Elliptic Curve with Cremona label 36980a4 (LMFDB label 36980.a2)

• Label: 36980a4
• Conductor: 36980
• Discriminant: -25285452196000000
• j-invariant: \( -\frac{20720464}{15625} \)
• CM: no
• Rank: 0
• Torsion Structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 = x^{3} - x^{2} - 67180 x + 10193400 \)

Mordell-Weil group structure

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

Torsion generators

\( \left(-315, 0\right) \)

Integral points

\( \left(-315, 0\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

Conductor:	\( 36980 \)	=	\(2^{2} \cdot 5 \cdot 43^{2}\)
Discriminant:	\(-25285452196000000 \)	=	\(-1 \cdot 2^{8} \cdot 5^{6} \cdot 43^{6} \)
j-invariant:	\( -\frac{20720464}{15625} \)	=	\(-1 \cdot 2^{4} \cdot 5^{-6} \cdot 109^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.346806941554\)
Tamagawa product:	\( 36 \)  = \( 3\cdot( 2 \cdot 3 )\cdot2 \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 36980.2.a.a

\( q + 2q^{3} + q^{5} - 2q^{7} + q^{9} + 2q^{13} + 2q^{15} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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

Modular degree:	231336
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 3.12126247399 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(3\)	\( IV^{*} \)	Additive	-1	2	8	0
\(5\)	\(6\)	\( I_{6} \)	Split multiplicative	-1	1	6	6
\(43\)	\(2\)	\( I_0^{*} \)	Additive	-1	2	6	0

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 X10.

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

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

prime	Image of Galois representation
\(2\)	B
\(3\)	B

$p$-adic data

$p$-adic regulators

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$	2	3	5	43
Reduction type	add	ordinary	split	add
$\lambda$-invariant(s)	-	2	1	-
$\mu$-invariant(s)	-	0	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.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 36980a consists of 4 curves linked by isogenies of degrees dividing 6.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base-change curve
2	\(\Q(\sqrt{129}) \)	\(\Z/6\Z\)	Not in database
	\(\Q(\sqrt{-1}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
4	\(\Q(i, \sqrt{129})\)	\(\Z/2\Z \times \Z/6\Z\)	Not in database
	4.2.739600.1	\(\Z/4\Z\)	Not in database
6	6.0.927369648.3	\(\Z/6\Z\)	Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.