Invariants
Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $144$ | ||
Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $6^{4}$ | Cusp orbits | $1^{2}\cdot2$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 6D1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.576 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&6\\15&17\end{bmatrix}$, $\begin{bmatrix}8&3\\9&16\end{bmatrix}$, $\begin{bmatrix}11&3\\3&2\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.24.1.d.1 for the level structure with $-I$) |
Cyclic 24-isogeny field degree: | $12$ |
Cyclic 24-torsion field degree: | $96$ |
Full 24-torsion field degree: | $1536$ |
Jacobian
Conductor: | $2^{4}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 144.2.a.a |
Models
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} - 1 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Weierstrass model |
---|
$(0:1:0)$, $(1:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{(y^{2}+z^{2})(y^{2}+9z^{2})^{3}}{z^{2}y^{6}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
24.16.0-12.b.1.1 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
24.16.0-12.b.1.3 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
24.24.0-3.a.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0-3.a.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
24.144.1-12.l.1.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
24.192.5-12.e.1.2 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
72.144.3-36.k.1.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
72.144.3-36.k.1.2 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
72.144.3-36.n.1.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
72.144.3-36.q.1.1 | $72$ | $3$ | $3$ | $3$ | $?$ | not computed |
72.144.4-36.r.1.1 | $72$ | $3$ | $3$ | $4$ | $?$ | not computed |
72.144.4-36.r.1.3 | $72$ | $3$ | $3$ | $4$ | $?$ | not computed |
72.144.5-36.d.1.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
120.240.9-60.co.1.3 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
120.288.9-60.ep.1.6 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
120.480.17-60.kc.1.7 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
168.384.13-84.n.1.2 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |