Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| 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: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.977 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}9&14\\4&9\end{bmatrix}$, $\begin{bmatrix}9&14\\16&7\end{bmatrix}$, $\begin{bmatrix}11&14\\0&7\end{bmatrix}$, $\begin{bmatrix}21&22\\8&9\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 8.48.1.n.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 44x + 112 $ |
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 |
|---|
| $(0:1:0)$, $(4:0:1)$, $(6:8:1)$, $(6:-8:1)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^4}\cdot\frac{48x^{2}y^{14}-356896x^{2}y^{12}z^{2}+701893632x^{2}y^{10}z^{4}-723570779136x^{2}y^{8}z^{6}+443515503378432x^{2}y^{6}z^{8}-165078270613192704x^{2}y^{4}z^{10}+35042697816563515392x^{2}y^{2}z^{12}-3279970130870308700160x^{2}z^{14}-1264xy^{14}z+5240064xy^{12}z^{3}-8619949824xy^{10}z^{5}+7846356996096xy^{8}z^{7}-4372637482614784xy^{6}z^{9}+1496846267870871552xy^{4}z^{11}-293187273636914921472xy^{2}z^{13}+25114253234762353213440xz^{15}-y^{16}+22656y^{14}z^{2}-57368832y^{12}z^{4}+71446622208y^{10}z^{6}-51548108275712y^{8}z^{8}+22961101307117568y^{6}z^{10}-6169291573720252416y^{4}z^{12}+893442882532622204928y^{2}z^{14}-47977490845124490428416z^{16}}{z^{2}y^{4}(x^{2}y^{8}-22688x^{2}y^{6}z^{2}+40288320x^{2}y^{4}z^{4}-19413336064x^{2}y^{2}z^{6}+2710594125824x^{2}z^{8}-48xy^{8}z+347536xy^{6}z^{3}-428539648xy^{4}z^{5}+169198223360xy^{2}z^{7}-20754624151552xz^{9}+1224y^{8}z^{2}-3698176y^{6}z^{4}+2583689472y^{4}z^{6}-598711730176y^{2}z^{8}+39648990593024z^{10})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.48.0-8.e.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-8.e.1.13 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-8.e.2.3 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-8.e.2.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-8.d.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1-8.d.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.192.1-8.f.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-8.f.2.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-8.j.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-8.j.2.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bg.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bg.2.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bo.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bo.2.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.288.9-24.ei.1.14 | $24$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.384.9-24.cm.1.18 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 120.192.1-40.bg.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-40.bg.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-40.bo.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-40.bo.2.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.eg.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.eg.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ew.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ew.2.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.480.17-40.bo.1.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-56.bg.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-56.bg.2.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-56.bo.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-56.bo.2.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.eg.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.eg.2.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ew.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ew.2.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-88.bg.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-88.bg.2.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-88.bo.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-88.bo.2.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.eg.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.eg.2.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ew.1.10 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ew.2.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-104.bg.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-104.bg.2.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-104.bo.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-104.bo.2.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.eg.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.eg.2.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ew.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ew.2.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |