Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $192$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.1687 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&8\\12&1\end{bmatrix}$, $\begin{bmatrix}1&13\\12&7\end{bmatrix}$, $\begin{bmatrix}11&15\\12&11\end{bmatrix}$, $\begin{bmatrix}19&23\\12&11\end{bmatrix}$, $\begin{bmatrix}23&13\\12&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.1035865 |
| Contains $-I$: | no $\quad$ (see 24.48.1.es.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{6}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 17x + 15 $ |
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 |
|---|
| $(3:0:1)$, $(1:0:1)$, $(-5:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2}\cdot\frac{1408x^{2}y^{14}+2837390880x^{2}y^{12}z^{2}+852859782144x^{2}y^{10}z^{4}+40684227015680x^{2}y^{8}z^{6}-14198899790643200x^{2}y^{6}z^{8}+2044197213532913664x^{2}y^{4}z^{10}-114381597197698334720x^{2}y^{2}z^{12}+2271857249241291292672x^{2}z^{14}+674384xy^{14}z+28810315200xy^{12}z^{3}+3536050837248xy^{10}z^{5}-134678248503296xy^{8}z^{7}+89133784780374016xy^{6}z^{9}-10270549770837295104xy^{4}z^{11}+504857792897761673216xy^{2}z^{13}-9087432106146290204672xz^{15}+y^{16}+115447536y^{14}z^{2}+139252204320y^{12}z^{4}+3977931004160y^{10}z^{6}+1845366839198720y^{8}z^{8}-361850542509588480y^{6}z^{10}+25975269007566635008y^{4}z^{12}-769119329672038973440y^{2}z^{14}+6815581356742225690624z^{16}}{y^{2}(x^{2}y^{12}-63648x^{2}y^{10}z^{2}+95435264x^{2}y^{8}z^{4}-18043782144x^{2}y^{6}z^{6}+743109054464x^{2}y^{4}z^{8}-8589934592x^{2}y^{2}z^{10}+68719476736x^{2}z^{12}-78xy^{12}z+951216xy^{10}z^{3}-666194432xy^{8}z^{5}+87721647104xy^{6}z^{7}-2970884202496xy^{4}z^{9}-30064771072xy^{2}z^{11}+274877906944xz^{13}+2825y^{12}z^{2}-10362000y^{10}z^{4}+3154535680y^{8}z^{6}-193857198080y^{6}z^{8}+2231113568256y^{4}z^{10}+73014444032y^{2}z^{12}-343597383680z^{14})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.0-12.g.1.11 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.g.1.21 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.y.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.y.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-24.et.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1-24.et.1.7 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.192.1-24.cy.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.cy.2.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.cy.3.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.cy.4.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.3-24.dg.1.15 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.du.1.12 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.eg.1.14 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.eg.2.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.eh.1.14 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.eh.2.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.eo.1.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.es.1.10 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.288.5-24.dq.1.12 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 72.288.5-72.w.1.17 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.bo.1.10 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.bu.1.9 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.1-120.qu.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.qu.2.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.qu.3.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.qu.4.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.3-120.ji.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.jm.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.kk.1.27 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.kk.2.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.kl.1.23 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.kl.2.3 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.kw.1.24 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.la.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.480.17-120.qw.1.11 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.qu.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.qu.2.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.qu.3.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.qu.4.11 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.3-168.hi.1.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.hm.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ik.1.16 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ik.2.14 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.il.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.il.2.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.is.1.15 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.iw.1.22 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.1-264.qu.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.qu.2.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.qu.3.12 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.qu.4.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.3-264.hi.1.20 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.hm.1.24 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ik.1.15 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ik.2.27 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.il.1.3 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.il.2.23 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.is.1.24 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.iw.1.12 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.1-312.qu.1.9 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.qu.2.9 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.qu.3.11 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.qu.4.11 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.3-312.ji.1.12 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.jm.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.kk.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.kk.2.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.kl.1.18 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.kl.2.18 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.kw.1.14 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.la.1.12 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |