Invariants
| Level: | $16$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $96$ | $\PSL_2$-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 | $4^{8}\cdot8^{8}$ | Cusp orbits | $4^{2}\cdot8$ | ||
| 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: | 8K1 |
| Rouse and Zureick-Brown (RZB) label: | X484 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.1.388 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}9&14\\10&7\end{bmatrix}$, $\begin{bmatrix}11&11\\14&5\end{bmatrix}$, $\begin{bmatrix}13&8\\8&13\end{bmatrix}$, $\begin{bmatrix}13&10\\2&11\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.(C_2\times D_4)$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 16-isogeny field degree: | $4$ |
| Cyclic 16-torsion field degree: | $32$ |
| Full 16-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x w + y^{2} + 2 y z - z^{2} $ |
| $=$ | $x^{2} + 2 y^{2} + 2 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 4 x^{3} y + 2 x^{2} y^{2} + 8 x^{2} z^{2} + 4 x y^{3} + y^{4} + 8 y^{2} z^{2} - 8 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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^8\,\frac{z^{3}(76230xyz^{18}w-4686580xyz^{16}w^{3}+73765926xyz^{14}w^{5}-461432704xyz^{12}w^{7}+1355509664xyz^{10}w^{9}-1990725984xyz^{8}w^{11}+1454706704xyz^{6}w^{13}-488602240xyz^{4}w^{15}+60296400xyz^{2}w^{17}-1241728xyw^{19}-41376xz^{19}w+3892276xz^{17}w^{3}-86258040xz^{15}w^{5}+730256000xz^{13}w^{7}-2862186368xz^{11}w^{9}+5638855248xz^{9}w^{11}-5681466832xz^{7}w^{13}+2803363456xz^{5}w^{15}-590447136xz^{3}w^{17}+36005888xzw^{19}-13860yz^{20}+2714520yz^{18}w^{2}-76100176yz^{16}w^{4}+722725336yz^{14}w^{6}-3028107168yz^{12}w^{8}+6236520560yz^{10}w^{10}-6516537872yz^{8}w^{12}+3355680192yz^{6}w^{14}-763690624yz^{4}w^{16}+58039168yz^{2}w^{18}-558144yw^{20}+5741z^{21}-1244952z^{19}w^{2}+38349628z^{17}w^{4}-400459408z^{15}w^{6}+1867019136z^{13}w^{8}-4402542512z^{11}w^{10}+5570322680z^{9}w^{12}-3854385216z^{7}w^{14}+1426722592z^{5}w^{16}-254460288z^{3}w^{18}+15272832zw^{20})}{w^{4}(85608xyz^{17}w-3457424xyz^{15}w^{3}+34820436xyz^{13}w^{5}-135858632xyz^{11}w^{7}+242409476xyz^{9}w^{9}-209185440xyz^{7}w^{11}+85351700xyz^{5}w^{13}-14524120xyz^{3}w^{15}+704220xyzw^{17}-48912xz^{18}w+3214468xz^{16}w^{3}-47467312xz^{14}w^{5}+258797240xz^{12}w^{7}-634089488xz^{10}w^{9}+756601776xz^{8}w^{11}-442630592xz^{6}w^{13}+118219176xz^{4}w^{15}-11599248xz^{2}w^{17}+192060xw^{19}-19024yz^{19}+2470484yz^{17}w^{2}-44782360yz^{15}w^{4}+268422912yz^{13}w^{6}-693886960yz^{11}w^{8}+860375680yz^{9}w^{10}-523771000yz^{7}w^{12}+148900512yz^{5}w^{14}-16801664yz^{3}w^{16}+473356yzw^{18}+7880z^{20}-1157268z^{18}w^{2}+23462513z^{16}w^{4}-157741408z^{14}w^{6}+467183392z^{12}w^{8}-695891824z^{10}w^{10}+555175562z^{8}w^{12}-239989280z^{6}w^{14}+53059736z^{4}w^{16}-4877148z^{2}w^{18}+86329w^{20})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1.x.1 | $8$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 16.192.5.bp.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 16.192.5.bq.1 | $16$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 16.192.5.br.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 16.192.5.bs.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 16.192.9.ck.1 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 16.192.9.dg.1 | $16$ | $2$ | $2$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 48.192.5.ga.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 48.192.5.ge.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 48.192.5.gg.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 48.192.5.gi.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 48.192.9.jp.1 | $48$ | $2$ | $2$ | $9$ | $5$ | $1^{6}\cdot2$ |
| 48.192.9.kw.1 | $48$ | $2$ | $2$ | $9$ | $3$ | $1^{6}\cdot2$ |
| 48.288.17.kl.1 | $48$ | $3$ | $3$ | $17$ | $8$ | $1^{8}\cdot2^{4}$ |
| 48.384.17.mh.1 | $48$ | $4$ | $4$ | $17$ | $5$ | $1^{16}$ |
| 80.192.5.ky.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5.lc.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5.le.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.5.lg.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.192.9.ol.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.192.9.ps.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.192.5.ga.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5.ge.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5.gg.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.5.gi.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.192.9.ix.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.192.9.ke.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.192.5.ga.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.5.ge.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.5.gg.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.5.gi.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.192.9.ix.1 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.192.9.ke.1 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.192.5.ky.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.192.5.lc.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.192.5.le.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.192.5.lg.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.192.9.ol.1 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.192.9.ps.1 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.192.5.bpo.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5.bpy.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5.bqc.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.5.bqk.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.192.9.bzb.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.192.9.cce.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.192.5.ky.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.192.5.lc.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.192.5.le.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.192.5.lg.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.192.9.ol.1 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.192.9.ps.1 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.192.5.ga.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.192.5.ge.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.192.5.gg.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.192.5.gi.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.192.9.ix.1 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 304.192.9.ke.1 | $304$ | $2$ | $2$ | $9$ | $?$ | not computed |