Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $128$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $8^{2}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16J1 |
| Sutherland and Zywina (SZ) label: | 16J1-16g |
| Rouse and Zureick-Brown (RZB) label: | X295 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.48.1.99 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&11\\2&13\end{bmatrix}$, $\begin{bmatrix}5&13\\6&7\end{bmatrix}$, $\begin{bmatrix}7&8\\0&7\end{bmatrix}$, $\begin{bmatrix}15&4\\8&3\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 16-isogeny field degree: | $8$ |
| Cyclic 16-torsion field degree: | $32$ |
| Full 16-torsion field degree: | $512$ |
Jacobian
| Conductor: | $2^{7}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 128.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 9x + 7 $ |
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^8}\cdot\frac{16x^{2}y^{14}+288x^{2}y^{12}z^{2}-3072x^{2}y^{10}z^{4}+3718144x^{2}y^{8}z^{6}-271187968x^{2}y^{6}z^{8}+98238529536x^{2}y^{4}z^{10}-6201943261184x^{2}y^{2}z^{12}+98629624791040x^{2}z^{14}+112xy^{14}z+1472xy^{12}z^{3}+206592xy^{10}z^{5}-11036672xy^{8}z^{7}+2412118016xy^{6}z^{9}-454208126976xy^{4}z^{11}+21152694009856xy^{2}z^{13}-278966724198400xz^{15}+y^{16}+192y^{14}z^{2}-480y^{12}z^{4}-964352y^{10}z^{6}+26242048y^{8}z^{8}-16082272256y^{6}z^{10}+1485590560768y^{4}z^{12}-35377569071104y^{2}z^{14}+180337116184576z^{16}}{z^{5}y^{8}(20x^{2}z+xy^{2}-56xz^{2}-5y^{2}z+36z^{3})}$ |
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 |
|---|---|---|---|---|---|---|
| 8.24.0.bl.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.24.0.o.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.24.1.l.1 | $16$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 16.96.3.ex.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 16.96.3.ey.2 | $16$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 16.96.3.ez.2 | $16$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 16.96.3.fa.2 | $16$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 16.96.5.w.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 16.96.5.bd.3 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 16.96.5.ca.1 | $16$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 16.96.5.ce.2 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.96.3.yz.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 48.96.3.za.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 48.96.3.zb.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 48.96.3.zc.2 | $48$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 48.96.5.tq.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.96.5.tu.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.ty.2 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.96.5.uc.2 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.144.5.ff.1 | $48$ | $3$ | $3$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 48.192.13.rq.2 | $48$ | $4$ | $4$ | $13$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 80.96.3.bcz.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.bda.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.bdb.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.bdc.2 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.5.um.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.uq.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.uu.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.uy.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.240.17.nx.1 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 80.288.17.bpb.1 | $80$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 112.96.3.xz.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.ya.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.yb.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.yc.2 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.5.sy.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.tc.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.tg.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.tk.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.3.xz.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.ya.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.yb.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.yc.2 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.5.sy.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.5.tc.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.5.tg.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.5.tk.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.3.bcz.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.bda.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.bdb.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.bdc.2 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.5.um.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.5.uq.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.5.uu.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.5.uy.2 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.3.fxz.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fya.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fyb.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fyc.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.ekk.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.eko.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.ela.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.ele.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.3.bcz.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.bda.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.bdb.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.bdc.2 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.5.um.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.5.uq.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.5.uu.2 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.5.uy.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.3.xz.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.ya.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.yb.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.yc.2 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.5.sy.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.5.tc.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.5.tg.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.5.tk.2 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |