Elliptic curve with Cremona label 2304p1 (LMFDB label 2304.a2)

• Label: 2304p1
• Conductor: $2304$
• Discriminant: $-373248$
• j-invariant: \( 1728 \)
• CM: yes (\(D=-4\))
• Rank: $1$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2=x^3+18x\)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(3, 9\right) \)
\(\hat{h}(P)\)	≈	$0.71474134868387035118438908301$

Torsion generators

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

Integral points

\( \left(0, 0\right) \), \((3,\pm 9)\), \((6,\pm 18)\), \((72,\pm 612)\)

Invariants

Conductor:	\( 2304 \)	=	\(2^{8} \cdot 3^{2}\)
Discriminant:	\(-373248 \)	=	\(-1 \cdot 2^{9} \cdot 3^{6} \)
j-invariant:	\( 1728 \)	=	\(2^{6} \cdot 3^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z[\sqrt{-1}]\)	(potential complex multiplication)
Sato-Tate group:	$N(\mathrm{U}(1))$

BSD invariants

Analytic rank:	\(1\)
Regulator:	\(0.71474134868387035118438908301\)
Real period:	\(1.8002759999214539912407190766\)
Tamagawa product:	\( 8 \)  = \( 2\cdot2^{2} \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form   2304.2.a.a

\( q - 4q^{5} + 4q^{13} + 2q^{17} + O(q^{20}) \)

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

Modular degree:	256
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 2.5734633923741266002289673990323402740 \)

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\(III\)	Additive	-1	8	9	0
\(3\)	\(4\)	\(I_0^{*}\)	Additive	-1	2	6	0

Galois representations

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

The image is a Borel subgroup if \(p=2\), 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

Note: \(p\)-adic regulator data only exists for primes \(p\ge 5\) of good ordinary reduction.

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	5,1	1,1	1	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	0,0	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 2304p consists of 2 curves linked by isogenies of degree 2.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 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
$2$	\(\Q(\sqrt{-2}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$4$	4.2.18432.3	\(\Z/4\Z\)	Not in database
$8$	8.0.1358954496.9	\(\Z/4\Z \times \Z/4\Z\)	Not in database
$8$	8.4.5435817984.2	\(\Z/8\Z\)	Not in database
$8$	8.0.5435817984.7	\(\Z/8\Z\)	Not in database
$8$	8.2.36691771392.4	\(\Z/6\Z\)	Not in database
$8$	8.0.169869312000.8	\(\Z/10\Z\)	Not in database
$16$	16.0.118192468620711297024.15	\(\Z/4\Z \times \Z/8\Z\)	Not in database
$16$	16.0.1346286087882789617664.2	\(\Z/3\Z \times \Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/10\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/10\Z\)	Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.