Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $1^{4}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.1.290 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&2\\12&13\end{bmatrix}$, $\begin{bmatrix}11&4\\12&7\end{bmatrix}$, $\begin{bmatrix}19&6\\4&17\end{bmatrix}$, $\begin{bmatrix}19&22\\4&21\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^3\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.x.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $384$ |
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} - 9x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(-3:0:1)$, $(3:0:1)$, $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^4}\cdot\frac{57348x^{2}y^{28}z^{2}-28217745630x^{2}y^{24}z^{6}+1702325241245511x^{2}y^{20}z^{10}+8119633115233342035x^{2}y^{16}z^{14}+25859339264100689920584x^{2}y^{12}z^{18}+35988833860673726883403485x^{2}y^{8}z^{22}+9178730331116150925790415541x^{2}y^{4}z^{26}+148695418365105736174136457735x^{2}z^{30}-72xy^{30}z+1060165746xy^{26}z^{5}-3078357112152xy^{22}z^{9}+559343178786328857xy^{18}z^{13}+2580026717665216890576xy^{14}z^{17}+4488406598071779348409785xy^{10}z^{21}+2719623699147462323753465100xy^{6}z^{25}+181738856485713392638390465137xy^{2}z^{29}+y^{32}-3026808y^{28}z^{4}+6663859867548y^{24}z^{8}+29627258969642526y^{20}z^{12}+129176662264973890524y^{16}z^{16}+247842484655981377266576y^{12}z^{20}+211526419194369812018721726y^{8}z^{24}+18357457744392591677917918842y^{4}z^{28}+79766443076872509863361z^{32}}{z^{2}y^{8}(x^{2}y^{20}+114453x^{2}y^{16}z^{4}-8832549420x^{2}y^{12}z^{8}+120605935328145x^{2}y^{8}z^{12}+277643203126256493x^{2}y^{4}z^{16}+13493075341822822215x^{2}z^{20}-3969xy^{18}z^{3}+11809800xy^{14}z^{7}+704459589165xy^{10}z^{11}+41131405877993172xy^{6}z^{15}+10494797169090723633xy^{2}z^{19}-54y^{20}z^{2}+8017542y^{16}z^{6}-155465624376y^{12}z^{10}+2285105998608162y^{8}z^{14}+999466727430619458y^{4}z^{18}+1853020188851841z^{22})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.c.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-8.c.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.b.1.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.b.1.23 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.v.2.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.v.2.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.w.2.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.w.2.16 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.o.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.o.2.10 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.be.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.be.1.14 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bf.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bf.1.14 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.384.5-24.bf.1.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.bg.1.8 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.bi.1.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.bj.4.8 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.576.17-24.pf.1.21 | $24$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.fv.1.17 | $24$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.384.5-48.a.2.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.g.2.7 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.p.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.r.2.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.cq.2.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.cs.2.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.db.2.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.dh.2.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 120.384.5-120.hn.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ho.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ht.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hu.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hn.1.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ho.1.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ht.1.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hu.1.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.cw.2.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.dc.2.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.fx.2.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.fz.2.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ma.2.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.mc.2.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ox.2.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.pd.2.10 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.hn.1.16 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ho.1.16 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ht.1.16 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.hu.1.16 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.hn.1.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ho.1.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ht.1.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.hu.1.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |