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 | $2^{3}$ | ||
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: | 24.36.1.104 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&7\\22&19\end{bmatrix}$, $\begin{bmatrix}5&19\\14&11\end{bmatrix}$, $\begin{bmatrix}9&7\\4&21\end{bmatrix}$, $\begin{bmatrix}19&2\\10&1\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 24-isogeny field degree: | $8$ |
Cyclic 24-torsion field degree: | $64$ |
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 $ | $=$ | $ x^{2} + 2 x y - y^{2} - y z $ |
$=$ | $6 x^{2} + 4 x z + 11 y^{2} - 4 y z + z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 4 x^{3} z + 3 x^{2} z^{2} - 2 x z^{3} + 2 y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle 2y$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^3\cdot3\,\frac{93927787656202027008xz^{8}-1365461332066767168xz^{6}w^{2}-165531976938472417056xz^{4}w^{4}+60658886256961171424xz^{2}w^{6}-2967995500522123442xw^{8}+398174619160980218880y^{2}z^{7}+788398899552842766048y^{2}z^{5}w^{2}-909359966880953870688y^{2}z^{3}w^{4}+149148405420464313640y^{2}zw^{6}+228714259376949265152yz^{8}-161477119996980411744yz^{6}w^{2}-293245071303094931664yz^{4}w^{4}+132648372342114213256yz^{2}w^{6}-6936813914422569401yw^{8}+16442817423024798720z^{9}+31552337827227751488z^{7}w^{2}-18392258797185108528z^{5}w^{4}-20900069956656032632z^{3}w^{6}+5287274306665259895zw^{8}}{1304552606336139264xz^{8}+2951622016809474630xz^{6}w^{2}+2240269861701749108xz^{4}w^{4}+526968476775471304xz^{2}w^{6}-40859739273807888xw^{8}+5530203043902503040y^{2}z^{7}+10389932126473827711y^{2}z^{5}w^{2}+7876981184696918644y^{2}z^{3}w^{4}+2471276853485338220y^{2}zw^{6}+3176586935790962016yz^{8}+7112363382800512749yz^{6}w^{2}+5052267504658251430yz^{4}w^{4}+1045253967327783836yz^{2}w^{6}-93875117917190904yw^{8}+228372464208677760z^{9}+853116892279077522z^{7}w^{2}+1015730202809736092z^{5}w^{4}+492668418457117784z^{3}w^{6}+93391736569067216zw^{8}}$ |
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.c.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.0.c.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.bg.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.eh.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.hu.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.hz.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.ms.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.mv.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.mz.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.nc.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
72.108.7.by.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.324.19.bu.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
120.72.3.bmm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bmo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bna.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bnc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.boq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bos.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bpe.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.bpg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.cs.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.fk.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.bke.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bkg.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bks.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bku.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bmi.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bmk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bmw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.bmy.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.19.tj.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
264.72.3.bke.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bkg.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bks.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bku.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bmi.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bmk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bmw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.bmy.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bke.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bkg.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bks.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bku.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bmi.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bmk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bmw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.bmy.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |