Elliptic Curve with LMFDB label 33600.ce2 (Cremona label 33600em4)

• Label: 33600.ce2
• Conductor: 33600
• Discriminant: 88510464000000
• j-invariant: \( \frac{13027640977}{21609} \)
• CM: no
• Rank: 0
• Torsion Structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 = x^{3} - x^{2} - 78433 x + 8468737 \)

Mordell-Weil group structure

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

Torsion generators

\( \left(157, 0\right) \), \( \left(167, 0\right) \)

Integral points

\( \left(-323, 0\right) \), \( \left(157, 0\right) \), \( \left(167, 0\right) \)

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

Invariants

Conductor:	\( 33600 \)	=	\(2^{6} \cdot 3 \cdot 5^{2} \cdot 7\)
Discriminant:	\(88510464000000 \)	=	\(2^{18} \cdot 3^{2} \cdot 5^{6} \cdot 7^{4} \)
j-invariant:	\( \frac{13027640977}{21609} \)	=	\(3^{-2} \cdot 7^{-4} \cdot 13^{3} \cdot 181^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.604308029248\)
Tamagawa product:	\( 64 \)  = \( 2^{2}\cdot2\cdot2^{2}\cdot2 \)
Torsion order:	\(4\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 33600.2.a.ce

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

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

Modular degree:	131072
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 2.41723211699 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(4\)	\( I_8^{*} \)	Additive	-1	6	18	0
\(3\)	\(2\)	\( I_{2} \)	Non-split multiplicative	1	1	2	2
\(5\)	\(4\)	\( I_0^{*} \)	Additive	1	2	6	0
\(7\)	\(2\)	\( I_{4} \)	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

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

Iwasawa invariants

$p$	2	3	5	7
Reduction type	add	nonsplit	add	nonsplit
$\lambda$-invariant(s)	-	0	-	0
$\mu$-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 33600.ce 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base-change curve
2	\(\Q(\sqrt{10}) \)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
4	\(\Q(\sqrt{3}, \sqrt{-10})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
	\(\Q(\sqrt{-3}, \sqrt{-10})\)	\(\Z/2\Z \times \Z/4\Z\)	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.