Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $4$ 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: | 12L1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.12 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&20\\4&3\end{bmatrix}$, $\begin{bmatrix}11&6\\18&1\end{bmatrix}$, $\begin{bmatrix}17&21\\18&7\end{bmatrix}$, $\begin{bmatrix}21&2\\8&9\end{bmatrix}$, $\begin{bmatrix}21&5\\20&9\end{bmatrix}$ |
| 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.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 156x - 560 $ |
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 |
|---|
| $(0:1:0)$, $(-10:0:1)$, $(-4:0:1)$, $(14:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^6\cdot3^6}\cdot\frac{36x^{2}y^{10}-28512x^{2}y^{8}z^{2}-409826304x^{2}y^{6}z^{4}+1280310810624x^{2}y^{4}z^{6}+13560944718200832x^{2}y^{2}z^{8}+18932127234638413824x^{2}z^{10}-252xy^{10}z-72576xy^{8}z^{3}-6256569600xy^{6}z^{5}+42859393007616xy^{4}z^{7}+230957502683922432xy^{2}z^{9}+265452280319874170880xz^{11}-y^{12}+144y^{10}z^{2}-35385984y^{8}z^{4}+15190447104y^{6}z^{6}+790571790323712y^{4}z^{8}+1743857282426535936y^{2}z^{10}+760894090152386494464z^{12}}{z^{4}y^{6}(12x^{2}+204xz+y^{2}+624z^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| 6.18.0.b.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.0.n.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.i.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.bx.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.by.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.cg.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.ch.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.3.bl.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.cz.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.hd.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.hg.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.lo.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.lr.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.mx.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.na.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.qm.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.qn.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.qv.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.qw.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 72.108.5.b.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.108.5.bc.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 120.72.1.oe.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.of.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.oh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.oi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.emk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eml.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.emy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.emz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eoo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eop.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epu.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epv.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.epy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.brf.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.bwz.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.1.fy.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.fz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.gb.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.gc.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.3.eas.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eat.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ebg.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ebh.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ecw.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ecx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.edk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.edl.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eec.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eed.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eef.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eeg.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.21.bbz.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
| 264.72.1.fu.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.fv.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.fx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.fy.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.3.eas.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eat.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ebg.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ebh.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ecw.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ecx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.edk.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.edl.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eec.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eed.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eef.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eeg.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.1.fy.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.fz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.gb.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.gc.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.3.eas.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eat.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ebg.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ebh.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ecw.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ecx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.edk.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.edl.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eec.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eed.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eef.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eeg.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |