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: | $16$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.192.1.957 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}9&4\\50&27\end{bmatrix}$, $\begin{bmatrix}19&48\\2&9\end{bmatrix}$, $\begin{bmatrix}39&36\\40&35\end{bmatrix}$, $\begin{bmatrix}53&8\\22&39\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.96.1.g.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $8$ |
| 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
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x $ |
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.
| Weierstrass model |
|---|
| $(1:0:1)$, $(-1:0:1)$, $(0:1:0)$, $(0:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{174228x^{2}y^{28}z^{2}+3016602930x^{2}y^{24}z^{6}+161742749031x^{2}y^{20}z^{10}+574925030955x^{2}y^{16}z^{14}+272387551224x^{2}y^{12}z^{18}+34999202565x^{2}y^{8}z^{22}+1509948741x^{2}y^{4}z^{26}+16777215x^{2}z^{30}+712xy^{30}z+324760626xy^{26}z^{5}+67075202952xy^{22}z^{9}+588642243017xy^{18}z^{13}+597507073344xy^{14}z^{17}+126615719865xy^{10}z^{21}+8808039100xy^{6}z^{25}+184549377xy^{2}z^{29}+y^{32}+15867528y^{28}z^{4}+15769425308y^{24}z^{8}+271032482654y^{20}z^{12}+399504719004y^{16}z^{16}+97855028624y^{12}z^{20}+7449250366y^{8}z^{24}+167772858y^{4}z^{28}+z^{32}}{zy^{4}(31x^{2}y^{24}z-998x^{2}y^{20}z^{5}+194106x^{2}y^{16}z^{9}+1831018x^{2}y^{12}z^{13}-13958543x^{2}y^{8}z^{17}-27787305x^{2}y^{4}z^{21}-1048575x^{2}z^{25}-xy^{26}+1760xy^{22}z^{4}+54206xy^{18}z^{8}-1438316xy^{14}z^{12}+5176757xy^{10}z^{16}-47710168xy^{6}z^{20}-9437185xy^{2}z^{24}-380y^{24}z^{3}-26676y^{20}z^{7}-598511y^{16}z^{11}+6033852y^{12}z^{15}-27263640y^{8}z^{19}-8388566y^{4}z^{23}-z^{27})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.96.0-8.b.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-8.b.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-8.c.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-8.c.1.6 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-8.k.1.4 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-8.k.1.6 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-8.l.1.3 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.0-8.l.1.6 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.96.1-8.h.1.5 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-8.h.1.6 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-8.i.2.2 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-8.i.2.7 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-8.k.2.2 | $56$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 56.96.1-8.k.2.5 | $56$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 56.384.5-8.d.1.2 | $56$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 56.384.5-8.d.2.3 | $56$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 56.384.5-56.bb.2.1 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 56.384.5-56.bb.4.3 | $56$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 56.1536.49-56.fw.2.7 | $56$ | $8$ | $8$ | $49$ | $3$ | $1^{20}\cdot2^{6}\cdot4^{4}$ |
| 56.4032.145-56.pk.2.5 | $56$ | $21$ | $21$ | $145$ | $19$ | $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$ |
| 56.5376.193-56.qe.2.7 | $56$ | $28$ | $28$ | $193$ | $22$ | $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
| 112.384.5-16.a.1.7 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.d.1.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.i.1.5 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.k.1.5 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.k.2.3 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.k.6.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.l.1.9 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.l.2.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.l.3.7 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.l.1.2 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.m.1.12 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.m.2.12 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.u.1.9 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-16.x.1.10 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bn.1.10 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bn.2.10 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bn.5.7 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bo.1.11 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bo.2.13 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bo.3.9 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bp.1.11 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.bp.2.10 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.cw.1.12 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.5-112.cz.2.11 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.384.9-16.bq.3.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-16.bq.4.1 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.ga.3.10 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 112.384.9-112.ga.4.10 | $112$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.5-24.bj.2.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-24.bj.4.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hv.2.15 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hv.4.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-40.bb.2.8 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-40.bb.4.6 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hn.3.14 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.384.5-280.hn.4.14 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |