Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.192.1.7 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&0\\0&3\end{bmatrix}$, $\begin{bmatrix}1&0\\0&5\end{bmatrix}$, $\begin{bmatrix}1&4\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3$ |
| Contains $-I$: | no $\quad$ (see 8.96.1.g.2 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $1$ |
| Cyclic 8-torsion field degree: | $1$ |
| Full 8-torsion field degree: | $8$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(1:0:1)$, $(-1:0:1)$, $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{708x^{2}y^{28}z^{2}-477870x^{2}y^{24}z^{6}+39545991x^{2}y^{20}z^{10}+258743115x^{2}y^{16}z^{14}+1130374344x^{2}y^{12}z^{18}+2157968565x^{2}y^{8}z^{22}+754974741x^{2}y^{4}z^{26}+16777215x^{2}z^{30}-8xy^{30}z+161586xy^{26}z^{5}-643608xy^{22}z^{9}+160418057xy^{18}z^{13}+1015012944xy^{14}z^{17}+2422211385xy^{10}z^{21}+2013265900xy^{6}z^{25}+184549377xy^{2}z^{29}+y^{32}-4152y^{28}z^{4}+12539228y^{24}z^{8}+76473134y^{20}z^{12}+457376604y^{16}z^{16}+1203755024y^{12}z^{20}+1409287006y^{8}z^{24}+167772138y^{4}z^{28}+z^{32}}{z^{2}y^{8}(x^{2}y^{20}+157x^{2}y^{16}z^{4}-16620x^{2}y^{12}z^{8}+311305x^{2}y^{8}z^{12}+983053x^{2}y^{4}z^{16}+65535x^{2}z^{20}-49xy^{18}z^{3}+200xy^{14}z^{7}+16365xy^{10}z^{11}+1310708xy^{6}z^{15}+458753xy^{2}z^{19}-6y^{20}z^{2}+1222y^{16}z^{6}-32504y^{12}z^{10}+655362y^{8}z^{14}+393202y^{4}z^{18}+z^{22})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.b.2.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.b.2.9 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.c.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.c.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.k.2.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.k.2.5 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| $X_1(2,8)$ | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.l.2.5 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.h.1.5 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.h.1.11 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.i.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.i.1.4 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.k.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.96.1-8.k.1.4 | $8$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{arith}}(8)$ | $8$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 8.384.5-8.d.3.1 | $8$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.384.5-16.a.2.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.d.2.2 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.k.3.6 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.k.4.2 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.k.5.2 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.l.4.5 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.l.5.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.l.6.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.m.3.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.384.5-16.m.4.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 16.384.5-16.u.2.2 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.5-16.x.2.1 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 16.384.9-16.bq.1.2 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 16.384.9-16.bq.2.2 | $16$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.384.5-24.bj.1.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.bj.3.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.576.17-24.pg.1.1 | $24$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.fw.1.2 | $24$ | $4$ | $4$ | $17$ | $0$ | $1^{8}\cdot2^{4}$ |
| 40.384.5-40.bb.1.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.384.5-40.bb.3.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 40.960.33-40.cw.1.1 | $40$ | $5$ | $5$ | $33$ | $5$ | $1^{14}\cdot2^{9}$ |
| 40.1152.33-40.jy.1.1 | $40$ | $6$ | $6$ | $33$ | $1$ | $1^{14}\cdot2\cdot4^{4}$ |
| 40.1920.65-40.ns.2.13 | $40$ | $10$ | $10$ | $65$ | $9$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |
| 48.384.5-48.i.2.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.l.2.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.bn.3.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.bn.4.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.bn.6.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.bo.4.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.bo.5.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.bo.6.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.bp.3.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.bp.4.2 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 48.384.5-48.cw.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.cz.2.9 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.9-48.ga.1.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 48.384.9-48.ga.2.2 | $48$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 56.384.5-56.bb.1.1 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 56.384.5-56.bb.3.1 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 56.1536.49-56.fw.1.2 | $56$ | $8$ | $8$ | $49$ | $3$ | $1^{20}\cdot2^{6}\cdot4^{4}$ |
| 56.4032.145-56.pk.1.1 | $56$ | $21$ | $21$ | $145$ | $19$ | $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
| 56.5376.193-56.qe.1.7 | $56$ | $28$ | $28$ | $193$ | $22$ | $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
| 80.384.5-80.ba.2.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.bd.2.9 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.cz.2.9 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.cz.5.9 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.cz.6.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.da.3.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.da.4.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.da.6.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.db.3.9 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.db.4.9 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.eu.2.10 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.5-80.ex.1.11 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 80.384.9-80.jo.1.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 80.384.9-80.jo.2.1 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 88.384.5-88.bb.1.1 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 88.384.5-88.bb.3.1 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.bb.1.1 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.bb.3.1 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.i.2.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.l.2.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bn.3.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bn.4.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bn.6.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bo.4.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bo.5.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bo.6.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bp.3.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bp.4.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.cw.2.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.cz.1.9 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.9-112.ga.1.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.ga.2.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.5-120.hv.1.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hv.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 136.384.5-136.bb.1.1 | $136$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 136.384.5-136.bb.3.1 | $136$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 152.384.5-152.bb.1.1 | $152$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 152.384.5-152.bb.3.1 | $152$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hv.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hv.3.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.i.2.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.l.2.3 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bn.3.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bn.4.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bn.6.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bo.4.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bo.5.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bo.6.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bp.3.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.bp.4.2 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.cw.2.1 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.5-176.cz.2.9 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 176.384.9-176.ga.1.2 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 176.384.9-176.ga.2.2 | $176$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 184.384.5-184.bb.1.1 | $184$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 184.384.5-184.bb.3.1 | $184$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.v.2.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.y.1.9 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.cz.1.9 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.cz.5.9 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.cz.6.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.da.3.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.da.4.1 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.da.6.5 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.db.2.9 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.db.4.9 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.ez.2.10 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.fc.1.11 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.9-208.jo.1.1 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 208.384.9-208.jo.3.1 | $208$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 232.384.5-232.bb.1.1 | $232$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 232.384.5-232.bb.3.1 | $232$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.de.1.22 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.dh.2.18 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.hz.2.17 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.hz.4.17 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.hz.6.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ia.3.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ia.4.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ia.6.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ib.3.17 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ib.4.17 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.os.2.17 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ov.1.17 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.9-240.bja.1.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.384.9-240.bja.2.1 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 248.384.5-248.bb.1.1 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.384.5-248.bb.3.1 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.hv.1.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.hv.3.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.i.2.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.l.2.9 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.cz.2.9 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.cz.5.9 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.cz.6.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.da.3.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.da.4.1 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.da.6.5 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.db.3.9 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.db.4.9 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.fm.2.10 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.5-272.fp.1.11 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 272.384.9-272.jo.1.2 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 272.384.9-272.jo.2.2 | $272$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 280.384.5-280.hn.1.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hn.2.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
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| 328.384.5-328.bb.3.1 | $328$ | $2$ | $2$ | $5$ | $?$ | not computed |