Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
Index: | $24$ | $\PSL_2$-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 | $4^{2}\cdot8^{2}$ | 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: | 8B1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.13 |
Level structure
Jacobian
Conductor: | $2^{5}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 288.2.a.d |
Models
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} + 36x $ |
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 |
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$(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{2^2}{3^2}\cdot\frac{937008x^{2}y^{4}z^{2}-247608990720x^{2}z^{6}-1656xy^{6}z+13761093888xy^{2}z^{5}+y^{8}-193155840y^{4}z^{4}+2176782336z^{8}}{zy^{4}(36x^{2}z+xy^{2}+1296z^{3})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
4.12.0.d.1 | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.12.0.ca.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.12.1.bz.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
24.48.1.di.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.dj.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.eb.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.ec.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.eu.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.fb.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.fk.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.1.fn.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.72.5.by.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
24.96.5.be.1 | $24$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
120.48.1.kj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.kn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.ld.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.ll.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.mj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.mr.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.nl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.48.1.np.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.120.9.y.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
120.144.9.czc.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
120.240.17.su.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
168.48.1.kj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.kn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.ld.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.ll.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.mj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.mr.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.nl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.48.1.np.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.192.13.bc.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
264.48.1.kj.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.kn.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.ld.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.ll.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.mj.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.mr.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.nl.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.48.1.np.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.288.21.bc.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
312.48.1.kj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.kn.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.ld.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.ll.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.mj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.mr.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.nl.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.48.1.np.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |