Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| 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: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.96.1.53 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}1&2\\0&7\end{bmatrix}$, $\begin{bmatrix}1&4\\4&5\end{bmatrix}$, $\begin{bmatrix}5&2\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2\times D_4$ |
| Contains $-I$: | no $\quad$ (see 8.48.1.m.2 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| Full 8-torsion field degree: | $16$ |
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 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^2}\cdot\frac{48x^{2}y^{14}+6796576x^{2}y^{12}z^{2}+129476692992x^{2}y^{10}z^{4}+602704419234816x^{2}y^{8}z^{6}+900603377265672192x^{2}y^{6}z^{8}+548258718855667974144x^{2}y^{4}z^{10}+143534890256669922557952x^{2}y^{2}z^{12}+13434757656044767260180480x^{2}z^{14}+4144xy^{14}z+210710784xy^{12}z^{3}+2600320614144xy^{10}z^{5}+8395457494142976xy^{8}z^{7}+9926549305151586304xy^{6}z^{9}+5151756278366582341632xy^{4}z^{11}+1200895072816696170381312xy^{2}z^{13}+102867981249586667464949760xz^{15}+y^{16}+160896y^{14}z^{2}+5567338752y^{12}z^{4}+40251395862528y^{10}z^{6}+82498549376663552y^{8}z^{8}+65797907214461042688y^{6}z^{10}+23398544187794441895936y^{4}z^{12}+3659542046852933524389888y^{2}z^{14}+196515802501630393713688576z^{16}}{zy^{4}(1148x^{2}y^{8}z+2137600x^{2}y^{6}z^{3}+587218176x^{2}y^{4}z^{5}+8192x^{2}y^{2}z^{7}+16384x^{2}z^{9}+xy^{10}+19120xy^{8}z^{2}+20819712xy^{6}z^{4}+4496236544xy^{4}z^{6}-28672xy^{2}z^{8}-65536xz^{10}+48y^{10}z+208736y^{8}z^{3}+99452928y^{6}z^{5}+8589457408y^{4}z^{7}-262144y^{2}z^{9}-458752z^{11})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.d.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.d.1.10 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.e.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.e.1.9 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.48.1-8.d.1.2 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.48.1-8.d.1.9 | $8$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 8.192.1-8.a.1.5 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.f.1.6 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.i.1.2 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 8.192.1-8.j.1.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bc.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.be.1.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bk.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bm.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.288.9-24.ee.2.4 | $24$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.384.9-24.ck.2.4 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 40.192.1-40.bc.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.be.1.5 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bk.1.3 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.192.1-40.bm.1.6 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 40.480.17-40.bm.2.4 | $40$ | $5$ | $5$ | $17$ | $4$ | $1^{6}\cdot2^{5}$ |
| 40.576.17-40.cu.1.16 | $40$ | $6$ | $6$ | $17$ | $0$ | $1^{6}\cdot2\cdot4^{2}$ |
| 40.960.33-40.gg.2.15 | $40$ | $10$ | $10$ | $33$ | $6$ | $1^{12}\cdot2^{6}\cdot4^{2}$ |
| 56.192.1-56.bc.1.7 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.be.1.5 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.bk.2.5 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.192.1-56.bm.1.5 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.768.25-56.ck.1.7 | $56$ | $8$ | $8$ | $25$ | $3$ | $1^{8}\cdot2^{4}\cdot4^{2}$ |
| 56.2016.73-56.ee.2.1 | $56$ | $21$ | $21$ | $73$ | $9$ | $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2688.97-56.ee.1.4 | $56$ | $28$ | $28$ | $97$ | $12$ | $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 88.192.1-88.bc.1.6 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.be.1.5 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.bk.1.3 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 88.192.1-88.bm.1.3 | $88$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.bc.1.7 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.be.1.5 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.bk.1.3 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.192.1-104.bm.1.6 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dy.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ea.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.eo.1.16 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.eq.1.15 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.bc.1.6 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.be.1.5 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.bk.1.5 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 136.192.1-136.bm.1.6 | $136$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.bc.1.6 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.be.1.5 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.bk.1.2 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 152.192.1-152.bm.1.5 | $152$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.dy.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ea.1.13 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.eo.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.eq.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.bc.1.6 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.be.1.5 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.bk.1.2 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 184.192.1-184.bm.1.3 | $184$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.bc.1.6 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.be.1.5 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.bk.1.5 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 232.192.1-232.bm.1.6 | $232$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.bc.1.6 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.be.1.5 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.bk.1.3 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.192.1-248.bm.1.3 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.dy.1.15 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ea.1.13 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.eo.2.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.eq.1.7 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.dy.1.15 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.ea.1.12 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.eo.1.16 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 280.192.1-280.eq.1.15 | $280$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.bc.1.6 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.be.1.5 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.bk.1.5 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 296.192.1-296.bm.1.7 | $296$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.dy.1.15 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ea.1.13 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.eo.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.eq.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.bc.1.7 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.be.1.5 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.bk.1.5 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 328.192.1-328.bm.1.7 | $328$ | $2$ | $2$ | $1$ | $?$ | dimension zero |