Elliptic curve with LMFDB label 576.c3 (Cremona label 576h2)

• Label: 576.c3
• Conductor: 576
• Discriminant: 2985984
• j-invariant: \( 1728 \)
• CM: yes (\(D=-4\))
• Rank: 1
• Torsion Structure: \(\Z/{2}\Z \times \Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 = x^{3} - 36 x \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(-3, 9\right) \)
\(\hat{h}(P)\)	≈	0.8886258748396192

Torsion generators

\( \left(0, 0\right) \), \( \left(6, 0\right) \)

Integral points

\( \left(-6, 0\right) \), \((-3,\pm 9)\), \((-2,\pm 8)\), \( \left(0, 0\right) \), \( \left(6, 0\right) \), \((12,\pm 36)\), \((18,\pm 72)\), \((294,\pm 5040)\)

Invariants

Conductor:	\( 576 \)	=	\(2^{6} \cdot 3^{2}\)
Discriminant:	\(2985984 \)	=	\(2^{12} \cdot 3^{6} \)
j-invariant:	\( 1728 \)	=	\(2^{6} \cdot 3^{3}\)
Endomorphism ring:	\(\Z[\sqrt{-1}]\)		( Complex Multiplication)
Sato-Tate Group:	$N(\mathrm{U}(1))$

BSD invariants

Rank:	\(1\)
Regulator:	\(0.88862587484\)
Real period:	\(2.14090102808\)
Tamagawa product:	\( 16 \)  = \( 2^{2}\cdot2^{2} \)
Torsion order:	\(4\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 576.2.a.c

\( q - 2q^{5} - 6q^{13} - 2q^{17} + O(q^{20}) \)

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

Modular degree:	64
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 1.90246004902 \)

Local data

This elliptic curve is not semistable.

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(4\)	\( I_2^{*} \)	Additive	-1	6	12	0
\(3\)	\(4\)	\( I_0^{*} \)	Additive	-1	2	6	0

Galois representations

The mod \( p \) Galois representation has maximal image for all primes \( p < 1000 \) except those listed.

prime	Image of Galois representation
\(2\)	Cs

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=-1\).

$p$-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	add	ordinary	ss	ss	ordinary	ordinary	ss	ss	ordinary	ss	ordinary	ordinary	ss	ss
$\lambda$-invariant(s)	-	-	3	1,1	1,3	1	1	1,3	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	0,0	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.
Its isogeny class 576.c consists of 4 curves linked by isogenies of degrees dividing 4.

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
4	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
	\(\Q(i, \sqrt{6})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
	\(\Q(\sqrt{2}, \sqrt{3})\)	\(\Z/2\Z \times \Z/4\Z\)	4.4.2304.1-64.1-b2

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.