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: | 48.192.3.6968 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&10\\24&17\end{bmatrix}$, $\begin{bmatrix}5&1\\36&41\end{bmatrix}$, $\begin{bmatrix}11&25\\12&23\end{bmatrix}$, $\begin{bmatrix}17&18\\0&29\end{bmatrix}$, $\begin{bmatrix}23&16\\36&1\end{bmatrix}$, $\begin{bmatrix}25&31\\0&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.gj.4 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 |
|---|
| $(0:1:1)$, $(0:-2:1)$, $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{63x^{2}y^{22}+1350x^{2}y^{21}z-216486x^{2}y^{20}z^{2}+15175584x^{2}y^{19}z^{3}-271541052x^{2}y^{18}z^{4}+2556685512x^{2}y^{17}z^{5}-15569079336x^{2}y^{16}z^{6}+64486620288x^{2}y^{15}z^{7}-255059253216x^{2}y^{14}z^{8}+1055184814656x^{2}y^{13}z^{9}-3473800181568x^{2}y^{12}z^{10}+7871314475520x^{2}y^{11}z^{11}-12207364753536x^{2}y^{10}z^{12}+13159730209536x^{2}y^{9}z^{13}-9977521828608x^{2}y^{8}z^{14}+5302808819712x^{2}y^{7}z^{15}-1938806537472x^{2}y^{6}z^{16}+469117066752x^{2}y^{5}z^{17}-69306379776x^{2}y^{4}z^{18}+1491664896x^{2}y^{3}z^{19}+5005145088x^{2}y^{2}z^{20}-2683607040x^{2}yz^{21}+536868864x^{2}z^{22}-32y^{24}-321y^{23}z+22533y^{22}z^{2}+10786y^{21}z^{3}+10117482y^{20}z^{4}-300608052y^{19}z^{5}+2668321444y^{18}z^{6}-9182583864y^{17}z^{7}+35348006520y^{16}z^{8}-207206728928y^{15}z^{9}+723051684192y^{14}z^{10}-1114660001856y^{13}z^{11}-399100627520y^{12}z^{12}+4983317317248y^{11}z^{13}-10394259911808y^{10}z^{14}+12408610445056y^{9}z^{15}-9782555592960y^{8}z^{16}+5272769809152y^{7}z^{17}-1929652365056y^{6}z^{18}+461553682944y^{5}z^{19}-71185055232y^{4}z^{20}+10886904832y^{3}z^{21}-3047918592y^{2}z^{22}+537618432yz^{23}-2048z^{24}}{zy(y-z)^{3}(y+2z)^{3}(x^{2}y^{14}-34x^{2}y^{13}z+66x^{2}y^{12}z^{2}+592x^{2}y^{11}z^{3}+5844x^{2}y^{10}z^{4}-664x^{2}y^{9}z^{5}-27864x^{2}y^{8}z^{6}-168960x^{2}y^{7}z^{7}+42096x^{2}y^{6}z^{8}+1568x^{2}y^{5}z^{9}-13088x^{2}y^{4}z^{10}-10752x^{2}y^{3}z^{11}-4672x^{2}y^{2}z^{12}-1152x^{2}yz^{13}-128x^{2}z^{14}+33y^{15}z+191y^{14}z^{2}+1470y^{13}z^{3}+5242y^{12}z^{4}+21460y^{11}z^{5}+27684y^{10}z^{6}+26216y^{9}z^{7}-165048y^{8}z^{8}+80496y^{7}z^{9}+34128y^{6}z^{10}-1376y^{5}z^{11}-13728y^{4}z^{12}-10816y^{3}z^{13}-4672y^{2}z^{14}-1152yz^{15}-128z^{16})}$ |
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.1.17 | $48$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 48.96.1-24.ir.1.26 | $48$ | $2$ | $2$ | $1$ | $0$ | $2$ |
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.3.14 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.ge.2.15 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gf.2.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gg.3.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gl.3.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gm.2.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gn.2.15 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.go.3.14 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.7-24.dr.3.12 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.dv.2.15 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.dy.2.18 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.ed.4.16 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.ej.1.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.en.1.14 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-24.ep.2.12 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.es.3.16 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-24.ey.3.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ey.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.ez.2.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fa.2.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fa.2.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fb.3.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fc.2.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fe.2.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fg.3.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fg.3.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fh.2.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fi.2.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fi.2.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fj.3.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fk.2.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fm.3.9 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gm.1.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.go.2.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hg.1.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hi.2.5 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hk.2.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hm.4.1 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ie.2.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ig.4.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bao.4.5 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.baq.2.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbi.4.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbk.2.1 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfe.2.3 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfg.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfy.2.1 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bga.1.1 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bhi.3.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhk.2.11 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhm.2.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bho.3.9 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhq.2.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhs.2.5 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhu.2.7 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhw.2.7 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.576.13-24.lk.1.2 | $48$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2\cdot4$ |
| 240.384.5-120.bfb.4.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfc.3.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfd.3.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfe.4.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfr.4.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfs.3.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bft.3.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfu.4.14 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-120.lj.2.8 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.lp.4.25 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.lv.4.28 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mb.4.25 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mn.2.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mt.3.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mz.4.28 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nf.3.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.ns.4.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nt.3.10 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nu.1.14 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nv.3.14 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.oi.4.14 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.oj.3.14 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.ok.3.10 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.ol.4.6 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uu.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uw.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uy.3.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.va.4.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vk.4.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vm.3.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vo.3.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vq.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xw.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.xy.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zg.2.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zi.4.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zs.2.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zu.4.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbc.2.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bbe.4.17 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.9-240.fsw.4.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fsy.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fug.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fui.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fus.3.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fuu.1.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwc.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fwe.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdi.3.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdk.1.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdm.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdo.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gdy.4.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gea.3.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gec.3.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gee.4.17 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |