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: | 24Y3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.3.622 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&46\\12&7\end{bmatrix}$, $\begin{bmatrix}7&29\\0&5\end{bmatrix}$, $\begin{bmatrix}19&21\\24&5\end{bmatrix}$, $\begin{bmatrix}25&7\\12&13\end{bmatrix}$, $\begin{bmatrix}29&20\\24&25\end{bmatrix}$, $\begin{bmatrix}35&39\\24&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.gf.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: | $2^{13}\cdot3^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 24.2.a.a, 96.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ - x^{3} z + x^{2} y^{2} + x z^{3} + 2 y^{4} + y^{2} z^{2} $ |
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 |
|---|
| $(1:0:1)$, $(0:0:1)$, $(-1:0:1)$, $(1:0:0)$ |
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{x^{24}-708x^{23}z+172116x^{22}z^{2}-15534740x^{21}z^{3}+296705322x^{20}z^{4}-3050610540x^{19}z^{5}+23919719348x^{18}z^{6}-172917010332x^{17}z^{7}+1222145208999x^{16}z^{8}-8548431082856x^{15}z^{9}+59575015665768x^{14}z^{10}-415026950151816x^{13}z^{11}+2889882985019180x^{12}z^{12}-20003222458287256x^{11}z^{13}+136086081493720392x^{10}z^{14}-893678695642968056x^{9}z^{15}+5509313012986118111x^{8}z^{16}-30445070335735881492x^{7}z^{17}+137939075840609738692x^{6}z^{18}-408222258336931878180x^{5}z^{19}+178910324986689457738x^{4}z^{20}+1383332585634595150660x^{3}z^{21}-322497750626045843836x^{2}z^{22}+4696545239040xy^{22}z-92345415557120xy^{20}z^{3}+1361438510940160xy^{18}z^{5}-18164631480483840xy^{16}z^{7}+221724283157954560xy^{14}z^{9}-2461214014448844800xy^{12}z^{11}+24033878260670955520xy^{10}z^{13}-187687912911954247680xy^{8}z^{15}+862328430601509642240xy^{6}z^{17}+133001631891311206400xy^{4}z^{19}-943751151292285747200xy^{2}z^{21}-943751151292366324268xz^{23}+782757785600y^{24}-28178895421440y^{22}z^{2}+437675871109120y^{20}z^{4}-6093651481518080y^{18}z^{6}+77261542687395840y^{16}z^{8}-895745251472998400y^{14}z^{10}+9309993301810872320y^{12}z^{12}-81650533729180794880y^{10}z^{14}+504385072705139281920y^{8}z^{16}-968679385139214827520y^{6}z^{18}-2508755703250972508160y^{4}z^{20}-943751151292366315520y^{2}z^{22}+729z^{24}}{z(x^{23}+34x^{22}z+464x^{21}z^{2}+3142x^{20}z^{3}+9827x^{19}z^{4}+2632x^{18}z^{5}-61104x^{17}z^{6}-93640x^{16}z^{7}+149290x^{15}z^{8}+316988x^{14}z^{9}-221840x^{13}z^{10}-470700x^{12}z^{11}+252198x^{11}z^{12}+54696x^{10}z^{13}+1713008x^{9}z^{14}-11749608x^{8}z^{15}+64587813x^{7}z^{16}-290882126x^{6}z^{17}+860366848x^{5}z^{18}-377280234x^{4}z^{19}-2917060313x^{3}z^{20}+680098816x^{2}z^{21}+2048xy^{22}-13312xy^{20}z^{2}-185344xy^{18}z^{4}-423936xy^{16}z^{6}-628736xy^{14}z^{8}+7004160xy^{12}z^{10}-48295936xy^{10}z^{12}+395491328xy^{8}z^{14}-1820209152xy^{6}z^{16}-281111552xy^{4}z^{18}+1990263808xy^{2}z^{20}+1990263808xz^{22}+24576y^{22}z+157696y^{20}z^{3}+650240y^{18}z^{5}+1521664y^{16}z^{7}+4896768y^{14}z^{9}-17766400y^{12}z^{11}+171151360y^{10}z^{13}-1065129984y^{8}z^{15}+2042488832y^{6}z^{17}+5290692608y^{4}z^{19}+1990263808y^{2}z^{21})}$ |
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 |
| 16.48.0-8.ba.2.4 | $16$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.48.0-8.ba.2.4 | $16$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 48.96.1-24.ir.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 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.fz.2.15 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.fz.4.15 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gd.1.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gd.3.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gh.2.15 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gh.4.15 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gl.2.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.5-24.gl.4.16 | $48$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 48.384.7-24.dj.1.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.du.2.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.ea.2.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.eb.1.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.eh.1.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-24.el.1.2 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-24.eo.2.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ep.3.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.ep.4.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.er.1.3 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.et.1.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.et.2.17 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.eu.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.eu.2.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.ey.2.11 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.ey.4.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fc.1.5 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fc.2.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fg.1.10 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-24.fg.3.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fn.3.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fn.4.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fr.2.2 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.fr.4.3 | $48$ | $2$ | $2$ | $7$ | $0$ | $2^{2}$ |
| 48.384.7-48.gr.2.6 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.gt.2.18 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.gz.1.19 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hb.1.11 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{2}\cdot2$ |
| 48.384.7-48.hn.2.4 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ht.2.18 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.hv.1.19 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.7-48.ib.1.7 | $48$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2$ |
| 48.384.9-48.bar.1.7 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bax.1.19 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.baz.2.18 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bbf.2.4 | $48$ | $2$ | $2$ | $9$ | $0$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfj.1.11 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfl.1.19 | $48$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2$ |
| 48.384.9-48.bfr.2.18 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bft.2.6 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2$ |
| 48.384.9-48.bgz.2.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bgz.4.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhd.3.2 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhd.4.3 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhx.1.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bhx.2.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bib.3.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.384.9-48.bib.4.17 | $48$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 48.576.13-24.ll.2.4 | $48$ | $3$ | $3$ | $13$ | $0$ | $1^{4}\cdot2^{3}$ |
| 240.384.5-120.bet.2.10 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bet.4.10 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bex.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bex.4.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfj.1.10 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfj.3.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfn.1.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-120.bfn.3.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.7-120.lf.1.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.ll.2.13 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.lr.1.11 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.lx.1.23 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mj.1.13 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mp.1.21 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.mv.1.27 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nb.1.27 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nk.1.21 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.nk.2.19 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.no.1.13 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.no.3.11 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.oa.1.21 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.oa.2.21 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.oe.2.13 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-120.oe.4.13 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uh.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.uh.2.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ul.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.ul.2.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vv.2.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vv.4.3 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vz.1.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.vz.3.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.yf.2.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.yh.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.yv.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.yx.2.9 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.zz.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.baf.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bap.1.33 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.7-240.bav.2.5 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.384.9-240.ftd.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ftj.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ftt.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.ftz.2.5 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fvb.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fvd.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fvr.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.fvt.2.9 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcv.1.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcv.3.2 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcz.2.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gcz.4.3 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gej.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gej.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gen.1.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.gen.2.33 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |