Elliptic curve with Cremona label 116610y3 (LMFDB label 116610.bb3)

• Label: 116610y3
• Conductor: $116610$
• Discriminant: $1.899\times 10^{19}$
• j-invariant: \( \frac{39248884582600321}{3935264062500} \)
• CM: no
• Rank: $2$
• Torsion structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2+xy+y=x^3-1196524x+457952822\)

Mordell-Weil group structure

\(\Z^2 \times \Z/{2}\Z \times \Z/{2}\Z\)

Infinite order Mordell-Weil generators and heights

\(P\)	=	\( \left(820, 4913\right) \)	\( \left(-60, 23041\right) \)
\(\hat{h}(P)\)	≈	$3.6233441940368527413377660639$	$3.9038633292458396175999541424$

Torsion generators

\( \left(469, -235\right) \), \( \left(781, -391\right) \)

Integral points

\( \left(-60, 23041\right) \), \( \left(-60, -22982\right) \), \( \left(469, -235\right) \), \( \left(781, -391\right) \), \( \left(820, 4913\right) \), \( \left(820, -5734\right) \), \( \left(1119, 22255\right) \), \( \left(1119, -23375\right) \), \( \left(1483, 43367\right) \), \( \left(1483, -44851\right) \), \( \left(2656, 125234\right) \), \( \left(2656, -127891\right) \), \( \left(17031, 2209609\right) \), \( \left(17031, -2226641\right) \)

Invariants

Conductor:	\( 116610 \)	=	\(2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 23\)
Discriminant:	\(18994767994251562500 \)	=	\(2^{2} \cdot 3^{2} \cdot 5^{8} \cdot 13^{6} \cdot 23^{4} \)
j-invariant:	\( \frac{39248884582600321}{3935264062500} \)	=	\(2^{-2} \cdot 3^{-2} \cdot 5^{-8} \cdot 23^{-4} \cdot 339841^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	\(2\)
Regulator:	\(11.322148552108623153670251606\)
Real period:	\(0.21105864774380100075524264248\)
Tamagawa product:	\( 64 \)  = \( 2\cdot2\cdot2\cdot2^{2}\cdot2 \)
Torsion order:	\(4\)
Analytic order of Ш:	\(1\) (rounded)

Modular invariants

Modular form 116610.2.a.bb

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

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

Modular degree:	3538944
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L^{(2)}(E,1)/2! \) ≈ \( 9.5585494518499216944177271577408696129 \)

Local data

This elliptic curve is semistable. There are 5 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\(I_{2}\)	Non-split multiplicative	1	1	2	2
\(3\)	\(2\)	\(I_{2}\)	Split multiplicative	-1	1	2	2
\(5\)	\(2\)	\(I_{8}\)	Non-split multiplicative	1	1	8	8
\(13\)	\(4\)	\(I_0^{*}\)	Additive	1	2	6	0
\(23\)	\(2\)	\(I_{4}\)	Non-split multiplicative	1	1	4	4

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X101.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 6 \\ 4 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 7 \end{array}\right)$ and has index 24.

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime	Image of Galois representation
\(2\)	Cs

$p$-adic data

$p$-adic regulators

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	nonsplit	split	nonsplit	ss	ordinary	add	ordinary	ordinary	nonsplit	ordinary	ss	ordinary	ordinary	ordinary	ss
$\lambda$-invariant(s)	4	5	2	2,2	2	-	2	2	2	2	2,2	2	2	2	2,2
$\mu$-invariant(s)	1	0	0	0,0	0	-	0	0	0	0	0,0	0	0	0	0,0

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

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 116610y consists of 3 curves linked by isogenies of degrees dividing 8.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{-13}) \)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{6}, \sqrt{13})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{-6}, \sqrt{13})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	8.0.151613669376.3	\(\Z/4\Z \times \Z/4\Z\)	Not in database
$8$	8.0.523799022862336.53	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$8$	8.0.165733284577536.33	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$8$	8.4.87329473560576.7	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/4\Z \times \Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/12\Z\)	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.