Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $288$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (all of which are rational) | Cusp widths | $2^{2}\cdot4\cdot6^{2}\cdot12\cdot16\cdot48$ | Cusp orbits | $1^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48D5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.5.3853 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&21\\24&23\end{bmatrix}$, $\begin{bmatrix}5&16\\0&5\end{bmatrix}$, $\begin{bmatrix}7&31\\0&1\end{bmatrix}$, $\begin{bmatrix}13&32\\0&13\end{bmatrix}$, $\begin{bmatrix}17&31\\24&19\end{bmatrix}$, $\begin{bmatrix}31&18\\12&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.5.oq.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $2$ |
| Cyclic 48-torsion field degree: | $16$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{23}\cdot3^{9}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}$ |
| Newforms: | 24.2.a.a, 288.2.a.b, 288.2.a.c, 288.2.a.d$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x w + y w - y t $ |
| $=$ | $x^{2} - x y - 2 y^{2} + w^{2} + w t$ | |
| $=$ | $2 x w + x t - y w + y t + 3 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 9 x^{6} + 3 x^{5} z + 14 x^{4} z^{2} - 2 x^{3} z^{3} - 5 x^{2} z^{4} + 27 x y^{4} z + \cdots - 9 y^{4} z^{2} $ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:1)$, $(2:1:0:0:0)$, $(0:1:0:1:1)$, $(-1:1:0:0:0)$, $(0:0:0:-1:1)$, $(0:-1:0:1:1)$, $(4/9:-1/9:0:-1/3:1)$, $(-4/9:1/9:0:-1/3:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1539xy^{9}t^{2}-6534xy^{7}t^{4}+8748xy^{5}t^{6}-2619xy^{3}t^{8}+135xyt^{10}-729y^{12}+1458y^{10}w^{2}-2187y^{10}wt+4455y^{10}t^{2}-2241y^{8}w^{2}t^{2}+1512y^{8}wt^{3}-10314y^{8}t^{4}+3186y^{6}w^{2}t^{4}+2430y^{6}wt^{5}+11448y^{6}t^{6}-3186y^{4}w^{2}t^{6}-3888y^{4}wt^{7}-4239y^{4}t^{8}+801y^{2}w^{2}t^{8}+1008y^{2}wt^{9}+459y^{2}t^{10}-18w^{2}t^{10}-18wt^{11}-t^{12}}{t^{2}y^{6}(3xy^{3}+xyt^{2}+3y^{4}-2y^{2}w^{2}+2y^{2}wt-4y^{2}t^{2}+w^{2}t^{2})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 48.96.5.oq.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ -9X^{6}+3X^{5}Z+27XY^{4}Z+14X^{4}Z^{2}-9Y^{4}Z^{2}-2X^{3}Z^{3}-5X^{2}Z^{4}-XZ^{5} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.1-24.ir.1.13 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{4}$ |
| 48.48.1-48.b.1.11 | $48$ | $4$ | $4$ | $1$ | $0$ | $1^{4}$ |
| 48.96.1-24.ir.1.17 | $48$ | $2$ | $2$ | $1$ | $0$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.384.9-48.hr.1.35 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.jb.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.384.9-48.mk.1.8 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.mq.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.384.9-48.bex.1.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bex.2.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bex.3.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bex.4.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bey.1.10 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bey.2.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bey.3.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bey.4.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bff.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bff.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bff.3.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bff.4.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfg.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfg.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfg.3.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfg.4.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfp.1.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfp.2.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfq.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfq.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfr.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfr.2.18 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfs.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfs.2.4 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bft.1.4 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bft.2.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfu.1.10 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfu.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfv.1.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfv.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfw.1.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfw.2.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfz.1.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfz.2.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfz.3.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bfz.4.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bga.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bga.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bga.3.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bga.4.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bgd.1.18 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bgd.2.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bgd.3.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bgd.4.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bge.1.10 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bge.2.11 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bge.3.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bge.4.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bgj.1.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.bgm.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.384.9-48.bgn.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.bgq.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.576.17-48.cae.1.5 | $48$ | $3$ | $3$ | $17$ | $3$ | $1^{12}$ |
| 240.384.9-240.flb.1.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.flc.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.flf.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.flg.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmh.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmh.2.4 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmh.3.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmh.4.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmi.1.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmi.2.7 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmi.3.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmi.4.11 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmp.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmp.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmp.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmp.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmq.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmq.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmq.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmq.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmz.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fmz.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fna.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fna.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnb.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnb.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnc.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnc.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnd.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnd.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fne.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fne.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnf.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnf.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fng.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fng.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnj.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnj.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnj.3.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnj.4.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnk.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnk.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnk.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnk.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnn.1.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnn.2.37 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnn.3.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fnn.4.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fno.1.34 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fno.2.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fno.3.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fno.4.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fol.1.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fom.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fop.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.foq.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |