Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot16^{4}$ | Cusp orbits | $2^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16M1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.1.1842 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}1&38\\24&13\end{bmatrix}$, $\begin{bmatrix}7&22\\40&3\end{bmatrix}$, $\begin{bmatrix}9&46\\44&3\end{bmatrix}$, $\begin{bmatrix}21&8\\16&17\end{bmatrix}$, $\begin{bmatrix}21&14\\40&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 16.96.1.c.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $64$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - x y - 2 x w + y w - 2 z^{2} + w^{2} $ |
| $=$ | $2 x^{2} - x y - 2 x w + y^{2} - y w + 2 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 2 x^{2} y^{2} - 6 x^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8}\cdot\frac{1993605120xz^{22}w-10038607872xz^{20}w^{3}-187870199808xz^{18}w^{5}-43214462976xz^{16}w^{7}+1512342716416xz^{14}w^{9}+1111590584320xz^{12}w^{11}-3401923123200xz^{10}w^{13}-2972840902144xz^{8}w^{15}+2808085754048xz^{6}w^{17}+2610884446800xz^{4}w^{19}-763677556692xz^{2}w^{21}-730920968190xw^{23}+8355250176y^{2}z^{20}w^{2}-376089137152y^{2}z^{16}w^{6}+2361846308864y^{2}z^{12}w^{10}-4681553935872y^{2}z^{8}w^{14}+3574958589264y^{2}z^{4}w^{18}-921840357375y^{2}w^{22}+10038607872yz^{20}w^{3}+43214462976yz^{16}w^{7}-1111590584320yz^{12}w^{11}+2972840902144yz^{8}w^{15}-2610884446800yz^{4}w^{19}+730920968190yw^{23}-287965184z^{24}+31427395584z^{22}w^{2}+92518072320z^{20}w^{4}-1316486348800z^{18}w^{6}-713092941824z^{16}w^{8}+7935042519040z^{14}w^{10}+156629397504z^{12}w^{12}-15324292620288z^{10}w^{14}+3114492709088z^{8}w^{16}+11491748603008z^{6}w^{18}-3765712539708z^{4}w^{20}-2923683872808z^{2}w^{22}+1191841146877w^{24}}{z^{16}(960xz^{6}w+784xz^{4}w^{3}-2228xz^{2}w^{5}-2142xw^{7}+1808y^{2}z^{4}w^{2}-2703y^{2}w^{6}-784yz^{4}w^{3}+2142yw^{7}-288z^{8}+6272z^{6}w^{2}+132z^{4}w^{4}-8584z^{2}w^{6}+3493w^{8})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 16.96.1.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 4z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-2X^{2}Y^{2}-6X^{2}Z^{2}+Z^{4} $ |
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.l.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.d.2.3 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.d.2.12 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-8.l.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.z.2.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.z.2.5 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.bb.2.2 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-16.bb.2.7 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.1-16.a.1.9 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.a.1.12 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.r.2.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.r.2.7 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.t.2.4 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 48.96.1-16.t.2.5 | $48$ | $2$ | $2$ | $1$ | $0$ | 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.m.2.9 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-16.n.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-16.z.1.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-16.bc.1.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-16.bd.1.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-16.bg.1.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.ef.1.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 48.384.5-48.eg.2.8 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.384.5-48.ff.1.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.fi.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.fj.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.fm.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.576.17-48.eg.2.33 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-48.ih.2.26 | $48$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 240.384.5-80.hl.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.hm.1.5 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.jj.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.jq.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.jr.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.jy.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bgl.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bgm.2.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bnh.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bno.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bnp.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.bnw.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |