Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $288$ | ||
| 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: | 24.72.1.75 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&6\\12&5\end{bmatrix}$, $\begin{bmatrix}7&4\\14&1\end{bmatrix}$, $\begin{bmatrix}7&13\\16&17\end{bmatrix}$, $\begin{bmatrix}7&23\\2&5\end{bmatrix}$, $\begin{bmatrix}11&1\\22&13\end{bmatrix}$ |
| 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: | $2^{5}\cdot3^{2}$ |
| 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 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\cdot3^3\,\frac{360x^{2}y^{22}-158683536x^{2}y^{20}z^{2}+3334857039360x^{2}y^{18}z^{4}-8483029723441152x^{2}y^{16}z^{6}+347116677002035200x^{2}y^{14}z^{8}+428407793302513582080x^{2}y^{12}z^{10}+124890533346189194035200x^{2}y^{10}z^{12}+13654933349880724142948352x^{2}y^{8}z^{14}-185843675399805070548664320x^{2}y^{6}z^{16}-105048744534819543156686585856x^{2}y^{4}z^{18}-5215708558403373819586255257600x^{2}y^{2}z^{20}-75107635800008373354201266257920x^{2}z^{22}-49896xy^{22}z+5650508160xy^{20}z^{3}-54429771585024xy^{18}z^{5}+61278125485916160xy^{16}z^{7}+18164104616438267904xy^{14}z^{9}+4245414386650959052800xy^{12}z^{11}+908607282493181550133248xy^{10}z^{13}+196685091901884930006712320xy^{8}z^{15}+21575703395366213958569558016xy^{6}z^{17}+1014183238936455901365959393280xy^{4}z^{19}+16690514105385204561103432974336xy^{2}z^{21}-y^{24}+3533760y^{22}z^{2}-151406247168y^{20}z^{4}+846575001169920y^{18}z^{6}-469769333793693696y^{16}z^{8}-121822020625870356480y^{14}z^{10}-27263639287585535164416y^{12}z^{12}-4869930211487757977518080y^{10}z^{14}-511455626793531129081102336y^{8}z^{16}-24129931760214704323370680320y^{6}z^{18}-405981030470833621380580245504y^{4}z^{20}+2865117999580704318381096960y^{2}z^{22}-10314424798490535546171949056z^{24}}{72x^{2}y^{22}-983664x^{2}y^{20}z^{2}+4274062848x^{2}y^{18}z^{4}-4258149046272x^{2}y^{16}z^{6}-6250952641609728x^{2}y^{14}z^{8}+11330268560270622720x^{2}y^{12}z^{10}-4193942785011839139840x^{2}y^{10}z^{12}-388232167383374197948416x^{2}y^{8}z^{14}+350283779038005744994615296x^{2}y^{6}z^{16}-30183645984484523426325725184x^{2}y^{4}z^{18}-1043141711680674763917251051520x^{2}y^{2}z^{20}+75107635800008373354201266257920x^{2}z^{22}+1512xy^{22}z-8553600xy^{20}z^{3}-11787545088xy^{18}z^{5}+112112997433344xy^{16}z^{7}-151283586070609920xy^{14}z^{9}+38804502190332837888xy^{12}z^{11}+30188907551916872957952xy^{10}z^{13}-14491600135767887104180224xy^{8}z^{15}+961423524952417000149221376xy^{6}z^{17}+202836647787291180273191878656xy^{4}z^{19}-16690514105385204561103432974336xy^{2}z^{21}+y^{24}-12096y^{22}z^{2}+195022080y^{20}z^{4}-827419152384y^{18}z^{6}+1035630317740032y^{16}z^{8}-56136325678497792y^{14}z^{10}-512338294685678174208y^{12}z^{12}+218496766759501767376896y^{10}z^{14}-10870780251233438517952512y^{8}z^{16}-4829392527991230682336591872y^{6}z^{18}+405683907122728955747562946560y^{4}z^{20}+573023599916140863676219392y^{2}z^{22}+10314424798490535546171949056z^{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 |
|---|---|---|---|---|---|---|
| 12.36.0.t.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.ci.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gs.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.bcw.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bcx.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bcy.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bcz.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bfe.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bff.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bfg.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bfh.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bhe.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bhf.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bhg.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.bhh.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bhu.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bhv.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bhw.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.5.bhx.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.9.u.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.nz.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bhk.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bhp.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.ejr.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.ejt.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.ekh.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.ekj.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 72.216.13.oa.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.leg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.leh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lei.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lej.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lew.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lex.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ley.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lez.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lgs.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lgt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lgu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lgv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lhi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lhj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lhk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lhl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bgjf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgjh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgjv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgjx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bglr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bglt.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgmh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgmj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.ief.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ieg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ieh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iei.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iev.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iew.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iex.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iey.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.igr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.igs.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.igt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.igu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ihh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ihi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ihj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ihk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bcfz.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcgb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcgp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcgr.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcil.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcin.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcjb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcjd.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.ieg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ieh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iei.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iej.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iew.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iex.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iey.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iez.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.igs.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.igt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.igu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.igv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ihi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ihj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ihk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ihl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bclz.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcmb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcmp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcmr.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcol.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcon.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcpb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcpd.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.ieg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ieh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iei.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iej.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iew.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iex.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iey.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iez.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.igs.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.igt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.igu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.igv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ihi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ihj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ihk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ihl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bcgh.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcgj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcgx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcgz.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcit.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bciv.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcjj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcjl.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |