Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $192$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4^{2}$ | ||
| 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: | 24J1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.1.2708 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&6\\4&19\end{bmatrix}$, $\begin{bmatrix}7&3\\4&19\end{bmatrix}$, $\begin{bmatrix}11&9\\4&7\end{bmatrix}$, $\begin{bmatrix}23&18\\12&11\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $S_3\times D_4:D_4$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.dq.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $4$ |
| Full 24-torsion field degree: | $384$ |
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} + 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 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^{12}}\cdot\frac{16x^{2}y^{30}+1152x^{2}y^{28}z^{2}-384000x^{2}y^{26}z^{4}+12181504x^{2}y^{24}z^{6}-90636288x^{2}y^{22}z^{8}-2991063040x^{2}y^{20}z^{10}+98738110464x^{2}y^{18}z^{12}-1551926034432x^{2}y^{16}z^{14}+15005273554944x^{2}y^{14}z^{16}-85076859682816x^{2}y^{12}z^{18}+168689135517696x^{2}y^{10}z^{20}+1469084973662208x^{2}y^{8}z^{22}-14307944812249088x^{2}y^{6}z^{24}+53489041668046848x^{2}y^{4}z^{26}-89579411338166272x^{2}y^{2}z^{28}+51228445761339392x^{2}z^{30}+23040xy^{28}z^{3}-724992xy^{26}z^{5}-14057472xy^{24}z^{7}+1011351552xy^{22}z^{9}-24930942976xy^{20}z^{11}+331031248896xy^{18}z^{13}-2495644434432xy^{16}z^{15}+3904125272064xy^{14}z^{17}+151912993259520xy^{12}z^{19}-1995819762843648xy^{10}z^{21}+13307938986786816xy^{8}z^{23}-51747415249649664xy^{6}z^{25}+107453072359292928xy^{4}z^{27}-89790517570699264xy^{2}z^{29}-y^{32}-784y^{30}z^{2}+1920y^{28}z^{4}+3599360y^{26}z^{6}-141762560y^{24}z^{8}+2683240448y^{22}z^{10}-28036300800y^{20}z^{12}+14759755776y^{18}z^{14}+3821984022528y^{16}z^{16}-63409554980864y^{14}z^{18}+596349766598656y^{12}z^{20}-3701213837131776y^{10}z^{22}+14901268774387712y^{8}z^{24}-34448798809849856y^{6}z^{26}+30566423252172800y^{4}z^{28}+26247541578268672y^{2}z^{30}-51509920738050048z^{32}}{z^{8}y^{8}(y^{2}-8z^{2})^{3}(480x^{2}y^{6}z^{2}-10496x^{2}y^{4}z^{4}+36864x^{2}y^{2}z^{6}-48xy^{8}z+2112xy^{6}z^{3}-11520xy^{4}z^{5}-36864xy^{2}z^{7}+147456xz^{9}+y^{10}-120y^{8}z^{2}-160y^{6}z^{4}+22016y^{4}z^{6}-110592y^{2}z^{8}+147456z^{10})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.0-12.c.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-12.c.1.23 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bs.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bs.1.21 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.ix.1.12 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.ix.1.22 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.384.5-24.de.3.9 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.dk.3.8 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.eq.1.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.er.3.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.fm.3.5 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.fo.3.7 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.fu.2.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.fw.3.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.576.9-24.bd.2.3 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 120.384.5-120.baq.4.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bas.3.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bbg.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bbi.3.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdc.4.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bde.3.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bds.2.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdu.3.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.baq.2.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bas.3.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bbg.2.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bbi.2.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdc.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bde.2.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bds.3.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdu.2.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.baq.3.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bas.3.15 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bbg.1.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bbi.3.13 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdc.3.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bde.3.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bds.1.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdu.2.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.baq.3.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bas.3.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bbg.3.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bbi.3.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdc.3.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bde.3.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bds.2.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdu.3.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |