Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $3^{4}\cdot12^{2}$ | Cusp orbits | $2\cdot4$ | ||
| 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: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 12K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.150 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&18\\6&5\end{bmatrix}$, $\begin{bmatrix}7&9\\6&11\end{bmatrix}$, $\begin{bmatrix}17&23\\4&5\end{bmatrix}$, $\begin{bmatrix}23&18\\18&13\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $2048$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 60x - 176 $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{252x^{2}y^{10}+66936672x^{2}y^{8}z^{2}+844089818112x^{2}y^{6}z^{4}+2085267153364992x^{2}y^{4}z^{6}+1573080151280762880x^{2}y^{2}z^{8}+352672937374046552064x^{2}z^{10}+27036xy^{10}z+2006930304xy^{8}z^{3}+14247883369728xy^{6}z^{5}+25755860651950080xy^{4}z^{7}+16013496045508116480xy^{2}z^{9}+3148735662138496647168xz^{11}+y^{12}+1650672y^{10}z^{2}+44090787456y^{8}z^{4}+167160141573120y^{6}z^{6}+179558357655859200y^{4}z^{8}+66164048683934220288y^{2}z^{10}+6952175650564081975296z^{12}}{36x^{2}y^{10}-19872x^{2}y^{8}z^{2}-580608x^{2}y^{6}z^{4}-10450944x^{2}y^{4}z^{6}-35831808x^{2}y^{2}z^{8}+429981696x^{2}z^{10}+252xy^{10}z+131328xy^{8}z^{3}+2923776xy^{6}z^{5}+41803776xy^{4}z^{7}+107495424xy^{2}z^{9}-1719926784xz^{11}+y^{12}-3888y^{10}z^{2}+452736y^{8}z^{4}+20984832y^{6}z^{6}+462827520y^{4}z^{8}+1863254016y^{2}z^{10}-18919194624z^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.18.0.a.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.0.j.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.d.1 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.72.3.j.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.dp.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.iw.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.ja.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.ko.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.kq.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.lc.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.le.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.7.bi.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.324.19.bk.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
| 120.72.3.bik.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bim.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.biy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bja.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bko.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bkq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.blc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ble.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.ci.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.fa.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.3.bgc.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bge.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bgq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bgs.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.big.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bii.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.biu.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.biw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.19.sz.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
| 264.72.3.bgc.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bge.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bgq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bgs.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.big.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bii.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.biu.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.biw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bgc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bge.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bgq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bgs.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.big.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bii.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.biu.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.biw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |