Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.192.1.1078 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}7&2\\18&43\end{bmatrix}$, $\begin{bmatrix}23&42\\4&33\end{bmatrix}$, $\begin{bmatrix}37&46\\48&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.96.1.t.2 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $16128$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 y^{2} - y z + z^{2} + w^{2} $ |
| $=$ | $14 x^{2} - 3 y^{2} - 2 y z + 2 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} + 3 x^{2} y^{2} + 112 x^{2} z^{2} + 2 y^{4} + 84 y^{2} z^{2} + 882 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^4\cdot7^2}\cdot\frac{29274322430115yz^{23}+184010026703580yz^{21}w^{2}+501845527373400yz^{19}w^{4}+778372654701600yz^{17}w^{6}+750369003943104yz^{15}w^{8}+454500231365376yz^{13}w^{10}+160383273103872yz^{11}w^{12}+22473470638080yz^{9}w^{14}-4032000145152yz^{7}w^{16}-1543916473344yz^{5}w^{18}-5313337344yz^{3}w^{20}+19224059904yzw^{22}+13384524723367z^{24}+95961644164533z^{22}w^{2}+300431070677934z^{20}w^{4}+539371153594840z^{18}w^{6}+607568288240760z^{16}w^{8}+433541233312320z^{14}w^{10}+179600049469568z^{12}w^{12}+25608227301888z^{10}w^{14}-11123540577408z^{8}w^{16}-4765186935040z^{6}w^{18}+35068432896z^{4}w^{20}+198026078208z^{2}w^{22}-14723188736w^{24}}{w^{8}(10941357yz^{15}+43765428yz^{13}w^{2}+69667416yz^{11}w^{4}+56142240yz^{9}w^{6}+24211488yz^{7}w^{8}+5507712yz^{5}w^{10}+622848yz^{3}w^{12}+27648yzw^{14}+10470761z^{16}+49429387z^{14}w^{2}+94700242z^{12}w^{4}+94385368z^{10}w^{6}+52418632z^{8}w^{8}+16408672z^{6}w^{10}+2892992z^{4}w^{12}+269568z^{2}w^{14}+10368w^{16})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 56.96.1.t.2 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{7}w$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}+3X^{2}Y^{2}+2Y^{4}+112X^{2}Z^{2}+84Y^{2}Z^{2}+882Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.i.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.0-56.h.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.h.1.16 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.j.2.7 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.j.2.14 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.t.2.6 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.t.2.9 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.v.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-56.v.1.13 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.1-8.i.2.3 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.p.1.5 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.p.1.9 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.s.1.7 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-56.s.1.9 | $56$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 56.1536.49-56.fm.2.10 | $56$ | $8$ | $8$ | $49$ | $5$ | $1^{20}\cdot2^{6}\cdot4^{4}$ |
| 56.4032.145-56.ow.1.5 | $56$ | $21$ | $21$ | $145$ | $19$ | $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
| 56.5376.193-56.pq.2.1 | $56$ | $28$ | $28$ | $193$ | $24$ | $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
| 112.384.9-112.fk.2.6 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.fl.2.6 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.fp.2.6 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.fq.2.6 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |