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: | $4$ (of which $2$ are rational) | Cusp widths | $12^{2}\cdot24^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $16$ 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: | 24H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.88 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&8\\10&5\end{bmatrix}$, $\begin{bmatrix}13&20\\10&7\end{bmatrix}$, $\begin{bmatrix}17&22\\22&7\end{bmatrix}$, $\begin{bmatrix}17&23\\22&23\end{bmatrix}$, $\begin{bmatrix}23&8\\14&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| 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 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 |
|---|
| $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{21x^{2}y^{20}z^{2}-2178x^{2}y^{16}z^{6}+196005x^{2}y^{12}z^{10}-251658162x^{2}y^{8}z^{14}+352321551x^{2}y^{4}z^{18}-16777215x^{2}z^{22}+6xy^{22}z-1466xy^{18}z^{5}-393864xy^{14}z^{9}+100663362xy^{10}z^{13}-587202546xy^{6}z^{17}+134217729xy^{2}z^{21}+y^{24}-712y^{20}z^{4}+193746y^{16}z^{8}-16777764y^{12}z^{12}+335544411y^{8}z^{16}-117440496y^{4}z^{20}+z^{24}}{z^{4}y^{12}(3x^{2}y^{4}z^{2}+x^{2}z^{6}+2xy^{6}z+xy^{2}z^{5}+y^{8}+4y^{4}z^{4}+z^{8})}$ |
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 |
|---|---|---|---|---|---|---|
| 12.36.0.o.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.cj.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.xa.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.xb.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.xc.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.xd.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.zm.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.zn.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.zo.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.zp.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bac.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.bad.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bae.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.baf.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bas.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bat.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bau.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bav.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.9.bp.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.ug.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bcb.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bct.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dhk.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.dhn.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
| 24.144.9.dhs.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.dhv.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 48.144.3.f.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.144.3.f.2 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.144.3.v.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.3.v.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.7.bab.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.bab.2 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.bac.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
| 48.144.7.bac.2 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
| 48.144.7.bag.1 | $48$ | $2$ | $2$ | $7$ | $5$ | $1^{6}$ |
| 48.144.7.bag.2 | $48$ | $2$ | $2$ | $7$ | $5$ | $1^{6}$ |
| 48.144.7.bah.1 | $48$ | $2$ | $2$ | $7$ | $4$ | $1^{6}$ |
| 48.144.7.bah.2 | $48$ | $2$ | $2$ | $7$ | $4$ | $1^{6}$ |
| 48.144.11.jx.1 | $48$ | $2$ | $2$ | $11$ | $2$ | $2^{3}\cdot4$ |
| 48.144.11.jx.2 | $48$ | $2$ | $2$ | $11$ | $2$ | $2^{3}\cdot4$ |
| 48.144.11.rv.1 | $48$ | $2$ | $2$ | $11$ | $2$ | $2^{3}\cdot4$ |
| 48.144.11.rv.2 | $48$ | $2$ | $2$ | $11$ | $2$ | $2^{3}\cdot4$ |
| 72.216.13.nq.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.kxu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kxv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kxw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kxx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kyk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kyl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kym.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kyn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lag.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lah.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lai.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.laj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.law.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lax.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lay.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.laz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bged.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgef.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bget.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgev.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bggp.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bggr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bghf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bghh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hxt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hyj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hyk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hyl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hym.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iaf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iag.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iah.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iai.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iav.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iaw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iax.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iay.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bcax.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcaz.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcbn.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcbp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdj.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdl.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdz.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bceb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.144.3.bl.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.bl.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.cb.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.cb.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.7.dxk.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxk.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxl.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxl.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxo.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxo.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxp.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxp.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.11.cth.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cth.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cuf.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cuf.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 264.144.5.hxu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hxv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hxw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hxx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hyk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hyl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hym.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hyn.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iag.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iah.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iai.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iaj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iaw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iax.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iay.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iaz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bcgx.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcgz.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bchn.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bchp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcjj.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcjl.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcjz.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bckb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hxu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hxv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hxw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hxx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hyk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hyl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hym.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hyn.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iag.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iah.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iai.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iaj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iaw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iax.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iay.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iaz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bcbf.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcbh.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcbv.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcbx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcdr.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcdt.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bceh.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcej.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |