Elliptic Curve with LMFDB label 24.a4 (Cremona label 24a1)

• Label: 24.a4
• Conductor: 24
• Discriminant: 2304
• j-invariant: \( \frac{35152}{9} \)
• CM: no
• Rank: 0
• Torsion Structure: \(\Z/{2}\Z \times \Z/{4}\Z\)

This is a model for the modular curve $X_0(24)$.

Minimal Weierstrass equation

\( y^2 = x^{3} - x^{2} - 4 x + 4 \)

Mordell-Weil group structure

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

Torsion generators

\( \left(1, 0\right) \), \( \left(4, 6\right) \)

Integral points

\( \left(-2, 0\right) \), \( \left(0, 2\right) \), \( \left(1, 0\right) \), \( \left(2, 0\right) \), \( \left(4, 6\right) \)

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

Invariants

Conductor:	\( 24 \)	=	\(2^{3} \cdot 3\)
Discriminant:	\(2304 \)	=	\(2^{8} \cdot 3^{2} \)
j-invariant:	\( \frac{35152}{9} \)	=	\(2^{4} \cdot 3^{-2} \cdot 13^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(4.313031295\)
Tamagawa product:	\( 8 \)  = \( 2^{2}\cdot2 \)
Torsion order:	\(8\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 24.2.a.a

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

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

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

Special L-value

\( L(E,1) \) ≈ \( 0.539128911875 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(4\)	\( I_1^{*} \)	Additive	-1	3	8	0
\(3\)	\(2\)	\( I_{2} \)	Non-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 X190c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 0 \\ 4 & 5 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 5 \end{array}\right)$ 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
Reduction type	add	nonsplit
$\lambda$-invariant(s)	-	0
$\mu$-invariant(s)	-	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 24.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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{4}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base-change curve
4	\(\Q(\sqrt{2}, \sqrt{3})\)	\(\Z/2\Z \times \Z/8\Z\)	4.4.2304.1-72.1-b4
	\(\Q(\zeta_{8})\)	\(\Z/2\Z \times \Z/8\Z\)	Not in database
	\(\Q(\zeta_{12})\)	\(\Z/4\Z \times \Z/4\Z\)	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.