Invariants
| Level: | $24$ | $\SL_2$-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: | $0$ | ||||||
| $\Q$-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: | 24.192.1.948 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&16\\4&1\end{bmatrix}$, $\begin{bmatrix}5&22\\4&19\end{bmatrix}$, $\begin{bmatrix}9&20\\16&17\end{bmatrix}$, $\begin{bmatrix}23&22\\8&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^3\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 8.96.1.f.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{6}$ |
| 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 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 |
|---|
| $(0:0:1)$, $(0:1:0)$, $(2:0:1)$, $(-2:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{11328x^{2}y^{28}z^{2}-489338880x^{2}y^{24}z^{6}+2591686066176x^{2}y^{20}z^{10}+1085247282216960x^{2}y^{16}z^{14}+303432552482340864x^{2}y^{12}z^{18}+37073617649884200960x^{2}y^{8}z^{22}+830103506406674006016x^{2}y^{4}z^{26}+1180591550348667125760x^{2}z^{30}-32xy^{30}z+41366016xy^{26}z^{5}-10544873472xy^{22}z^{9}+168210524536832xy^{18}z^{13}+68116365617135616xy^{14}z^{17}+10403318682573864960xy^{10}z^{21}+553402316713728409600xy^{6}z^{25}+3246626974565067128832xy^{2}z^{29}+y^{32}-265728y^{28}z^{4}+51360677888y^{24}z^{8}+20046973239296y^{20}z^{12}+7673506078654464y^{16}z^{16}+1292522115118923776y^{12}z^{20}+96845465623164092416y^{8}z^{24}+737869666191358820352y^{4}z^{28}+281474976710656z^{32}}{z^{2}y^{8}(x^{2}y^{20}+10048x^{2}y^{16}z^{4}-68075520x^{2}y^{12}z^{8}+81606737920x^{2}y^{8}z^{12}+16492892520448x^{2}y^{4}z^{16}+70367670435840x^{2}z^{20}-784xy^{18}z^{3}+204800xy^{14}z^{7}+1072496640xy^{10}z^{11}+5497507807232xy^{6}z^{15}+123145570746368xy^{2}z^{19}-24y^{20}z^{2}+312832y^{16}z^{6}-532545536y^{12}z^{10}+687196864512y^{8}z^{14}+26387339542528y^{4}z^{18}+4294967296z^{22})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.0-8.b.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-8.b.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-8.c.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-8.c.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-8.h.2.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-8.h.2.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-8.i.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-8.i.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-8.g.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-8.g.1.11 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-8.m.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-8.m.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-8.n.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-8.n.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.384.5-8.c.1.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-8.d.2.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.bh.2.6 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 24.384.5-24.bi.2.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.576.17-24.oz.1.17 | $24$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 24.768.17-24.fr.1.17 | $24$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
| 48.384.5-16.b.2.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-16.i.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.j.2.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-16.o.2.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-16.v.2.4 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.bl.2.3 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.bs.2.2 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 48.384.5-48.cx.2.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 120.384.5-40.z.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-40.ba.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hp.1.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.hr.2.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-56.z.2.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-56.ba.2.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hp.1.14 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.hr.2.15 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.bb.2.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.cx.2.2 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.de.2.7 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.df.2.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-80.ev.2.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.hx.2.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ie.2.6 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.384.5-240.ot.2.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-88.z.2.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-88.ba.2.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.hp.1.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.hr.2.15 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-104.z.2.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-104.ba.2.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.hp.1.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.hr.2.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |