Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $96$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2\cdot3^{2}\cdot4\cdot6\cdot8^{2}\cdot12\cdot24^{2}$ | Cusp orbits | $1^{4}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24X3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.3.6871 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&35\\12&13\end{bmatrix}$, $\begin{bmatrix}11&38\\12&5\end{bmatrix}$, $\begin{bmatrix}35&46\\24&31\end{bmatrix}$, $\begin{bmatrix}37&16\\36&11\end{bmatrix}$, $\begin{bmatrix}41&46\\24&41\end{bmatrix}$, $\begin{bmatrix}43&27\\24&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.gj.3 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: | $2^{13}\cdot3^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 24.2.a.a, 96.2.f.a |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ 2 x^{4} - x^{2} y^{2} - 2 x^{2} y z + 2 x^{2} z^{2} + y^{3} z - y^{2} z^{2} - 2 y z^{3} $ |
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.
| Canonical model |
|---|
| $(0:2:1)$, $(0:-1:1)$, $(0:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{524286x^{2}y^{22}-5241420x^{2}y^{21}z+19551348x^{2}y^{20}z^{2}+11653632x^{2}y^{19}z^{3}-1082912184x^{2}y^{18}z^{4}+14659908336x^{2}y^{17}z^{5}-121175408592x^{2}y^{16}z^{6}+662851102464x^{2}y^{15}z^{7}-2494380457152x^{2}y^{14}z^{8}+6579865104768x^{2}y^{13}z^{9}-12207364753536x^{2}y^{12}z^{10}+15742628951040x^{2}y^{11}z^{11}-13895200726272x^{2}y^{10}z^{12}+8441478517248x^{2}y^{9}z^{13}-4080948051456x^{2}y^{8}z^{14}+2063571849216x^{2}y^{7}z^{15}-996421077504x^{2}y^{6}z^{16}+327255745536x^{2}y^{5}z^{17}-69514509312x^{2}y^{4}z^{18}+7769899008x^{2}y^{3}z^{19}-221681664x^{2}y^{2}z^{20}+2764800x^{2}yz^{21}+258048x^{2}z^{22}+y^{24}-525018y^{23}z+5952966y^{22}z^{2}-42526972y^{21}z^{3}+556133244y^{20}z^{4}-7211776296y^{19}z^{5}+60301636408y^{18}z^{6}-329548113072y^{17}z^{7}+1222819449120y^{16}z^{8}-3102152611264y^{15}z^{9}+5197129955904y^{14}z^{10}-4983317317248y^{13}z^{11}+798201255040y^{12}z^{12}+4458640007424y^{11}z^{13}-5784413473536y^{10}z^{14}+3315307662848y^{9}z^{15}-1131136208640y^{8}z^{16}+587685367296y^{7}z^{17}-341545144832y^{6}z^{18}+76955661312y^{5}z^{19}-5180150784y^{4}z^{20}-11044864y^{3}z^{21}-46147584y^{2}z^{22}+1314816yz^{23}+262144z^{24}}{zy(y-2z)^{3}(y+z)^{3}(2x^{2}y^{14}+36x^{2}y^{13}z+292x^{2}y^{12}z^{2}+1344x^{2}y^{11}z^{3}+3272x^{2}y^{10}z^{4}-784x^{2}y^{9}z^{5}-42096x^{2}y^{8}z^{6}+337920x^{2}y^{7}z^{7}+111456x^{2}y^{6}z^{8}+5312x^{2}y^{5}z^{9}-93504x^{2}y^{4}z^{10}-18944x^{2}y^{3}z^{11}-4224x^{2}y^{2}z^{12}+4352x^{2}yz^{13}-256x^{2}z^{14}-y^{16}-18y^{15}z-146y^{14}z^{2}-676y^{13}z^{3}-1716y^{12}z^{4}-344y^{11}z^{5}+17064y^{10}z^{6}+80496y^{9}z^{7}-330096y^{8}z^{8}+104864y^{7}z^{9}+221472y^{6}z^{10}+343360y^{5}z^{11}+167744y^{4}z^{12}+94080y^{3}z^{13}+24448y^{2}z^{14}+8448yz^{15})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.96.1-24.ir.1.1 | $48$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 48.96.1-24.ir.1.17 | $48$ | $2$ | $2$ | $1$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.384.5-24.gd.4.14 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.ge.4.14 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gf.4.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gg.4.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gl.4.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gm.4.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gn.4.14 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.go.4.14 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.7-24.dr.4.15 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.dv.2.15 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.dy.4.13 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.ed.3.14 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.ej.2.15 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.en.2.14 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-24.ep.4.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.es.2.14 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-24.ey.4.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ey.4.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.ez.4.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fa.4.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fa.4.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fb.4.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fc.4.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fe.4.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fg.4.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fg.4.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fh.4.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fi.4.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fi.4.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fj.4.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fk.4.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fm.4.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gm.3.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.go.1.2 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hg.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hi.1.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hk.4.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hm.3.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ie.4.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ig.3.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bao.3.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.baq.4.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbi.3.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbk.4.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfe.1.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfg.3.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfy.1.5 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bga.3.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bhi.4.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhk.4.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhm.4.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bho.4.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhq.4.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhs.4.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhu.4.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhw.4.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.576.13-24.lk.2.4 | $48$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 240.384.5-120.bfb.3.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfc.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfd.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfe.3.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfr.3.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfs.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bft.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfu.3.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-120.lj.4.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.lp.3.26 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.lv.2.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mb.3.22 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mn.4.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mt.2.20 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mz.2.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nf.2.20 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.ns.3.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nt.1.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nu.3.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nv.4.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.oi.3.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.oj.1.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.ok.1.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.ol.3.4 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uu.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uw.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uy.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.va.3.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vk.3.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vm.1.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vo.1.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vq.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xw.4.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xy.3.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zg.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zi.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zs.4.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zu.3.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbc.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbe.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.9-240.fsw.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fsy.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fug.3.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fui.4.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fus.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fuu.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwc.1.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwe.3.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdi.4.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdk.3.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdm.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdo.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdy.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gea.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gec.1.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gee.3.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |