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: | $4$ (all of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{4}$ | ||
| 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: | 8C1 |
| Rouse and Zureick-Brown (RZB) label: | X127 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.1.13 |
Level structure
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 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 |
|---|
| $(0:1:0)$, $(1:-2:1)$, $(1:2:1)$, $(2:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\,\frac{12x^{2}y^{6}-2787x^{2}y^{4}z^{2}+141696x^{2}y^{2}z^{4}-2054593x^{2}z^{6}-98xy^{6}z+14304xy^{4}z^{3}-604801xy^{2}z^{5}+7865854xz^{7}-y^{8}+528y^{6}z^{2}-39940y^{4}z^{4}+995324y^{2}z^{6}-7513337z^{8}}{z^{2}(x^{2}y^{4}-472x^{2}y^{2}z^{2}+19024x^{2}z^{4}-12xy^{4}z+2384xy^{2}z^{3}-72832xz^{5}+84y^{4}z^{2}-6144y^{2}z^{4}+69568z^{6})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(4)$ | $4$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.0.r.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.h.2 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.y.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bb.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bg.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gd.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gh.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gt.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gx.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.ep.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.cj.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 40.48.1.ff.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fj.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fv.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fz.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.cd.1 | $40$ | $5$ | $5$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 40.144.9.dr.1 | $40$ | $6$ | $6$ | $9$ | $0$ | $1^{6}\cdot2$ |
| 40.240.17.np.1 | $40$ | $10$ | $10$ | $17$ | $3$ | $1^{12}\cdot2^{2}$ |
| 56.48.1.ff.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fj.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fv.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fz.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.13.cj.1 | $56$ | $8$ | $8$ | $13$ | $2$ | $1^{12}$ |
| 56.504.37.ep.1 | $56$ | $21$ | $21$ | $37$ | $10$ | $1^{8}\cdot2^{12}\cdot4$ |
| 56.672.49.ep.1 | $56$ | $28$ | $28$ | $49$ | $12$ | $1^{20}\cdot2^{12}\cdot4$ |
| 88.48.1.ff.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fj.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fv.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fz.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.288.21.cj.1 | $88$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 104.48.1.ff.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fj.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fv.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fz.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.uj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.un.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.uz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.vd.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.ff.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fj.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fv.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fz.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.ff.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fj.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fv.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fz.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.uh.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ul.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ux.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.vb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.ff.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fj.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fv.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fz.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.ff.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fj.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fv.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fz.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.ff.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fj.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fv.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fz.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.uh.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.ul.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.ux.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.vb.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.tl.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.tp.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.ub.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.uf.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.ff.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fj.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fv.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fz.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.uj.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.un.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.uz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.vd.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.ff.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fj.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fv.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fz.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |