Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | 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: | $6$ (of which $2$ are rational) | Cusp widths | $6^{6}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6E1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.4 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&0\\0&19\end{bmatrix}$, $\begin{bmatrix}11&9\\0&23\end{bmatrix}$, $\begin{bmatrix}15&16\\8&15\end{bmatrix}$, $\begin{bmatrix}21&23\\10&9\end{bmatrix}$, $\begin{bmatrix}23&21\\12&13\end{bmatrix}$ |
| 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
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 60x - 176 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(-4:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3}\cdot\frac{36x^{2}y^{10}-6048x^{2}y^{8}z^{2}-339987456x^{2}y^{6}z^{4}+489225111552x^{2}y^{4}z^{6}+5479268696064x^{2}y^{2}z^{8}-204093134624194560x^{2}z^{10}+828xy^{10}z-1804032xy^{8}z^{3}-1698091776xy^{6}z^{5}+5348838924288xy^{4}z^{7}-1090381493551104xy^{2}z^{9}-1822184991384403968xz^{11}+y^{12}+7632y^{10}z^{2}-34577280y^{8}z^{4}+27646424064y^{6}z^{6}+20808248537088y^{4}z^{8}-20235852515966976y^{2}z^{10}-4023249816710283264z^{12}}{z^{2}(36x^{2}y^{8}+184464x^{2}y^{6}z^{2}+222953472x^{2}y^{4}z^{4}+98977185792x^{2}y^{2}z^{6}+14763681644544x^{2}z^{8}+804xy^{8}z+2530752xy^{6}z^{3}+2474311680xy^{4}z^{5}+966103400448xy^{2}z^{7}+131813150294016xz^{9}+y^{10}+12592y^{8}z^{2}+21735232y^{6}z^{4}+13492813824y^{4}z^{6}+3422753390592y^{2}z^{8}+291033694863360z^{10})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 8.6.0.a.1 | $8$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.18.0.b.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.n.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.18.0.h.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.a.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 |
|---|---|---|---|---|---|---|
| 24.72.1.c.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.d.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.e.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.f.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.3.cc.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ce.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.cw.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.cx.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.cz.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.da.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.dl.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.dm.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.ds.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.dt.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.du.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.dw.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.4.a.1 | $72$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 72.108.7.m.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 120.72.1.e.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.f.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.k.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.l.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.hw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.hy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ie.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.if.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ig.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ii.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ks.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ku.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.la.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.lb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.lc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.le.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.k.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.w.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.1.c.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.d.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.e.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.f.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.3.hm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ho.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.hu.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.hv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.hw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.hy.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ka.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.kc.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ki.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.kj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.kk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.km.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.19.qf.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
| 264.72.1.c.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.d.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.e.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.f.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.3.hm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ho.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.hu.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.hv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.hw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.hy.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ka.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.kc.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ki.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.kj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.kk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.km.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.1.c.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.d.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.e.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.f.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.3.hm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ho.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.hu.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.hv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.hw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.hy.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ka.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.kc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ki.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.kj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.kk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.km.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |