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: | 8H1-16g |
| Rouse and Zureick-Brown (RZB) label: | X308 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.48.1.132 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&9\\6&7\end{bmatrix}$, $\begin{bmatrix}3&4\\0&11\end{bmatrix}$, $\begin{bmatrix}3&13\\6&9\end{bmatrix}$, $\begin{bmatrix}5&5\\0&11\end{bmatrix}$ |
| 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} - 3x + 1 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{864x^{2}y^{14}+128792x^{2}y^{12}z^{2}+5691264x^{2}y^{10}z^{4}-130287392x^{2}y^{8}z^{6}+2033523712x^{2}y^{6}z^{8}-14590764288x^{2}y^{4}z^{10}+50726686720x^{2}y^{2}z^{12}-68107053056x^{2}z^{14}+9864xy^{14}z+688464xy^{12}z^{3}-7427760xy^{10}z^{5}+350842048xy^{8}z^{7}-4062287872xy^{6}z^{9}+25428668928xy^{4}z^{11}-78788424704xy^{2}z^{13}+96318107648xz^{15}+27y^{16}+42040y^{14}z^{2}+116856y^{12}z^{4}+42431728y^{10}z^{6}-696158368y^{8}z^{8}+5934021632y^{6}z^{10}-25081016576y^{4}z^{12}+48011506688y^{2}z^{14}-28210944000z^{16}}{16x^{2}y^{14}-1576x^{2}y^{12}z^{2}+19072x^{2}y^{10}z^{4}-57760x^{2}y^{8}z^{6}+1792x^{2}y^{4}z^{10}-4096x^{2}y^{2}z^{12}-2048x^{2}z^{14}-104xy^{14}z+3792xy^{12}z^{3}-32752xy^{10}z^{5}+80832xy^{8}z^{7}+1024xy^{6}z^{9}+2560xy^{4}z^{11}-9216xy^{2}z^{13}-4096xz^{15}-y^{16}+440y^{14}z^{2}-7368y^{12}z^{4}+31024y^{10}z^{6}-24864y^{8}z^{8}+3072y^{6}z^{10}-5376y^{4}z^{12}+1024y^{2}z^{14}+2048z^{16}}$ |
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.k.2 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 16.24.1.f.2 | $16$ | $2$ | $2$ | $1$ | $1$ | 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.b.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 16.96.3.bj.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 16.96.3.dh.2 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.96.3.di.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.96.4.b.1 | $16$ | $2$ | $2$ | $4$ | $2$ | $1^{3}$ |
| 16.96.4.e.1 | $16$ | $2$ | $2$ | $4$ | $1$ | $1^{3}$ |
| 16.96.4.i.1 | $16$ | $2$ | $2$ | $4$ | $2$ | $1^{3}$ |
| 16.96.4.n.1 | $16$ | $2$ | $2$ | $4$ | $3$ | $1^{3}$ |
| 32.96.3.bj.1 | $32$ | $2$ | $2$ | $3$ | $3$ | $2$ |
| 32.96.3.bk.1 | $32$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 32.96.5.bm.1 | $32$ | $2$ | $2$ | $5$ | $5$ | $4$ |
| 32.96.5.bn.1 | $32$ | $2$ | $2$ | $5$ | $1$ | $4$ |
| 48.96.3.th.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.tl.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.ur.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.uv.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{2}$ |
| 48.96.4.r.1 | $48$ | $2$ | $2$ | $4$ | $3$ | $1^{3}$ |
| 48.96.4.y.1 | $48$ | $2$ | $2$ | $4$ | $2$ | $1^{3}$ |
| 48.96.4.bg.1 | $48$ | $2$ | $2$ | $4$ | $1$ | $1^{3}$ |
| 48.96.4.bp.1 | $48$ | $2$ | $2$ | $4$ | $2$ | $1^{3}$ |
| 48.144.7.rm.1 | $48$ | $3$ | $3$ | $7$ | $7$ | $1^{2}\cdot2^{2}$ |
| 48.192.11.le.1 | $48$ | $4$ | $4$ | $11$ | $3$ | $1^{8}\cdot2$ |
| 80.96.3.wr.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.wv.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.yb.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.yf.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.4.j.1 | $80$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 80.96.4.q.1 | $80$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 80.96.4.y.1 | $80$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 80.96.4.bh.1 | $80$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 80.240.17.iw.1 | $80$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 80.288.17.bco.1 | $80$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 96.96.3.cx.1 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.3.cy.1 | $96$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 96.96.5.dq.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 96.96.5.dr.1 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.3.sh.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.sl.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.tr.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.tv.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.4.j.1 | $112$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 112.96.4.q.1 | $112$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 112.96.4.y.1 | $112$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 112.96.4.bh.1 | $112$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 160.96.3.dj.1 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.96.3.dk.1 | $160$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 160.96.5.du.1 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 160.96.5.dv.1 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.3.sh.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.sl.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.tr.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.tv.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.4.j.1 | $176$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 176.96.4.q.1 | $176$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 176.96.4.y.1 | $176$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 176.96.4.bh.1 | $176$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 208.96.3.wr.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.wv.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.yb.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.yf.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.4.j.1 | $208$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 208.96.4.q.1 | $208$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 208.96.4.y.1 | $208$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 208.96.4.bh.1 | $208$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 224.96.3.cx.1 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.96.3.cy.1 | $224$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 224.96.5.dq.1 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 224.96.5.dr.1 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.3.fhn.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fhv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fkh.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fkp.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.4.bx.1 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.96.4.cm.1 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.96.4.dc.1 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 240.96.4.dt.1 | $240$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 272.96.3.wr.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.wv.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.yb.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.yf.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.4.j.1 | $272$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 272.96.4.q.1 | $272$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 272.96.4.y.1 | $272$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 272.96.4.bh.1 | $272$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 304.96.3.sh.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.sl.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.tr.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.tv.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.4.j.1 | $304$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 304.96.4.q.1 | $304$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 304.96.4.y.1 | $304$ | $2$ | $2$ | $4$ | $?$ | not computed |
| 304.96.4.bh.1 | $304$ | $2$ | $2$ | $4$ | $?$ | not computed |