b3)

• Label: 162.c2
• Conductor: $162$
• Discriminant: $-169869312$
• j-invariant: $-\frac{1159088625}{2097152}$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{3}\Z$

Minimal Weierstrass equation

$y^2+xy+y=x^3-x^2-95x-697$

(, )

Mordell-Weil group structure

$\Z/{3}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(19, 54)$	$0$	$3$

Integral points

$\left(19, 54\right)$, $\left(19, -74\right)$

Invariants

Conductor:	$N$	=	$162$	=	$2 \cdot 3^{4}$
Discriminant:	$\Delta$	=	$-169869312$	=	$-1 \cdot 2^{21} \cdot 3^{4}$
j-invariant:	$j$	=	$-\frac{1159088625}{2097152}$	=	$-1 \cdot 2^{-21} \cdot 3^{2} \cdot 5^{3} \cdot 101^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$0.26936807887764018183084025072$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.096836017345063048634241494921$
$abc$ quality:	$Q$	≈	$1.1123490200903752$
Szpiro ratio:	$\sigma_{m}$	≈	$5.244672406658039$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$0.72170105890730159936550912938$
Tamagawa product:	$\prod_{p}c_p$	=	$21$  = $( 3 \cdot 7 )\cdot1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$3$
Special value:	$L(E,1)$	≈	$1.6839691374503703985195213019$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\begin{aligned} 1.683969137 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.721701 \cdot 1.000000 \cdot 21}{3^2} \\ & \approx 1.683969137\end{aligned}$

Modular invariants

Modular form   162.2.a.c

$q + q^{2} + q^{4} + 2 q^{7} + q^{8} - 3 q^{11} + 2 q^{13} + 2 q^{14} + q^{16} - 3 q^{17} - q^{19} + O(q^{20})$

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

Modular degree:	42
$\Gamma_0(N)$-optimal:	no
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$21$	$I_{21}$	split multiplicative	-1	1	21	21
$3$	$1$	$II$	additive	1	4	4	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
$2$	2G	8.2.0.1
$3$	3B.1.1	3.8.0.1
$7$	7B.2.3	7.16.0.2

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $504 = 2^{3} \cdot 3^{2} \cdot 7$, index $768$, genus $21$, and generators

$\left(\begin{array}{rr} 281 & 392 \\ 112 & 393 \end{array}\right),\left(\begin{array}{rr} 73 & 402 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 426 \\ 42 & 253 \end{array}\right),\left(\begin{array}{rr} 1 & 216 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 462 & 1 \end{array}\right),\left(\begin{array}{rr} 85 & 42 \\ 189 & 43 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 168 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 168 \\ 336 & 169 \end{array}\right),\left(\begin{array}{rr} 295 & 42 \\ 231 & 211 \end{array}\right),\left(\begin{array}{rr} 22 & 321 \\ 189 & 169 \end{array}\right),\left(\begin{array}{rr} 249 & 226 \\ 448 & 345 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 420 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 168 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 463 & 282 \\ 420 & 295 \end{array}\right)$.

The torsion field $K:=\Q(E[504])$ is a degree-$15676416$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.

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	$4$	$81 = 3^{4}$
$3$	additive	$8$	$1$
$7$	good	$2$	$81 = 3^{4}$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3, 7 and 21.
Its isogeny class 162.c consists of 4 curves linked by isogenies of degrees dividing 21.

Twists

The minimal quadratic twist of this elliptic curve is 162.b2, its twist by $-3$.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.648.1	$\Z/6\Z$	not in database
$6$	6.0.3359232.4	$\Z/2\Z \oplus \Z/6\Z$	not in database
$6$	6.0.177147.2	$\Z/3\Z \oplus \Z/3\Z$	not in database
$6$	6.0.110270727.2	$\Z/21\Z$	not in database
$9$	9.3.74384733888.1	$\Z/9\Z$	not in database
$12$	12.2.5777633090469888.10	$\Z/12\Z$	not in database
$18$	18.0.1062353018033006514536448.1	$\Z/3\Z \oplus \Z/6\Z$	not in database
$18$	18.0.351496200956998572502045949952.2	$\Z/42\Z$	not in database
$21$	21.3.12777737809210143774260519108727.3	$\Z/21\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

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

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

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

$p$-adic regulators

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