Invariants
| Level: | $104$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| 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 | $1^{4}\cdot2^{2}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
Level structure
| $\GL_2(\Z/104\Z)$-generators: | $\begin{bmatrix}9&64\\52&97\end{bmatrix}$, $\begin{bmatrix}17&84\\40&31\end{bmatrix}$, $\begin{bmatrix}29&20\\4&25\end{bmatrix}$, $\begin{bmatrix}71&100\\44&33\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.96.1.f.2 for the level structure with $-I$) |
| Cyclic 104-isogeny field degree: | $28$ |
| Cyclic 104-torsion field degree: | $672$ |
| Full 104-torsion field degree: | $209664$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 4x $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{11328x^{2}y^{28}z^{2}-489338880x^{2}y^{24}z^{6}+2591686066176x^{2}y^{20}z^{10}+1085247282216960x^{2}y^{16}z^{14}+303432552482340864x^{2}y^{12}z^{18}+37073617649884200960x^{2}y^{8}z^{22}+830103506406674006016x^{2}y^{4}z^{26}+1180591550348667125760x^{2}z^{30}-32xy^{30}z+41366016xy^{26}z^{5}-10544873472xy^{22}z^{9}+168210524536832xy^{18}z^{13}+68116365617135616xy^{14}z^{17}+10403318682573864960xy^{10}z^{21}+553402316713728409600xy^{6}z^{25}+3246626974565067128832xy^{2}z^{29}+y^{32}-265728y^{28}z^{4}+51360677888y^{24}z^{8}+20046973239296y^{20}z^{12}+7673506078654464y^{16}z^{16}+1292522115118923776y^{12}z^{20}+96845465623164092416y^{8}z^{24}+737869666191358820352y^{4}z^{28}+281474976710656z^{32}}{z^{2}y^{8}(x^{2}y^{20}+10048x^{2}y^{16}z^{4}-68075520x^{2}y^{12}z^{8}+81606737920x^{2}y^{8}z^{12}+16492892520448x^{2}y^{4}z^{16}+70367670435840x^{2}z^{20}-784xy^{18}z^{3}+204800xy^{14}z^{7}+1072496640xy^{10}z^{11}+5497507807232xy^{6}z^{15}+123145570746368xy^{2}z^{19}-24y^{20}z^{2}+312832y^{16}z^{6}-532545536y^{12}z^{10}+687196864512y^{8}z^{14}+26387339542528y^{4}z^{18}+4294967296z^{22})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 104.96.0-8.b.1.3 | $104$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 104.96.0-8.b.1.8 | $104$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 104.96.0-8.c.1.2 | $104$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 104.96.0-8.c.1.8 | $104$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 104.96.1-8.g.1.5 | $104$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 104.96.1-8.g.1.8 | $104$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 104.384.5-8.c.1.2 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-8.c.1.3 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-8.d.2.1 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-8.d.2.4 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.z.2.2 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.z.2.8 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.ba.2.2 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 104.384.5-104.ba.2.5 | $104$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-16.b.2.5 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-16.b.2.7 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-16.i.2.7 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-16.i.2.8 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-16.o.2.5 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-16.o.2.6 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-16.v.2.5 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-16.v.2.7 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.w.2.10 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.w.2.16 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.cx.2.10 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.cx.2.16 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.de.2.12 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.de.2.14 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.fa.2.11 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 208.384.5-208.fa.2.16 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-24.bh.2.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-24.bh.2.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-24.bi.2.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-24.bi.2.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.hp.1.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.hp.1.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.hr.2.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.hr.2.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |