Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $5 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $12^{2}\cdot24^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24A5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.5.210 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&1\\8&21\end{bmatrix}$, $\begin{bmatrix}5&22\\14&23\end{bmatrix}$, $\begin{bmatrix}9&7\\20&15\end{bmatrix}$, $\begin{bmatrix}23&1\\4&1\end{bmatrix}$, $\begin{bmatrix}23&9\\6&1\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{24}\cdot3^{10}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{5}$ |
| Newforms: | 36.2.a.a, 144.2.a.a, 576.2.a.a, 576.2.a.c, 576.2.a.i |
Models
Embedded model Embedded model in $\mathbb{P}^{7}$
| $ 0 $ | $=$ | $ - z r + t^{2} $ |
| $=$ | $ - x u + w^{2}$ | |
| $=$ | $ - y r + z t$ | |
| $=$ | $x r + z^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{12} + 3 y^{2} z^{10} - 64 z^{12} $ |
Weierstrass model Weierstrass model
| $ y^{2} + x^{6} y $ | $=$ | $ -x^{12} + 48 $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}r$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ | $=$ | $\displaystyle -t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{1}{2}t^{6}-\frac{3}{16}vr^{5}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}r$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{(v^{2}+r^{2})^{3}}{r^{2}(3v^{2}-r^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.36.2.bx.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{3}$ |
| 24.24.1.eo.1 | $24$ | $3$ | $3$ | $1$ | $1$ | $1^{4}$ |
| 24.36.0.cj.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.3.g.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.9.dhs.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.144.9.dht.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 24.144.9.dia.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.144.9.dib.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 24.144.9.edm.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.144.9.edn.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.144.9.edo.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 24.144.9.edp.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 24.144.9.ego.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.144.9.egp.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.144.9.egq.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 24.144.9.egr.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.144.9.ehe.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 24.144.9.ehf.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.144.9.ehg.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 24.144.9.ehh.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 24.144.9.ehu.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 24.144.9.ehv.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 24.144.9.ehw.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 24.144.9.ehx.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 24.144.9.ekc.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.144.9.ekd.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 24.144.9.ekk.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.144.9.ekl.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{4}$ |
| 48.144.11.fi.1 | $48$ | $2$ | $2$ | $11$ | $1$ | $2\cdot4$ |
| 48.144.11.fi.2 | $48$ | $2$ | $2$ | $11$ | $1$ | $2\cdot4$ |
| 48.144.11.rg.1 | $48$ | $2$ | $2$ | $11$ | $1$ | $2^{3}$ |
| 48.144.11.rg.2 | $48$ | $2$ | $2$ | $11$ | $1$ | $2^{3}$ |
| 48.144.11.ts.1 | $48$ | $2$ | $2$ | $11$ | $3$ | $1^{6}$ |
| 48.144.11.ts.2 | $48$ | $2$ | $2$ | $11$ | $3$ | $1^{6}$ |
| 48.144.11.tu.1 | $48$ | $2$ | $2$ | $11$ | $2$ | $1^{6}$ |
| 48.144.11.tu.2 | $48$ | $2$ | $2$ | $11$ | $2$ | $1^{6}$ |
| 48.144.11.tv.1 | $48$ | $2$ | $2$ | $11$ | $6$ | $1^{6}$ |
| 48.144.11.tv.2 | $48$ | $2$ | $2$ | $11$ | $6$ | $1^{6}$ |
| 48.144.11.tw.1 | $48$ | $2$ | $2$ | $11$ | $5$ | $1^{6}$ |
| 48.144.11.tw.2 | $48$ | $2$ | $2$ | $11$ | $5$ | $1^{6}$ |
| 48.144.11.xy.1 | $48$ | $2$ | $2$ | $11$ | $5$ | $2^{3}$ |
| 48.144.11.xy.2 | $48$ | $2$ | $2$ | $11$ | $5$ | $2^{3}$ |
| 48.144.11.yw.1 | $48$ | $2$ | $2$ | $11$ | $3$ | $2\cdot4$ |
| 48.144.11.yw.2 | $48$ | $2$ | $2$ | $11$ | $3$ | $2\cdot4$ |
| 72.216.17.pi.1 | $72$ | $3$ | $3$ | $17$ | $?$ | not computed |
| 120.144.9.nnw.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.nnx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.noe.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.nof.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qyk.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qyl.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qym.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qyn.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qza.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qzb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qzc.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qzd.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qzq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qzr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qzs.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.qzt.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.rag.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.rah.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.rai.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.raj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.ruk.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.rul.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.rus.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.rut.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.mdw.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.mdx.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.mee.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.mef.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pbu.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pbv.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pbw.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pbx.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pck.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pcl.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pcm.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pcn.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pda.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pdb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pdc.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pdd.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pdq.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pdr.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pds.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pdt.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.ppu.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.ppv.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pqc.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.pqd.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.144.11.bju.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bju.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bkk.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bkk.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bmu.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bmu.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bmv.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bmv.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bmw.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bmw.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bmx.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bmx.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.brc.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.brc.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bsa.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.bsa.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 264.144.9.mgy.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.mgz.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.mhg.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.mhh.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pgi.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pgj.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pgk.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pgl.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pgy.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pgz.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pha.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.phb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pho.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.php.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.phq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.phr.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pie.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pif.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pig.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pih.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pui.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.puj.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.puq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.pur.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.mee.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.mef.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.mem.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.men.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pcc.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pcd.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pce.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pcf.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pcs.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pct.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pcu.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pcv.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pdi.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pdj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pdk.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pdl.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pdy.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pdz.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pea.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.peb.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pqc.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pqd.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pqk.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.pql.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |