Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
| 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 $4$ are rational) | Cusp widths | $8^{12}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.192.3.58 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&0\\52&41\end{bmatrix}$, $\begin{bmatrix}15&48\\4&49\end{bmatrix}$, $\begin{bmatrix}47&44\\20&55\end{bmatrix}$, $\begin{bmatrix}53&4\\52&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.96.3.w.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $96$ |
| Full 56-torsion field degree: | $16128$ |
Jacobian
| Conductor: | $2^{18}\cdot7^{6}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2$ |
| Newforms: | 3136.2.a.m, 3136.2.b.b |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 2 x z w - x w t - y w^{2} + y w t $ |
| $=$ | $2 x z^{2} - x z t - y z w + y z t$ | |
| $=$ | $2 x z t - x t^{2} - y w t + y t^{2}$ | |
| $=$ | $2 x y z - x y t - y^{2} w + y^{2} t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} z + 7 x^{3} y^{2} + x^{3} z^{2} - 42 x^{2} y^{2} z + 2 x^{2} z^{3} + 70 x y^{2} z^{2} + \cdots - 28 y^{2} z^{3} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -7x^{7} + 49x^{5} - 49x^{3} + 7x $ |
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.
| Embedded model |
|---|
| $(0:0:1/2:1:1)$, $(0:1:0:0:0)$, $(1:1:0:0:0)$, $(1:0:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{823543x^{14}-705894x^{12}t^{2}+1176490x^{10}t^{4}-1709512x^{8}t^{6}+2762522x^{6}t^{8}-4495456x^{4}t^{10}+7336308x^{2}t^{12}-1647086xy^{13}+5411854xy^{11}t^{2}-1747928xy^{9}t^{4}+14425208xy^{7}t^{6}-2730966xy^{5}t^{8}+24969910xy^{3}t^{10}-4659424xyt^{12}-2352980y^{12}t^{2}-2958032y^{10}t^{4}-7049336y^{8}t^{6}-5718496y^{6}t^{8}-12071052y^{4}t^{10}-9209312y^{2}t^{12}+640zw^{13}+128zw^{12}t+4352zw^{11}t^{2}+256zw^{10}t^{3}+43904zw^{9}t^{4}+4480zw^{8}t^{5}+232960zw^{7}t^{6}+3592zw^{6}t^{7}+1216704zw^{5}t^{8}+12128zw^{4}t^{9}+4893120zw^{3}t^{10}-214360zw^{2}t^{11}+13187584zwt^{12}+640zt^{13}-224w^{14}-64w^{13}t-1568w^{12}t^{2}-384w^{11}t^{3}-15456w^{10}t^{4}-4032w^{9}t^{5}-83104w^{8}t^{6}-19840w^{7}t^{7}-425024w^{6}t^{8}-120488w^{5}t^{9}-1615232w^{4}t^{10}-821832w^{3}t^{11}-3403488w^{2}t^{12}-3182088wt^{13}-224t^{14}}{t^{4}(343x^{6}t^{4}-1078x^{4}t^{6}+2506x^{2}t^{8}-686xy^{5}t^{4}+3038xy^{3}t^{6}-2240xyt^{8}-980y^{4}t^{6}-1904y^{2}t^{8}+40zw^{9}+8zw^{8}t+272zw^{7}t^{2}+16zw^{6}t^{3}+904zw^{5}t^{4}-88zw^{4}t^{5}+2048zw^{3}t^{6}-504zw^{2}t^{7}+3456zwt^{8}-14w^{10}-4w^{9}t-98w^{8}t^{2}-24w^{7}t^{3}-322w^{6}t^{4}-68w^{5}t^{5}-686w^{4}t^{6}-136w^{3}t^{7}-984w^{2}t^{8}-740wt^{9})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 56.96.3.w.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{7}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Equation of the image curve:
| $0$ | $=$ | $ 7X^{3}Y^{2}-X^{4}Z-42X^{2}Y^{2}Z+X^{3}Z^{2}+70XY^{2}Z^{2}+2X^{2}Z^{3}-28Y^{2}Z^{3}-2XZ^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 56.96.3.w.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x+y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x^{3}t-6x^{2}yt+10xy^{2}t-4y^{3}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
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.c.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-8.c.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.1-56.n.1.2 | $56$ | $2$ | $2$ | $1$ | $1$ | $2$ |
| 56.96.1-56.n.1.8 | $56$ | $2$ | $2$ | $1$ | $1$ | $2$ |
| 56.96.2-56.a.1.8 | $56$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 56.96.2-56.a.1.12 | $56$ | $2$ | $2$ | $2$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.384.5-56.z.1.1 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.384.5-56.z.1.6 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.384.5-56.z.2.2 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.384.5-56.z.2.8 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.384.5-56.bb.3.1 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.384.5-56.bb.3.7 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.384.5-56.bb.4.2 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.384.5-56.bb.4.8 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 56.1536.53-56.cw.1.20 | $56$ | $8$ | $8$ | $53$ | $8$ | $1^{20}\cdot2^{7}\cdot4^{4}$ |
| 56.4032.151-56.ez.1.22 | $56$ | $21$ | $21$ | $151$ | $24$ | $1^{16}\cdot2^{26}\cdot4^{4}\cdot6^{2}\cdot12^{3}\cdot16$ |
| 56.5376.201-56.fa.1.22 | $56$ | $28$ | $28$ | $201$ | $31$ | $1^{36}\cdot2^{33}\cdot4^{8}\cdot6^{2}\cdot12^{3}\cdot16$ |
| 112.384.7-112.a.1.4 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.a.1.11 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.d.1.7 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.d.1.13 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.l.1.4 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.l.1.11 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.m.1.3 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.m.1.15 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.x.1.3 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.x.1.15 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.y.1.2 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.y.1.14 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.bg.1.3 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.bg.1.15 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.bj.1.2 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 112.384.7-112.bj.1.14 | $112$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 168.384.5-168.hl.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hl.1.12 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hl.2.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hl.2.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hn.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hn.1.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hn.2.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hn.2.14 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hd.1.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hd.1.15 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hd.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hd.2.13 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hf.1.2 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hf.1.11 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hf.2.4 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hf.2.9 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |