Invariants
| Level: | $8$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $8^{12}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| 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: | 8B3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 8.192.3.29 |
Level structure
| $\GL_2(\Z/8\Z)$-generators: | $\begin{bmatrix}3&4\\4&7\end{bmatrix}$, $\begin{bmatrix}5&4\\0&5\end{bmatrix}$, $\begin{bmatrix}7&4\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: | $C_2^3$ |
| Contains $-I$: | no $\quad$ (see 8.96.3.f.1 for the level structure with $-I$) |
| Cyclic 8-isogeny field degree: | $2$ |
| Cyclic 8-torsion field degree: | $4$ |
| Full 8-torsion field degree: | $8$ |
Jacobian
| Conductor: | $2^{18}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 64.2.a.a, 64.2.b.a |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ y w^{2} + w^{2} t + w t^{2} $ |
| $=$ | $y w t + w t^{2} + t^{3}$ | |
| $=$ | $y^{2} w + y w t + y t^{2}$ | |
| $=$ | $y z w + z w t + z t^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} y^{2} - 2 x^{5} z^{2} - x^{4} y^{2} z - 2 x^{4} z^{3} + x^{3} y^{2} z^{2} - x^{2} y^{2} z^{3} + \cdots - 2 z^{7} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 2x^{8} - 2 $ |
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.
| Embedded model |
|---|
| $(1/2:0:1:0:0)$, $(0:0:0:-1:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{4096x^{2}z^{12}-13056x^{2}z^{10}t^{2}+1024x^{2}z^{8}t^{4}+12352x^{2}z^{6}t^{6}+14400x^{2}z^{4}t^{8}-3504x^{2}z^{2}t^{10}+2832x^{2}t^{12}+7168xz^{11}t^{2}-6144xz^{9}t^{4}-1920xz^{7}t^{6}-256xz^{5}t^{8}+9952xz^{3}t^{10}+6912xzt^{12}-256y^{2}z^{12}-2272y^{2}z^{10}t^{2}+1840y^{2}z^{8}t^{4}+400y^{2}z^{6}t^{6}-704y^{2}z^{4}t^{8}-862y^{2}z^{2}t^{10}+489y^{2}t^{12}+2048yz^{12}t-7680yz^{10}t^{3}+2336yz^{8}t^{5}+576yz^{6}t^{7}+112yz^{4}t^{9}-1728yz^{2}t^{11}+98yt^{13}-1024z^{14}+2880z^{12}t^{2}-2816z^{10}t^{4}+784z^{8}t^{6}-2832z^{6}t^{8}-908z^{4}t^{10}+2748z^{2}t^{12}+4w^{14}+24w^{13}t+56w^{12}t^{2}+56w^{11}t^{3}+28w^{10}t^{4}+112w^{9}t^{5}+336w^{8}t^{6}+368w^{7}t^{7}+276w^{6}t^{8}+1288w^{5}t^{9}+3528w^{4}t^{10}+3752w^{3}t^{11}+741w^{2}t^{12}-1360wt^{13}-727t^{14}}{t^{8}(64x^{2}z^{4}-16x^{2}z^{2}t^{2}+48xz^{3}t^{2}+32xzt^{4}-4y^{2}z^{4}-4y^{2}z^{2}t^{2}+5y^{2}t^{4}-8yz^{2}t^{3}-4yt^{5}-16z^{6}-4z^{4}t^{2}+16z^{2}t^{4}+w^{6}+6w^{5}t+14w^{4}t^{2}+14w^{3}t^{3}+6w^{2}t^{4}+6wt^{5}+5t^{6})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 8.96.3.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}Y^{2}-X^{4}Y^{2}Z-2X^{5}Z^{2}+X^{3}Y^{2}Z^{2}-2X^{4}Z^{3}-X^{2}Y^{2}Z^{3}-2XZ^{6}-2Z^{7} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 8.96.3.f.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -wt$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -2zw^{4}t^{3}+2zw^{3}t^{4}-2zw^{2}t^{5}+2zwt^{6}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t^{2}$ |
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.1.5 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.b.1.8 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.g.1.5 | $8$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 8.96.1-8.g.1.9 | $8$ | $2$ | $2$ | $1$ | $0$ | $2$ |
| 8.96.2-8.a.1.4 | $8$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 8.96.2-8.a.1.9 | $8$ | $2$ | $2$ | $2$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.384.5-8.a.1.1 | $8$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 8.384.5-8.b.3.4 | $8$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 8.384.5-8.c.1.3 | $8$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 8.384.5-8.d.2.2 | $8$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.p.2.8 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 24.384.5-24.q.2.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.384.5-24.x.2.4 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 24.384.5-24.y.2.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 24.576.19-24.cf.2.19 | $24$ | $3$ | $3$ | $19$ | $1$ | $1^{8}\cdot2^{2}\cdot4$ |
| 24.768.21-24.bh.2.16 | $24$ | $4$ | $4$ | $21$ | $1$ | $1^{8}\cdot2^{3}\cdot4$ |
| 40.384.5-40.h.2.8 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.384.5-40.i.2.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.384.5-40.p.2.4 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.384.5-40.q.2.6 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.960.35-40.w.2.12 | $40$ | $5$ | $5$ | $35$ | $7$ | $1^{14}\cdot2^{5}\cdot4^{2}$ |
| 40.1152.37-40.cy.2.3 | $40$ | $6$ | $6$ | $37$ | $2$ | $1^{14}\cdot2^{2}\cdot4^{4}$ |
| 40.1920.69-40.ea.2.2 | $40$ | $10$ | $10$ | $69$ | $12$ | $1^{28}\cdot2^{7}\cdot4^{6}$ |
| 56.384.5-56.h.2.8 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.384.5-56.i.2.1 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.384.5-56.p.2.4 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.384.5-56.q.2.6 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.1536.53-56.bh.2.5 | $56$ | $8$ | $8$ | $53$ | $6$ | $1^{20}\cdot2^{7}\cdot4^{4}$ |
| 56.4032.151-56.cf.2.7 | $56$ | $21$ | $21$ | $151$ | $23$ | $1^{16}\cdot2^{26}\cdot4^{4}\cdot6^{2}\cdot12^{3}\cdot16$ |
| 56.5376.201-56.cg.2.3 | $56$ | $28$ | $28$ | $201$ | $29$ | $1^{36}\cdot2^{33}\cdot4^{8}\cdot6^{2}\cdot12^{3}\cdot16$ |
| 88.384.5-88.h.2.8 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 88.384.5-88.i.2.1 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 88.384.5-88.p.2.4 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 88.384.5-88.q.2.6 | $88$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.h.2.8 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.i.2.1 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.p.2.4 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.q.2.6 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.cx.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.cz.2.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ed.2.15 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ef.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 136.384.5-136.h.2.8 | $136$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 136.384.5-136.i.2.1 | $136$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 136.384.5-136.p.2.4 | $136$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 136.384.5-136.q.2.6 | $136$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 152.384.5-152.h.2.8 | $152$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 152.384.5-152.i.2.1 | $152$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 152.384.5-152.p.2.4 | $152$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 152.384.5-152.q.2.6 | $152$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.cx.2.15 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.cz.2.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ed.2.8 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ef.2.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 184.384.5-184.h.2.8 | $184$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 184.384.5-184.i.2.1 | $184$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 184.384.5-184.p.2.4 | $184$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 184.384.5-184.q.2.6 | $184$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 232.384.5-232.h.2.8 | $232$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 232.384.5-232.i.2.1 | $232$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 232.384.5-232.p.2.4 | $232$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 232.384.5-232.q.2.6 | $232$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.384.5-248.h.2.8 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.384.5-248.i.2.1 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.384.5-248.p.2.4 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.384.5-248.q.2.6 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.cx.2.15 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.cz.2.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ed.2.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ef.2.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.cp.1.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.cr.2.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.dv.2.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.dx.2.7 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 296.384.5-296.h.2.8 | $296$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 296.384.5-296.i.2.1 | $296$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 296.384.5-296.p.2.4 | $296$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 296.384.5-296.q.2.6 | $296$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.cx.2.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.cz.2.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ed.2.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ef.2.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 328.384.5-328.h.2.8 | $328$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 328.384.5-328.i.2.1 | $328$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 328.384.5-328.p.2.4 | $328$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 328.384.5-328.q.2.6 | $328$ | $2$ | $2$ | $5$ | $?$ | not computed |