Elliptic curve with LMFDB label 336.d3 (Cremona label 336d4)

• Label: 336.d3
• Conductor: 336
• Discriminant: 258096513024
• j-invariant: \( \frac{124475734657}{63011844} \)
• CM: no
• Rank: 0
• Torsion Structure: \(\Z/{2}\Z \times \Z/{4}\Z\)

Minimal Weierstrass equation

\( y^2 = x^{3} + x^{2} - 1664 x - 9804 \)

Mordell-Weil group structure

\(\Z/{2}\Z \times \Z/{4}\Z\)

Torsion generators

\( \left(-6, 0\right) \), \( \left(106, 1008\right) \)

Integral points

\( \left(-38, 0\right) \), \((-20,\pm 126)\), \( \left(-6, 0\right) \), \( \left(43, 0\right) \), \((106,\pm 1008)\)

Invariants

Conductor:	\( 336 \)	=	\(2^{4} \cdot 3 \cdot 7\)
Discriminant:	\(258096513024 \)	=	\(2^{14} \cdot 3^{8} \cdot 7^{4} \)
j-invariant:	\( \frac{124475734657}{63011844} \)	=	\(2^{-2} \cdot 3^{-8} \cdot 7^{-4} \cdot 4993^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.788493385608\)
Tamagawa product:	\( 128 \)  = \( 2^{2}\cdot2^{3}\cdot2^{2} \)
Torsion order:	\(8\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 336.2.a.d

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

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

Modular degree:	384
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 1.57698677122 \)

Local data

This elliptic curve is not semistable.

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(4\)	\( I_6^{*} \)	Additive	-1	4	14	2
\(3\)	\(8\)	\( I_{8} \)	Split multiplicative	-1	1	8	8
\(7\)	\(4\)	\( I_{4} \)	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 X200a.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 2 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$ and has index 96.

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

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

Iwasawa invariants

$p$	2	3	7
Reduction type	add	split	split
$\lambda$-invariant(s)	-	3	1
$\mu$-invariant(s)	-	0	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 and 4.
Its isogeny class 336.d 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{4}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base-change curve
2	\(\Q(\sqrt{7}) \)	\(\Z/2\Z \times \Z/8\Z\)	2.2.28.1-126.1-a4
4	\(\Q(\zeta_{8})\)	\(\Z/4\Z \times \Z/4\Z\)	Not in database
	\(\Q(\sqrt{2}, \sqrt{-7})\)	\(\Z/2\Z \times \Z/8\Z\)	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.