Elliptic curve with Cremona label 196b1 (LMFDB label 196.b2)

• Label: 196b1
• Conductor: $196$
• Discriminant: $92236816$
• j-invariant: $1792$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{3}\Z$

Minimal Weierstrass equation

$y^2=x^3+x^2-114x-127$

(, )

Mordell-Weil group structure

$\Z/{3}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(16, 49)$	$0$	$3$

Integral points

$(16,\pm 49)$

Invariants

Conductor:	$N$	=	$196$	=	$2^{2} \cdot 7^{2}$
Discriminant:	$\Delta$	=	$92236816$	=	$2^{4} \cdot 7^{8}$
j-invariant:	$j$	=	$1792$	=	$2^{8} \cdot 7$
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:	$h_{\mathrm{Faltings}}$	≈	$0.21861415198622039674399074368$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.3097083409039702431319884591$
$abc$ quality:	$Q$	≈	$0.8915192755066496$
Szpiro ratio:	$\sigma_{m}$	≈	$4.89397430255579$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$1.5672749721048733828823104029$
Tamagawa product:	$\prod_{p}c_p$	=	$9$  = $3\cdot3$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$3$
Special value:	$L(E,1)$	≈	$1.5672749721048733828823104029$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\displaystyle 1.567274972 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.567275 \cdot 1.000000 \cdot 9}{3^2} \approx 1.567274972$

Modular invariants

Modular form   196.2.a.b

$q + q^{3} + 3 q^{5} - 2 q^{9} - 3 q^{11} + 2 q^{13} + 3 q^{15} + 3 q^{17} - q^{19} + O(q^{20})$

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

Modular degree:	42
$\Gamma_0(N)$-optimal:	yes
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	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$3$	$IV$	additive	-1	2	4	0
$7$	$3$	$IV^{*}$	additive	1	2	8	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
$2$	2Cn	4.4.0.2
$3$	3B.1.1	9.24.0.2

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $252 = 2^{2} \cdot 3^{2} \cdot 7$, index $864$, genus $28$, and generators

$\left(\begin{array}{rr} 29 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 217 & 36 \\ 216 & 37 \end{array}\right),\left(\begin{array}{rr} 127 & 36 \\ 18 & 145 \end{array}\right),\left(\begin{array}{rr} 238 & 243 \\ 75 & 205 \end{array}\right),\left(\begin{array}{rr} 31 & 6 \\ 112 & 103 \end{array}\right),\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 42 & 199 \end{array}\right)$.

The torsion field $K:=\Q(E[252])$ is a degree-$870912$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/252\Z)$.

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
$2$	additive	$2$	$49 = 7^{2}$
$7$	additive	$26$	$4 = 2^{2}$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 196b consists of 2 curves linked by isogenies of degree 3.

Twists

The minimal quadratic twist of this elliptic curve is 196a1, its twist by $-7$.

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
$3$	$\Q(\zeta_{7})^+$	$\Z/2\Z \oplus \Z/6\Z$	3.3.49.1-64.1-a7
$6$	6.0.1037232.1	$\Z/3\Z \oplus \Z/3\Z$	not in database
$9$	9.3.203297472.1	$\Z/2\Z \oplus \Z/18\Z$	not in database
$12$	12.4.1511207993344.1	$\Z/2\Z \oplus \Z/12\Z$	not in database
$18$	18.0.1115906277282951168.1	$\Z/6\Z \oplus \Z/18\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$	2	3	7
Reduction type	add	ord	add
$\lambda$-invariant(s)	-	2	-
$\mu$-invariant(s)	-	0	-

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.