Elliptic Curve with LMFDB label 336.a1 (Cremona label 336e5)

• Label: 336.a1
• Conductor: 336
• Discriminant: 602112
• j-invariant: \( \frac{53297461115137}{147} \)
• CM: no
• Rank: 1
• Torsion Structure: \(\Z/{4}\Z\)

Minimal Weierstrass equation

\( y^2 = x^{3} - x^{2} - 12544 x + 544960 \)

Mordell-Weil group structure

\(\Z\times \Z/{4}\Z\)

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(90, 370\right) \)
\(\hat{h}(P)\)	≈	2.98519683478

Torsion generators

\( \left(66, 14\right) \)

Integral points

\( \left(65, 0\right) \), \( \left(66, 14\right) \), \( \left(90, 370\right) \), \( \left(101, 546\right) \)

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

Invariants

Conductor:	\( 336 \)	=	\(2^{4} \cdot 3 \cdot 7\)
Discriminant:	\(602112 \)	=	\(2^{12} \cdot 3 \cdot 7^{2} \)
j-invariant:	\( \frac{53297461115137}{147} \)	=	\(3^{-1} \cdot 7^{-2} \cdot 37633^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(1\)
Regulator:	\(2.98519683478\)
Real period:	\(1.91098978075\)
Tamagawa product:	\( 4 \)  = \( 2\cdot1\cdot2 \)
Torsion order:	\(4\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 336.2.a.a

\( q - q^{3} - 2q^{5} + q^{7} + q^{9} - 4q^{11} - 2q^{13} + 2q^{15} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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

Modular degree:	256
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 1.4261701612 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\( I_4^{*} \)	Additive	-1	4	12	0
\(3\)	\(1\)	\( I_{1} \)	Non-split multiplicative	1	1	1	1
\(7\)	\(2\)	\( I_{2} \)	Split multiplicative	-1	1	2	2

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 X122d.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right)$ and has index 48.

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

$p$-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	add	nonsplit	ordinary	split	ordinary	ordinary	ordinary	ordinary	ss	ordinary	ss	ordinary	ordinary	ordinary	ss
$\lambda$-invariant(s)	-	1	1	2	1	1	1	1	1,1	1	1,1	1	1	1	1,1
$\mu$-invariant(s)	-	0	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, 4 and 8.
Its isogeny class 336.a consists of 6 curves linked by isogenies of degrees dividing 8.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base-change curve
2	\(\Q(\sqrt{3}) \)	\(\Z/2\Z \times \Z/4\Z\)	2.2.12.1-147.1-b7
	\(\Q(\sqrt{14}) \)	\(\Z/8\Z\)	Not in database
	\(\Q(\sqrt{42}) \)	\(\Z/8\Z\)	Not in database
4	\(\Q(\sqrt{3}, \sqrt{14})\)	\(\Z/2\Z \times \Z/8\Z\)	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.