Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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 | $2^{3}$ | ||
| Elliptic points: | $4$ 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: | 8I1 |
| Rouse and Zureick-Brown (RZB) label: | X267 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.48.1.72 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&3\\4&7\end{bmatrix}$, $\begin{bmatrix}7&0\\4&3\end{bmatrix}$, $\begin{bmatrix}7&7\\2&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^2.D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 8-isogeny field degree: | $4$ |
| Cyclic 8-torsion field degree: | $16$ |
| Full 8-torsion field degree: | $32$ |
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 $ | $=$ | $ x y + x z + 2 y z - y w + z w $ |
| $=$ | $x^{2} - 2 x y + 2 x z - 3 y^{2} + 2 y z + z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} + 2 x^{3} y - 2 x^{2} y^{2} + 2 x^{2} y z - 6 x^{2} z^{2} - 6 x y z^{2} - 2 y^{2} z^{2} + \cdots - 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2\cdot3^3\,\frac{10741051008xz^{11}-6456646080xz^{10}w-2767753152xz^{9}w^{2}+894917376xz^{8}w^{3}+411994944xz^{7}w^{4}+72471456xz^{6}w^{5}+24516000xz^{5}w^{6}+13516416xz^{4}w^{7}+6629064xz^{3}w^{8}+2023060xz^{2}w^{9}+282068xzw^{10}-7538909760y^{2}z^{10}+3955728960y^{2}z^{9}w+1934264448y^{2}z^{8}w^{2}-410396544y^{2}z^{7}w^{3}-176230944y^{2}z^{6}w^{4}+14885856y^{2}z^{5}w^{5}+38176704y^{2}z^{4}w^{6}+28466208y^{2}z^{3}w^{7}+13493772y^{2}z^{2}w^{8}+3978564y^{2}zw^{9}+555032y^{2}w^{10}+18604826496yz^{11}-18046058688yz^{10}w-518021568yz^{9}w^{2}+3126874752yz^{8}w^{3}+44515008yz^{7}w^{4}-193979232yz^{6}w^{5}-80670816yz^{5}w^{6}-51178752yz^{4}w^{7}-26134056yz^{3}w^{8}-10018236yz^{2}w^{9}-2541324yzw^{10}-308312yw^{11}+1711715328z^{12}+6921355392z^{11}w-2785969728z^{10}w^{2}-2533128768z^{9}w^{3}-53085456z^{8}w^{4}+252536832z^{7}w^{5}+100378656z^{6}w^{6}+30807936z^{5}w^{7}+8936568z^{4}w^{8}+2147960z^{3}w^{9}+325452z^{2}w^{10}+26244zw^{11}+2187w^{12}}{515957040xz^{11}-725226696xz^{10}w+404455032xz^{9}w^{2}-128420640xz^{8}w^{3}+27849096xz^{7}w^{4}-4391820xz^{6}w^{5}+514404xz^{5}w^{6}-31860xz^{4}w^{7}+3504xz^{3}w^{8}-3898xz^{2}w^{9}-4898xzw^{10}-362114712y^{2}z^{10}+481367448y^{2}z^{9}w-262317528y^{2}z^{8}w^{2}+84505680y^{2}z^{7}w^{3}-18764460y^{2}z^{6}w^{4}+3020004y^{2}z^{5}w^{5}-356454y^{2}z^{4}w^{6}+51408y^{2}z^{3}w^{7}-900y^{2}z^{2}w^{8}-5658y^{2}zw^{9}-8933y^{2}w^{10}+893660688yz^{11}-1585889928yz^{10}w+1171514664yz^{9}w^{2}-494584704yz^{8}w^{3}+139301208yz^{7}w^{4}-28338660yz^{6}w^{5}+4310496yz^{5}w^{6}-512208yz^{4}w^{7}+30564yz^{3}w^{8}+3846yz^{2}w^{9}+8796yzw^{10}+4898yw^{11}+82231200z^{12}+266394096z^{11}w-355308768z^{10}w^{2}+154822104z^{9}w^{3}-38483586z^{8}w^{4}+6651072z^{7}w^{5}-824850z^{6}w^{6}+83196z^{5}w^{7}-1512z^{4}w^{8}-1760z^{3}w^{9}-4035z^{2}w^{10}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0.bs.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.3.b.1 | $8$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 8.96.3.k.1 | $8$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 8.96.3.n.1 | $8$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 8.96.3.o.1 | $8$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3.ed.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.96.3.ej.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.96.3.fl.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 16.96.3.fn.1 | $16$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 16.96.5.ct.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.96.5.cv.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 16.96.5.dp.1 | $16$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 16.96.5.dv.1 | $16$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 24.96.3.id.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.96.3.ih.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.96.3.il.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.96.3.ip.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.144.7.con.1 | $24$ | $3$ | $3$ | $7$ | $2$ | $1^{6}$ |
| 24.192.11.j.1 | $24$ | $4$ | $4$ | $11$ | $1$ | $1^{10}$ |
| 40.96.3.cb.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.96.3.cf.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.96.3.cj.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.96.3.cn.1 | $40$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 40.240.17.bhz.1 | $40$ | $5$ | $5$ | $17$ | $8$ | $1^{14}\cdot2$ |
| 40.288.17.cvl.1 | $40$ | $6$ | $6$ | $17$ | $2$ | $1^{14}\cdot2$ |
| 40.480.33.ezl.1 | $40$ | $10$ | $10$ | $33$ | $17$ | $1^{28}\cdot2^{2}$ |
| 48.96.3.wz.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.xf.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.xz.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.yb.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.5.qf.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
| 48.96.5.qh.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.96.5.rb.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
| 48.96.5.rh.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 56.96.3.bg.1 | $56$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 56.96.3.bk.1 | $56$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 56.96.3.bo.1 | $56$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 56.96.3.bs.1 | $56$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 56.384.27.l.1 | $56$ | $8$ | $8$ | $27$ | $7$ | $1^{18}\cdot2^{4}$ |
| 56.1008.71.yt.1 | $56$ | $21$ | $21$ | $71$ | $38$ | $1^{14}\cdot2^{26}\cdot4$ |
| 56.1344.97.ewj.1 | $56$ | $28$ | $28$ | $97$ | $45$ | $1^{32}\cdot2^{30}\cdot4$ |
| 80.96.3.baz.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.bbf.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.bbz.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.3.bcb.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.96.5.rb.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.rd.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.rx.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.96.5.sd.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 88.96.3.bf.1 | $88$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 88.96.3.bj.1 | $88$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 88.96.3.bn.1 | $88$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 88.96.3.br.1 | $88$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 104.96.3.cc.1 | $104$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 104.96.3.cg.1 | $104$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 104.96.3.ck.1 | $104$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 104.96.3.co.1 | $104$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.vz.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.wf.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.wz.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.xb.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.5.pn.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.pp.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.qj.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.qp.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.96.3.ban.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3.bav.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3.bbd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.96.3.bbl.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 136.96.3.cb.1 | $136$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 136.96.3.cf.1 | $136$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 136.96.3.cj.1 | $136$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 136.96.3.cn.1 | $136$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 152.96.3.bg.1 | $152$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 152.96.3.bk.1 | $152$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 152.96.3.bo.1 | $152$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 152.96.3.bs.1 | $152$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.xz.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.yh.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.yp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.yx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.vz.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.wf.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.wz.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.3.xb.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.96.5.pn.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.5.pp.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.5.qj.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.96.5.qp.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 184.96.3.bf.1 | $184$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 184.96.3.bj.1 | $184$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 184.96.3.bn.1 | $184$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 184.96.3.br.1 | $184$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.baz.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.bbf.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.bbz.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.3.bcb.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.96.5.rb.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.5.rd.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.5.rx.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.96.5.sd.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 232.96.3.cb.1 | $232$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 232.96.3.cf.1 | $232$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 232.96.3.cj.1 | $232$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 232.96.3.cn.1 | $232$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fpf.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fpl.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fqv.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.fqx.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.dmj.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dml.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dnv.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.dob.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.96.3.bg.1 | $248$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 248.96.3.bk.1 | $248$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 248.96.3.bo.1 | $248$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 248.96.3.bs.1 | $248$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.xz.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.yh.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.yp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.96.3.yx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.baz.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.bbf.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.bbz.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.3.bcb.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.96.5.rb.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.5.rd.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.5.rx.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.96.5.sd.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.96.3.gf.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.96.3.gn.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.96.3.gv.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.96.3.hd.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 296.96.3.cc.1 | $296$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 296.96.3.cg.1 | $296$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 296.96.3.ck.1 | $296$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 296.96.3.co.1 | $296$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.vz.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.wf.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.wz.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.3.xb.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.96.5.pn.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.5.pp.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.5.qj.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 304.96.5.qp.1 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.96.3.ban.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3.bav.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3.bbd.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.96.3.bbl.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 328.96.3.cb.1 | $328$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 328.96.3.cf.1 | $328$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 328.96.3.cj.1 | $328$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 328.96.3.cn.1 | $328$ | $2$ | $2$ | $3$ | $?$ | not computed |