Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $192$ | ||
| 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: | 48.192.5.469 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}7&18\\24&35\end{bmatrix}$, $\begin{bmatrix}7&19\\0&5\end{bmatrix}$, $\begin{bmatrix}7&26\\36&37\end{bmatrix}$, $\begin{bmatrix}35&17\\24&41\end{bmatrix}$, $\begin{bmatrix}41&26\\24&1\end{bmatrix}$, $\begin{bmatrix}41&32\\36&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 48.96.5.mh.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: | $2^{27}\cdot3^{3}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}$ |
| Newforms: | 24.2.a.a, 64.2.a.a$^{2}$, 192.2.a.a, 192.2.a.c |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x w + y w + y t $ |
| $=$ | $x^{2} - x y - 2 y^{2} + w^{2} - w t$ | |
| $=$ | $2 x w - x t - y w - y t + 2 z^{2}$ |
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} + 12 x y^{4} z - x z^{5} + 4 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.mh.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{6}+3X^{5}Z+12XY^{4}Z-14X^{4}Z^{2}+4Y^{4}Z^{2}-2X^{3}Z^{3}+5X^{2}Z^{4}-XZ^{5} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 16.48.1-16.a.1.14 | $16$ | $4$ | $4$ | $1$ | $0$ | $1^{4}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.1-16.a.1.14 | $16$ | $4$ | $4$ | $1$ | $0$ | $1^{4}$ |
| 24.96.1-24.ir.1.31 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{4}$ |
| 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.hn.1.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.id.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.mg.1.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.mm.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 48.384.9-48.zp.1.17 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.384.9-48.zq.1.5 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.384.9-48.zt.1.3 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.384.9-48.zu.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 48.384.9-48.baf.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.baf.2.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.baf.3.13 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.baf.4.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bag.1.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bag.2.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bag.3.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bag.4.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.ban.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.ban.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.ban.3.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.ban.4.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bao.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bao.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bao.3.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bao.4.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bar.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bar.2.4 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bas.1.18 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bas.2.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bat.1.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bat.2.4 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bau.1.18 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bau.2.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bav.1.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bav.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.baw.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.baw.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bax.1.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bax.2.3 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bay.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bay.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bbh.1.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bbh.2.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bbh.3.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bbh.4.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bbi.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bbi.2.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bbi.3.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bbi.4.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $2^{2}$ |
| 48.384.9-48.bbl.1.25 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bbl.2.25 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bbl.3.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bbl.4.17 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bbm.1.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bbm.2.21 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bbm.3.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.384.9-48.bbm.4.9 | $48$ | $2$ | $2$ | $9$ | $1$ | $4$ |
| 48.576.17-48.ze.1.33 | $48$ | $3$ | $3$ | $17$ | $3$ | $1^{12}$ |
| 240.384.9-240.ein.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.eio.1.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.eir.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.eis.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ejd.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.eje.1.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ejh.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.eji.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ejt.1.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ejt.2.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ejt.3.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ejt.4.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.eju.1.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.eju.2.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.eju.3.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.eju.4.10 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekb.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekb.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekb.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekb.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekc.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekc.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekc.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekc.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekf.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekf.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekg.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekg.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekh.1.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekh.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.eki.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.eki.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekj.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekj.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekk.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekk.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekl.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekl.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekm.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekm.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekv.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekv.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekv.3.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekv.4.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekw.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekw.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekw.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekw.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekz.1.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekz.2.37 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekz.3.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ekz.4.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ela.1.35 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ela.2.37 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ela.3.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ela.4.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |