Invariants
| Level: | $16$ | $\SL_2$-level: | $8$ | Newform level: | $256$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $8^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8H1 |
| Sutherland and Zywina (SZ) label: | 8H1-16c |
| Rouse and Zureick-Brown (RZB) label: | X304 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.48.1.129 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&3\\2&3\end{bmatrix}$, $\begin{bmatrix}1&6\\2&15\end{bmatrix}$, $\begin{bmatrix}3&3\\8&5\end{bmatrix}$, $\begin{bmatrix}9&15\\12&15\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 16-isogeny field degree: | $8$ |
| Cyclic 16-torsion field degree: | $64$ |
| Full 16-torsion field degree: | $512$ |
Jacobian
| Conductor: | $2^{8}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 256.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 13x - 21 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{1728x^{2}y^{14}-2060672x^{2}y^{12}z^{2}+728481792x^{2}y^{10}z^{4}+133414289408x^{2}y^{8}z^{6}+16658626248704x^{2}y^{6}z^{8}+956220328378368x^{2}y^{4}z^{10}+26595393127055360x^{2}y^{2}z^{12}+285661685060993024x^{2}z^{14}-36000xy^{14}z+17909504xy^{12}z^{3}+3358470144xy^{10}z^{5}+985353093120xy^{8}z^{7}+99873776992256xy^{6}z^{9}+5245427150487552xy^{4}z^{11}+135806437476532224xy^{2}z^{13}+1379298218482860032xz^{15}-27y^{16}+298592y^{14}z^{2}+12491392y^{12}z^{4}+24355033088y^{10}z^{6}+3703403479040y^{8}z^{8}+277661171580928y^{6}z^{10}+10864044831408128y^{4}z^{12}+209898471623229440y^{2}z^{14}+1566937634473771008z^{16}}{32x^{2}y^{14}+25216x^{2}y^{12}z^{2}+2441216x^{2}y^{10}z^{4}+59146240x^{2}y^{8}z^{6}-117440512x^{2}y^{4}z^{10}-2147483648x^{2}y^{2}z^{12}+8589934592x^{2}z^{14}+480xy^{14}z+171776xy^{12}z^{3}+13266944xy^{10}z^{5}+283836416xy^{8}z^{7}-16777216xy^{6}z^{9}+100663296xy^{4}z^{11}+5368709120xy^{2}z^{13}-17179869184xz^{15}+y^{16}+3968y^{14}z^{2}+618112y^{12}z^{4}+26710016y^{10}z^{6}+326533120y^{8}z^{8}+83886080y^{6}z^{10}+1627389952y^{4}z^{12}+9663676416y^{2}z^{14}-60129542144z^{16}}$ |
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.bo.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.24.0.l.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.24.1.f.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.f.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 16.96.3.bj.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 16.96.3.dh.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.96.3.dj.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.96.4.c.1 | $16$ | $2$ | $2$ | $4$ | $1$ | $1^{3}$ |
| 16.96.4.h.1 | $16$ | $2$ | $2$ | $4$ | $2$ | $1^{3}$ |
| 16.96.4.l.1 | $16$ | $2$ | $2$ | $4$ | $3$ | $1^{3}$ |
| 16.96.4.o.1 | $16$ | $2$ | $2$ | $4$ | $2$ | $1^{3}$ |
| 48.96.3.tj.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.tn.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.ut.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.ux.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{2}$ |
| 48.96.4.w.1 | $48$ | $2$ | $2$ | $4$ | $2$ | $1^{3}$ |
| 48.96.4.bf.1 | $48$ | $2$ | $2$ | $4$ | $3$ | $1^{3}$ |
| 48.96.4.bn.1 | $48$ | $2$ | $2$ | $4$ | $2$ | $1^{3}$ |
| 48.96.4.bu.1 | $48$ | $2$ | $2$ | $4$ | $1$ | $1^{3}$ |
| 48.144.7.sa.1 | $48$ | $3$ | $3$ | $7$ | $5$ | $1^{2}\cdot2^{2}$ |
| 48.192.11.lk.1 | $48$ | $4$ | $4$ | $11$ | $3$ | $1^{8}\cdot2$ |
| 80.96.3.wt.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.wx.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.yd.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.yh.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.4.o.1 | $80$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 80.96.4.x.1 | $80$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 80.96.4.bf.1 | $80$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 80.96.4.bm.1 | $80$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 80.240.17.jc.1 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 80.288.17.bcy.1 | $80$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 112.96.3.sj.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.sn.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.tt.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.tx.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.4.o.1 | $112$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 112.96.4.x.1 | $112$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 112.96.4.bf.1 | $112$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 112.96.4.bm.1 | $112$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 176.96.3.sj.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.sn.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.tt.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.tx.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.4.o.1 | $176$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 176.96.4.x.1 | $176$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 176.96.4.bf.1 | $176$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 176.96.4.bm.1 | $176$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 208.96.3.wt.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.wx.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.yd.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.yh.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.4.o.1 | $208$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 208.96.4.x.1 | $208$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 208.96.4.bf.1 | $208$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 208.96.4.bm.1 | $208$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.96.3.fht.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fib.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fkn.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fkv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.4.ck.1 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.96.4.db.1 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.96.4.dr.1 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.96.4.eg.1 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 272.96.3.wt.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.wx.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.yd.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.yh.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.4.o.1 | $272$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 272.96.4.x.1 | $272$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 272.96.4.bf.1 | $272$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 272.96.4.bm.1 | $272$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 304.96.3.sj.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.sn.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.tt.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.tx.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.4.o.1 | $304$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 304.96.4.x.1 | $304$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 304.96.4.bf.1 | $304$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 304.96.4.bm.1 | $304$ | $2$ | $2$ | $4$ | $?$ | not computed |