Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8C1 |
| Rouse and Zureick-Brown (RZB) label: | X143 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.1.26 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&3\\4&7\end{bmatrix}$, $\begin{bmatrix}1&5\\0&7\end{bmatrix}$, $\begin{bmatrix}7&4\\6&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3.C_2^3$ |
| 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: | $64$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 11x + 14 $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| 32.a3 | $-4$ | $1728$ | $= 2^{6} \cdot 3^{3}$ | $7.455$ | $(1:-2:1)$, $(1:2:1)$, $(2:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{1404x^{2}y^{6}-3761589x^{2}y^{4}z^{2}+641729592x^{2}y^{2}z^{4}-19948109851x^{2}z^{6}-31626xy^{6}z+27474288xy^{4}z^{3}-3061839787xy^{2}z^{5}+76369887178xz^{7}-27y^{8}+410896y^{6}z^{2}-132195520y^{4}z^{4}+6979314188y^{2}z^{6}-72947334979z^{8}}{4x^{2}y^{6}+17x^{2}y^{4}z^{2}-8x^{2}y^{2}z^{4}-x^{2}z^{6}+2xy^{6}z+32xy^{4}z^{3}-17xy^{2}z^{5}-2xz^{7}-y^{8}-48y^{6}z^{2}-128y^{4}z^{4}+52y^{2}z^{6}+7z^{8}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.0.s.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.0.v.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.1.d.1 | $8$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 8.48.1.c.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.x.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bn.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.br.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 16.48.2.u.1 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 16.48.2.w.1 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 16.48.2.bk.1 | $16$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 16.48.2.bl.1 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 16.48.2.bm.1 | $16$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 16.48.2.bn.1 | $16$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 16.48.2.bs.1 | $16$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 16.48.2.bu.1 | $16$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 16.48.3.cf.1 | $16$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 16.48.3.cg.1 | $16$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.48.1.hi.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hm.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.hy.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.ic.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.gs.1 | $24$ | $3$ | $3$ | $5$ | $3$ | $1^{4}$ |
| 24.96.5.di.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 40.48.1.gm.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.gq.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.hc.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.hg.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.da.1 | $40$ | $5$ | $5$ | $9$ | $5$ | $1^{6}\cdot2$ |
| 40.144.9.fe.1 | $40$ | $6$ | $6$ | $9$ | $2$ | $1^{6}\cdot2$ |
| 40.240.17.ps.1 | $40$ | $10$ | $10$ | $17$ | $11$ | $1^{12}\cdot2^{2}$ |
| 48.48.2.bc.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.be.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.bk.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.bl.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.bm.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.bn.1 | $48$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 48.48.2.bs.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.2.bu.1 | $48$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 48.48.3.dd.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.48.3.de.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 56.48.1.gk.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.go.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.ha.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.he.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.13.di.1 | $56$ | $8$ | $8$ | $13$ | $4$ | $1^{12}$ |
| 56.504.37.gs.1 | $56$ | $21$ | $21$ | $37$ | $26$ | $1^{8}\cdot2^{12}\cdot4$ |
| 56.672.49.gs.1 | $56$ | $28$ | $28$ | $49$ | $30$ | $1^{20}\cdot2^{12}\cdot4$ |
| 80.48.2.bi.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.bk.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.bq.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.br.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.bs.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.bt.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.by.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.2.ca.1 | $80$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 80.48.3.dp.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 80.48.3.dq.1 | $80$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 88.48.1.gk.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.go.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.ha.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.he.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.288.21.di.1 | $88$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 104.48.1.gm.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.gq.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.hc.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.hg.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 112.48.2.bc.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2.be.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2.bk.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2.bl.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2.bm.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2.bn.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2.bs.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.2.bu.1 | $112$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 112.48.3.dd.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.48.3.de.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.48.1.xk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.xo.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.yq.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.yu.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.gm.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.gq.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.hc.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.hg.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.gk.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.go.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.ha.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.he.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.xi.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.xm.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.yo.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ys.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 176.48.2.bc.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2.be.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2.bk.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2.bl.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2.bm.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2.bn.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2.bs.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.2.bu.1 | $176$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 176.48.3.dd.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 176.48.3.de.1 | $176$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 184.48.1.gk.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.go.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.ha.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.he.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 208.48.2.bi.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2.bk.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2.bq.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2.br.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2.bs.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2.bt.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2.by.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.2.ca.1 | $208$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 208.48.3.dp.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 208.48.3.dq.1 | $208$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 232.48.1.gm.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.gq.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.hc.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.hg.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 240.48.2.bi.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.bk.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.bq.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.br.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.bs.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.bt.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.by.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.2.ca.1 | $240$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 240.48.3.fl.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.48.3.fm.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 248.48.1.gk.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.go.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.ha.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.he.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.xi.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.xm.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.yo.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.ys.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 272.48.2.bi.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2.bk.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2.bq.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2.br.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2.bs.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2.bt.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2.by.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.2.ca.1 | $272$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 272.48.3.dp.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 272.48.3.dq.1 | $272$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.48.1.wm.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.wq.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.xs.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.xw.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.gm.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.gq.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.hc.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.hg.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 304.48.2.bc.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2.be.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2.bk.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2.bl.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2.bm.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2.bn.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2.bs.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.2.bu.1 | $304$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 304.48.3.dd.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 304.48.3.de.1 | $304$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.48.1.xk.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.xo.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.yq.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.yu.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.gm.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.gq.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.hc.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.hg.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |