Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $64$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $12^{2}\cdot24^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 24H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.349 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&6\\18&11\end{bmatrix}$, $\begin{bmatrix}5&2\\2&19\end{bmatrix}$, $\begin{bmatrix}11&20\\10&17\end{bmatrix}$, $\begin{bmatrix}19&19\\8&13\end{bmatrix}$, $\begin{bmatrix}23&0\\0&7\end{bmatrix}$ |
| 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: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{1620x^{2}y^{22}+3305907x^{2}y^{20}z^{2}+321649020x^{2}y^{18}z^{4}+3787932654x^{2}y^{16}z^{6}+717584400x^{2}y^{14}z^{8}-4100162985x^{2}y^{12}z^{10}+5533749900x^{2}y^{10}z^{12}-2801082114x^{2}y^{8}z^{14}-176494140x^{2}y^{6}z^{16}+461869857x^{2}y^{4}z^{18}-106166700x^{2}y^{2}z^{20}+7077915x^{2}z^{22}+37422xy^{22}z+19619820xy^{20}z^{3}+874964178xy^{18}z^{5}+4560426720xy^{16}z^{7}-6258355848xy^{14}z^{9}+6771932100xy^{12}z^{11}-6709883346xy^{10}z^{13}+6724445040xy^{8}z^{15}-3415043322xy^{6}z^{17}+743179860xy^{4}z^{19}-56623077xy^{2}z^{21}+27y^{24}+441720y^{22}z^{2}+87619356y^{20}z^{4}+2268130040y^{18}z^{6}+5826844422y^{16}z^{8}-6995532960y^{14}z^{10}+7248115152y^{12}z^{12}-5993910360y^{10}z^{14}+2914350057y^{8}z^{16}-636555240y^{6}z^{18}+49582908y^{4}z^{20}+1620y^{2}z^{22}+27z^{24}}{12x^{2}y^{22}+759x^{2}y^{20}z^{2}+15268x^{2}y^{18}z^{4}+70422x^{2}y^{16}z^{6}-478608x^{2}y^{14}z^{8}-4016245x^{2}y^{12}z^{10}-6882540x^{2}y^{10}z^{12}+2949606x^{2}y^{8}z^{14}+12320796x^{2}y^{6}z^{16}+4915149x^{2}y^{4}z^{18}-786420x^{2}y^{2}z^{20}-262145x^{2}z^{22}-42xy^{22}z-1100xy^{20}z^{3}+7018xy^{18}z^{5}+309024xy^{16}z^{7}+1930520xy^{14}z^{9}+2292508xy^{12}z^{11}-8257002xy^{10}z^{13}-18350064xy^{8}z^{15}-5636146xy^{6}z^{17}+5505036xy^{4}z^{19}+2097151xy^{2}z^{21}-y^{24}-56y^{22}z^{2}-4180y^{20}z^{4}-82104y^{18}z^{6}-475762y^{16}z^{8}-119392y^{14}z^{10}+5044688y^{12}z^{12}+9960216y^{10}z^{14}+2294197y^{8}z^{16}-4718552y^{6}z^{18}-1835060y^{4}z^{20}+12y^{2}z^{22}-z^{24}}$ |
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 |
|---|---|---|---|---|---|---|
| 24.36.0.bm.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.ci.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gp.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 |
|---|---|---|---|---|---|---|
| 24.144.5.wo.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.wp.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.wq.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.wr.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.za.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.zb.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.zc.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.zd.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.zq.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.zr.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.zs.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.zt.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bag.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bah.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bai.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.baj.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.9.bl.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.mx.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bcg.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bck.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.dhh.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dhi.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.dhp.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.dhq.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 72.216.13.np.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.kxi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kxj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kxk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kxl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kxy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kxz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kya.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kyb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kzu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kzv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kzw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kzx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lak.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lal.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lam.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lan.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bgdy.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgea.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgeo.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgeq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bggk.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bggm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgha.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bghc.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hxh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hya.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hzt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hzu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hzv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hzw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iaj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iak.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ial.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iam.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bcas.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcau.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcbi.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcbk.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcde.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdg.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdu.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdw.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.hxi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hxj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hxk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hxl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hxy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hxz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hya.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hyb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hzu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hzv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hzw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hzx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iak.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ial.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iam.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ian.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bcgs.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcgu.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bchi.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bchk.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcje.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcjg.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcju.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcjw.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hxi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hxj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hxk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hxl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hxy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hxz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hya.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hyb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hzu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hzv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hzw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hzx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iak.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ial.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iam.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ian.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bcba.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcbc.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcbq.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcbs.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcdm.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcdo.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcec.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcee.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |