Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $192$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $8$ 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: | 12T1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.13 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&11\\16&21\end{bmatrix}$, $\begin{bmatrix}19&12\\0&1\end{bmatrix}$, $\begin{bmatrix}21&4\\10&9\end{bmatrix}$, $\begin{bmatrix}23&6\\18&23\end{bmatrix}$ |
| 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: | $1024$ |
Jacobian
| Conductor: | $2^{6}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} + 3x - 3 $ |
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 |
|---|
| $(0:1:0)$, $(1:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{24x^{2}y^{22}+3904x^{2}y^{20}z^{2}+547328x^{2}y^{18}z^{4}+14561280x^{2}y^{16}z^{6}-942735360x^{2}y^{14}z^{8}-59348877312x^{2}y^{12}z^{10}-525072334848x^{2}y^{10}z^{12}+8172886556672x^{2}y^{8}z^{14}+74850408333312x^{2}y^{6}z^{16}+36989332094976x^{2}y^{4}z^{18}-422684911468544x^{2}y^{2}z^{20}-338168545017856x^{2}z^{22}-192xy^{22}z-14848xy^{20}z^{3}+126222336xy^{16}z^{7}+6670516224xy^{14}z^{9}+70250397696xy^{12}z^{11}-2307001417728xy^{10}z^{13}-31426812575744xy^{8}z^{15}-26417270095872xy^{6}z^{17}+496661428174848xy^{4}z^{19}+929911959191552xy^{2}z^{21}-y^{24}+200y^{22}z^{2}-68032y^{20}z^{4}-8499712y^{18}z^{6}-420421632y^{16}z^{8}-358809600y^{14}z^{10}+379337572352y^{12}z^{12}+5116342042624y^{10}z^{14}+2008081760256y^{8}z^{16}-164673743749120y^{6}z^{18}-554766966980608y^{4}z^{20}-338142775214080y^{2}z^{22}+338099825541120z^{24}}{z^{4}y^{12}(8x^{2}y^{6}+192x^{2}y^{4}z^{2}-6656x^{2}y^{2}z^{4}-28672x^{2}z^{6}+1536xy^{4}z^{3}+20480xy^{2}z^{5}-y^{8}-168y^{6}z^{2}-3136y^{4}z^{4}+512y^{2}z^{6}+24576z^{8})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(6)$ | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.c.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.fc.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 |
|---|---|---|---|---|---|---|
| 24.144.5.d.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bf.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.cn.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.cp.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ib.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ig.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ip.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.iq.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.216.9.bc.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.216.9.bo.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.216.9.cm.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.144.5.ewi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ewj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ewp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ewq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eym.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eyn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eyt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eyu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cii.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cij.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cip.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ciq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ckm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ckn.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ckt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cku.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cii.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cij.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cip.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ciq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ckm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ckn.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ckt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cku.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cii.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cij.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cip.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ciq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ckm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ckn.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ckt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cku.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |