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: | none |
Other labels
| Cummins and Pauli (CP) label: | 12K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.110 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&19\\8&9\end{bmatrix}$, $\begin{bmatrix}5&3\\12&17\end{bmatrix}$, $\begin{bmatrix}7&13\\8&7\end{bmatrix}$, $\begin{bmatrix}17&15\\18&19\end{bmatrix}$ |
| 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
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} - 3 x y + 3 y^{2} - 2 w^{2} $ |
| $=$ | $3 x y + 4 y^{2} - 4 y z + 4 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 37 x^{4} - 22 x^{3} z - 6 x^{2} y^{2} + 15 x^{2} z^{2} - 4 x z^{3} + z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2z$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^3}{3^3}\cdot\frac{3316770362736xz^{8}+11209681665432xz^{6}w^{2}+13488993492012xz^{4}w^{4}+32808941607942xz^{2}w^{6}-6075640136512xw^{8}-1964460817728y^{2}z^{7}-22756015717488y^{2}z^{5}w^{2}-7407669962952y^{2}z^{3}w^{4}+43322792437848y^{2}zw^{6}-940571418096yz^{8}+1447875589680yz^{6}w^{2}-13309502107068yz^{4}w^{4}-40647274178304yz^{2}w^{6}-7898089742077yw^{8}+567296462592z^{9}-3549595874976z^{7}w^{2}+28835933953248z^{5}w^{4}+43745255477256z^{3}w^{6}}{568719198xz^{8}+100985432xz^{6}w^{2}-48692592xz^{4}w^{4}+7294032xz^{2}w^{6}-336841704y^{2}z^{7}-859231649y^{2}z^{5}w^{2}-25559230y^{2}z^{3}w^{4}+39509340y^{2}zw^{6}-161277678yz^{8}+821284951yz^{6}w^{2}+237136700yz^{4}w^{4}-51091080yz^{2}w^{6}+3647016yw^{8}+97273056z^{9}-293831504z^{7}w^{2}-98962272z^{5}w^{4}+9725376z^{3}w^{6}}$ |
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.f.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.0.a.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.f.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.el.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.hk.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.hp.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.ki.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.kk.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.kw.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ky.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.7.bk.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.324.19.bi.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
| 120.72.3.bie.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.big.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bis.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.biu.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bki.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bkk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bkw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bky.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.cg.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.ey.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.3.bfw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bfy.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bgk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bgm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bia.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bic.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bio.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.biq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.19.sx.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
| 264.72.3.bfw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bfy.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bgk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bgm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bia.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bic.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bio.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.biq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bfw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bfy.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bgk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bgm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bia.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bic.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bio.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.biq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |