Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot16^{4}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16M1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.1.2317 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&44\\40&7\end{bmatrix}$, $\begin{bmatrix}13&39\\24&35\end{bmatrix}$, $\begin{bmatrix}15&25\\16&21\end{bmatrix}$, $\begin{bmatrix}19&9\\40&13\end{bmatrix}$, $\begin{bmatrix}35&27\\24&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 16.96.1.m.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $4$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x $ |
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.
| Weierstrass model |
|---|
| $(1:0:1)$, $(-1:0:1)$, $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{720x^{2}y^{30}-395719188x^{2}y^{28}z^{2}+163957645440x^{2}y^{26}z^{4}-6180704673330x^{2}y^{24}z^{6}+68757623808480x^{2}y^{22}z^{8}-331581463950951x^{2}y^{20}z^{10}+827155650102480x^{2}y^{18}z^{12}-1179864481855275x^{2}y^{16}z^{14}+1016842368486960x^{2}y^{14}z^{16}-537716814073464x^{2}y^{12}z^{18}+169446488656320x^{2}y^{10}z^{20}-31281499154565x^{2}y^{8}z^{22}+4553235595440x^{2}y^{6}z^{24}-188743500741x^{2}y^{4}z^{26}+3019898160x^{2}y^{2}z^{28}-16777215x^{2}z^{30}-179272xy^{30}z+4531118400xy^{28}z^{3}-667384554546xy^{26}z^{5}+16119283078080xy^{24}z^{7}-137508659933832xy^{22}z^{9}+551380638884400xy^{20}z^{11}-1203935362225097xy^{18}z^{13}+1550846607837120xy^{16}z^{15}-1228922323844544xy^{14}z^{17}+611157563850240xy^{12}z^{19}-192642543358905xy^{10}z^{21}+35809977860640xy^{8}z^{23}-1693911792700xy^{6}z^{25}+30198989520xy^{4}z^{27}-184549377xy^{2}z^{29}-y^{32}+16948800y^{30}z^{2}-32028588168y^{28}z^{4}+2120901490080y^{26}z^{6}-32251300021148y^{24}z^{8}+191199903598080y^{22}z^{10}-554413269723614y^{20}z^{12}+883315939052880y^{18}z^{14}-825468334146204y^{16}z^{16}+468607928904000y^{14}z^{18}-162705985921424y^{12}z^{20}+31285229236560y^{10}z^{22}-1505712675646y^{8}z^{24}+27196030080y^{6}z^{26}-167951418y^{4}z^{28}+720y^{2}z^{30}-z^{32}}{y^{2}(x^{2}y^{28}+1304x^{2}y^{26}z^{2}-46040x^{2}y^{24}z^{4}+345488x^{2}y^{22}z^{6}+3274070x^{2}y^{20}z^{8}-38696304x^{2}y^{18}z^{10}-24569439x^{2}y^{16}z^{12}+593608208x^{2}y^{14}z^{14}+1776151887x^{2}y^{12}z^{16}+2348297576x^{2}y^{10}z^{18}+1759247244x^{2}y^{8}z^{20}+801111000x^{2}y^{6}z^{22}+217054999x^{2}y^{4}z^{24}+25165800x^{2}y^{2}z^{26}+4194303x^{2}z^{28}+24xy^{28}z+3300xy^{26}z^{3}-142592xy^{24}z^{5}+1601964xy^{22}z^{7}-891344xy^{20}z^{9}-65957489xy^{18}z^{11}+68035104xy^{16}z^{13}+1072372204xy^{14}z^{15}+2560087816xy^{12}z^{17}+3017278816xy^{10}z^{19}+2084570176xy^{8}z^{21}+880804074xy^{6}z^{23}+226492440xy^{4}z^{25}+41943041xy^{2}z^{27}+244y^{28}z^{2}-3168y^{26}z^{4}-129278y^{24}z^{6}+3142880y^{22}z^{8}-17161248y^{20}z^{10}-52203928y^{18}z^{12}+295715913y^{16}z^{14}+1197368928y^{14}z^{16}+1801129060y^{12}z^{18}+1459605056y^{10}z^{20}+697303945y^{8}z^{22}+201327632y^{6}z^{24}+37748968y^{4}z^{26}+24y^{2}z^{28}+z^{30})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.0-8.n.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.j.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.j.1.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-8.n.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.u.2.1 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.u.2.7 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.v.2.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.v.2.10 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.1-16.h.1.5 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.h.1.6 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.u.2.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.u.2.7 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.v.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.v.2.11 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.384.5-16.bu.1.5 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-16.bw.1.7 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-16.bx.1.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-16.bx.3.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-16.by.1.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-16.by.2.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.gr.1.8 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.gs.1.8 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.gt.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.gt.3.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.gu.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.gu.2.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.576.17-48.md.2.18 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.ov.2.18 | $48$ | $4$ | $4$ | $17$ | $0$ | $1^{8}\cdot2^{4}$ |
| 96.384.5-32.n.1.3 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-32.p.1.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-32.t.2.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-32.u.1.5 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-32.v.1.4 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-32.v.2.3 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-32.bc.1.4 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-32.be.1.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bf.1.19 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bh.1.20 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bp.2.6 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.bq.1.10 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.br.1.8 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.br.2.8 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.co.1.6 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.5-96.cq.1.2 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.384.9-32.bh.3.6 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-32.bh.4.6 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.cx.3.8 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 96.384.9-96.cx.4.8 | $96$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.5-80.lp.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.lq.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.lr.1.3 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.lr.3.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.ls.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.ls.2.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bri.1.19 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brk.1.19 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brl.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brl.3.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brm.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.brm.2.10 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |