Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8C1 |
| Rouse and Zureick-Brown (RZB) label: | X137 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.24.1.21 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&3\\0&3\end{bmatrix}$, $\begin{bmatrix}5&6\\6&7\end{bmatrix}$, $\begin{bmatrix}7&3\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3.D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 16.48.1-8.t.1.1, 16.48.1-8.t.1.2, 16.48.1-8.t.1.3, 16.48.1-8.t.1.4, 48.48.1-8.t.1.1, 48.48.1-8.t.1.2, 48.48.1-8.t.1.3, 48.48.1-8.t.1.4, 80.48.1-8.t.1.1, 80.48.1-8.t.1.2, 80.48.1-8.t.1.3, 80.48.1-8.t.1.4, 112.48.1-8.t.1.1, 112.48.1-8.t.1.2, 112.48.1-8.t.1.3, 112.48.1-8.t.1.4, 176.48.1-8.t.1.1, 176.48.1-8.t.1.2, 176.48.1-8.t.1.3, 176.48.1-8.t.1.4, 208.48.1-8.t.1.1, 208.48.1-8.t.1.2, 208.48.1-8.t.1.3, 208.48.1-8.t.1.4, 240.48.1-8.t.1.1, 240.48.1-8.t.1.2, 240.48.1-8.t.1.3, 240.48.1-8.t.1.4, 272.48.1-8.t.1.1, 272.48.1-8.t.1.2, 272.48.1-8.t.1.3, 272.48.1-8.t.1.4, 304.48.1-8.t.1.1, 304.48.1-8.t.1.2, 304.48.1-8.t.1.3, 304.48.1-8.t.1.4 |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $8$ |
| Full 8-torsion field degree: | $64$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 44x - 112 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(-4:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{24x^{2}y^{6}-38928x^{2}y^{4}z^{2}+77856768x^{2}y^{2}z^{4}-134649740288x^{2}z^{6}-328xy^{6}z+1483008xy^{4}z^{3}-424673024xy^{2}z^{5}-1030993481728xz^{7}+y^{8}-13056y^{6}z^{2}+20700160y^{4}z^{4}-14495516672y^{2}z^{6}-1969578078208z^{8}}{z(388x^{2}y^{4}z+385024x^{2}y^{2}z^{3}+77922304x^{2}z^{5}+xy^{6}+5104xy^{4}z^{2}+3538944xy^{2}z^{4}+596639744xz^{6}+24y^{6}z+42896y^{4}z^{3}+14680064y^{2}z^{5}+1139802112z^{7})}$ |
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 |
|---|---|---|---|---|---|---|
| 8.12.0.r.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.0.t.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.12.1.c.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.o.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.u.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bc.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1.bd.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fv.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.fz.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gl.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1.gp.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.5.el.1 | $24$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 24.96.5.cf.1 | $24$ | $4$ | $4$ | $5$ | $0$ | $1^{4}$ |
| 40.48.1.ex.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fb.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fn.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.48.1.fr.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.120.9.bz.1 | $40$ | $5$ | $5$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 40.144.9.dn.1 | $40$ | $6$ | $6$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 40.240.17.nl.1 | $40$ | $10$ | $10$ | $17$ | $2$ | $1^{12}\cdot2^{2}$ |
| 56.48.1.ex.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fb.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fn.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.48.1.fr.1 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.13.cf.1 | $56$ | $8$ | $8$ | $13$ | $3$ | $1^{12}$ |
| 56.504.37.el.1 | $56$ | $21$ | $21$ | $37$ | $5$ | $1^{8}\cdot2^{12}\cdot4$ |
| 56.672.49.el.1 | $56$ | $28$ | $28$ | $49$ | $8$ | $1^{20}\cdot2^{12}\cdot4$ |
| 88.48.1.ex.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fb.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fn.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.48.1.fr.1 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.288.21.cf.1 | $88$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 104.48.1.ex.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fb.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fn.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.48.1.fr.1 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ub.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.uf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.ur.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.uv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.ex.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fb.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fn.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.48.1.fr.1 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.ex.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fb.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fn.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.48.1.fr.1 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.tz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ud.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.up.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.ut.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.ex.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fb.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fn.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.48.1.fr.1 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.ex.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fb.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fn.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.48.1.fr.1 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.ex.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fb.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fn.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.48.1.fr.1 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.tz.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.ud.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.up.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.ut.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.td.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.th.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.tt.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.48.1.tx.1 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.ex.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fb.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fn.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.48.1.fr.1 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.ub.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.uf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.ur.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.uv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.ex.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fb.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fn.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.48.1.fr.1 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |