Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | 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: | yes $\quad(D =$ $-12$) |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.79 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&13\\22&15\end{bmatrix}$, $\begin{bmatrix}1&14\\12&1\end{bmatrix}$, $\begin{bmatrix}11&14\\2&9\end{bmatrix}$, $\begin{bmatrix}21&14\\16&17\end{bmatrix}$ |
| 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: | $3072$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 9x $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\cdot3^3\,\frac{39390x^{2}y^{14}-4687224x^{2}y^{13}z+11971641x^{2}y^{12}z^{2}+885593088x^{2}y^{11}z^{3}+2484027972x^{2}y^{10}z^{4}-44886999852x^{2}y^{9}z^{5}-306671939406x^{2}y^{8}z^{6}+447314598288x^{2}y^{7}z^{7}+4939440937434x^{2}y^{6}z^{8}+25122997460232x^{2}y^{5}z^{9}+36625154650290x^{2}y^{4}z^{10}-348952389432444x^{2}y^{3}z^{11}+15260493061710x^{2}y^{2}z^{12}+11506388523300x^{2}yz^{13}+3922632451125x^{2}z^{14}-3300xy^{15}+1360329xy^{14}z-19662756xy^{13}z^{2}-385196220xy^{12}z^{3}+1007551224xy^{11}z^{4}+34405914375xy^{10}z^{5}+110913801228xy^{9}z^{6}-844059974148xy^{8}z^{7}-5987095971804xy^{7}z^{8}-9305286915699xy^{6}z^{9}+141778546362468xy^{5}z^{10}+1695610340190xy^{4}z^{11}+1278487613700xy^{3}z^{12}+435848050125xy^{2}z^{13}+125y^{16}-282244y^{15}z+11938528y^{14}z^{2}+60536520y^{13}z^{3}-1445166846y^{12}z^{4}-10630968120y^{11}z^{5}+37966169826y^{10}z^{6}+435901423860y^{9}z^{7}+1062931089402y^{8}z^{8}-9384075346860y^{7}z^{9}+7098227854488y^{6}z^{10}+23129117984556y^{5}z^{11}+58848268297959y^{4}z^{12}+109347108497316y^{3}z^{13}+137344437555390y^{2}z^{14}+103557496709700yz^{15}+35303692060125z^{16}}{84x^{2}y^{14}-577773x^{2}y^{12}z^{2}-87663708x^{2}y^{10}z^{4}+1565270892x^{2}y^{8}z^{6}+826292969856x^{2}y^{6}z^{8}-17646357573135x^{2}y^{4}z^{10}-152307391681548x^{2}y^{2}z^{12}+2056557741475815x^{2}z^{14}-2970xy^{14}z+2317248xy^{12}z^{3}-166544424xy^{10}z^{5}-66374146548xy^{8}z^{7}+1487321606178xy^{6}z^{9}+67710253703508xy^{4}z^{11}-1142552999304081xy^{2}z^{13}-y^{16}+56484y^{14}z^{2}+4949910y^{12}z^{4}+2689248924y^{10}z^{6}-56562859953y^{8}z^{8}-5610594823884y^{6}z^{10}+101429783384112y^{4}z^{12}+292889889684y^{2}z^{14}-282429536481z^{16}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.0.z.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.12.0.m.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.by.1 | $24$ | $2$ | $2$ | $1$ | $1$ | 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.ki.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.kj.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.kk.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.kl.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.lw.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.lx.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.ly.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.lz.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.72.5.pa.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.gc.1 | $24$ | $4$ | $4$ | $5$ | $3$ | $1^{4}$ |
| 48.48.3.bn.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 48.48.3.bn.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 48.48.3.dz.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.48.3.dz.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.48.3.eb.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{2}$ |
| 48.48.3.eb.2 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{2}$ |
| 48.48.3.gf.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $2$ |
| 48.48.3.gf.2 | $48$ | $2$ | $2$ | $3$ | $3$ | $2$ |
| 120.48.1.bxm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bxn.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bxo.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bxp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.byc.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.byd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bye.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.byf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.xe.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.rim.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.gfw.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.bxk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bxl.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bxm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bxn.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bya.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.byb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.byc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.byd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.oe.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 240.48.3.gb.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.gb.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.hf.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.hf.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.hg.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.hg.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.jl.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.jl.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.48.1.bxk.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bxl.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bxm.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bxn.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bya.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.byb.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.byc.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.byd.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.mi.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.bxm.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bxn.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bxo.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bxp.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.byc.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.byd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bye.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.byf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |