Invariants
| Level: | $48$ | $\SL_2$-level: | $16$ | Newform level: | $576$ | ||
| 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 | $4^{8}\cdot8^{8}$ | Cusp orbits | $1^{4}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.192.1.674 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&32\\32&9\end{bmatrix}$, $\begin{bmatrix}23&44\\40&39\end{bmatrix}$, $\begin{bmatrix}25&34\\24&41\end{bmatrix}$, $\begin{bmatrix}31&20\\24&31\end{bmatrix}$, $\begin{bmatrix}43&0\\16&41\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.ch.1 for the level structure with $-I$) |
| Cyclic 48-isogeny field degree: | $8$ |
| Cyclic 48-torsion field degree: | $128$ |
| Full 48-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 36x $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2\cdot3^2}\cdot\frac{225799488x^{2}y^{28}z^{2}+182402443687495680x^{2}y^{24}z^{6}+456294071750523443675136x^{2}y^{20}z^{10}+75672612155865275602179194880x^{2}y^{16}z^{14}+1672716212006158924993632229392384x^{2}y^{12}z^{18}+10027684523995265550439026536994570240x^{2}y^{8}z^{22}+20184231549251471851472074073223846690816x^{2}y^{4}z^{26}+10463509855324846890187493327062814467031040x^{2}z^{30}+25632xy^{30}z+545473143599616xy^{26}z^{5}+5256292210903031611392xy^{22}z^{9}+2152169304984205947376238592xy^{18}z^{13}+101923789170541508083401898328064xy^{14}z^{17}+1007691605597443459543157619441008640xy^{10}z^{21}+3270594849088113290542405773904419225600xy^{6}z^{25}+3197183774796089395655041850743370381524992xy^{2}z^{29}+y^{32}+740315386368y^{28}z^{4}+34326606459325759488y^{24}z^{8}+27526047194073451149656064y^{20}z^{12}+1893005705099975015015418691584y^{16}z^{16}+21633194742292240823554217003188224y^{12}z^{20}+76834732705235098164857401404341354496y^{8}z^{24}+80737299470923390078318406624830436671488y^{4}z^{28}+22452257707354557240087211123792674816z^{32}}{zy^{4}(1116x^{2}y^{24}z-1676256768x^{2}y^{20}z^{5}+15210954436018176x^{2}y^{16}z^{9}+6694491914607230189568x^{2}y^{12}z^{13}-2381072387808954141804331008x^{2}y^{8}z^{17}-221149743653719175026868465172480x^{2}y^{4}z^{21}-389356127390779679171781053330227200x^{2}z^{25}-xy^{26}+82114560xy^{22}z^{4}+117994663305216xy^{18}z^{8}-146075310635489427456xy^{14}z^{12}+24529448761825798517686272xy^{10}z^{16}-10547473850294701588602440122368xy^{6}z^{20}-97339134991942904698300725052047360xy^{2}z^{24}-492480y^{24}z^{3}-1612995710976y^{20}z^{7}-1688465311821398016y^{16}z^{11}+794185880972071243087872y^{12}z^{15}-167424437795971522382518026240y^{8}z^{19}-2403423143727071050122484500135936y^{4}z^{23}-13367494538843734067838845976576z^{27})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 16.96.0-8.l.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-8.l.1.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-24.bb.2.4 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.0-24.bb.2.8 | $48$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 48.96.1-24.bv.1.2 | $48$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 48.96.1-24.bv.1.7 | $48$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 48.384.5-48.br.1.3 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.384.5-48.br.1.8 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.384.5-48.cl.1.10 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.cl.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.dl.1.15 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dl.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dm.1.14 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dm.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dp.1.15 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dp.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dq.1.12 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dq.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dt.1.14 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dt.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.du.1.14 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.du.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dx.1.12 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dx.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dy.1.14 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.dy.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 48.384.5-48.eg.2.4 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.384.5-48.eg.2.8 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.384.5-48.eo.1.14 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.eo.1.16 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.576.17-24.bln.2.16 | $48$ | $3$ | $3$ | $17$ | $2$ | $1^{8}\cdot2^{4}$ |
| 48.768.17-24.oi.2.17 | $48$ | $4$ | $4$ | $17$ | $3$ | $1^{8}\cdot2^{4}$ |
| 240.384.5-240.tr.1.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.tr.1.15 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ud.1.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ud.1.29 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ux.1.24 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ux.1.31 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.uy.1.26 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.uy.1.32 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.vj.1.24 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.vj.1.31 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.vk.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.vk.1.32 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.vn.2.26 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.vn.2.32 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.vo.2.28 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.vo.2.31 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.vz.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.vz.1.32 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.wa.1.24 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.wa.1.31 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.xz.2.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.xz.2.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.yl.1.19 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.yl.1.31 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |