Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | 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 | $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ | 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: | 12V1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.1.1660 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&6\\18&7\end{bmatrix}$, $\begin{bmatrix}13&6\\0&23\end{bmatrix}$, $\begin{bmatrix}17&8\\12&23\end{bmatrix}$, $\begin{bmatrix}19&10\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $S_3\times C_2^3:D_4$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.cl.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $384$ |
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 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2}\cdot\frac{16x^{2}y^{30}+2821248x^{2}y^{28}z^{2}+7304033280x^{2}y^{26}z^{4}+769548886016x^{2}y^{24}z^{6}+17661894131712x^{2}y^{22}z^{8}+176318749081600x^{2}y^{20}z^{10}+4497468800630784x^{2}y^{18}z^{12}+165409340818194432x^{2}y^{16}z^{14}+2983525021818814464x^{2}y^{14}z^{16}+25632090174401806336x^{2}y^{12}z^{18}+58059595351616126976x^{2}y^{10}z^{20}-798695387899424145408x^{2}y^{8}z^{22}-7667518083839345819648x^{2}y^{6}z^{24}-28540212271055842050048x^{2}y^{4}z^{26}-47709013045832310587392x^{2}y^{2}z^{28}-27262293209257745580032x^{2}z^{30}+2880xy^{30}z+26795520xy^{28}z^{3}+45526474752xy^{26}z^{5}+1785092702208xy^{24}z^{7}+36595786579968xy^{22}z^{9}+277634481127424xy^{20}z^{11}-7302939521581056xy^{18}z^{13}-140599674129088512xy^{16}z^{15}+1565556273133387776xy^{14}z^{17}+73169293279651430400xy^{12}z^{19}+983770238716999630848xy^{10}z^{21}+6941295060535901945856xy^{8}z^{23}+27475280191614420516864xy^{6}z^{25}+57080426600397451296768xy^{4}z^{27}+47709013256938543120384xy^{2}z^{29}+y^{32}+36656y^{30}z^{2}+974901120y^{28}z^{4}+219990256640y^{26}z^{6}+6264136048640y^{24}z^{8}+71883165728768y^{22}z^{10}+261565812572160y^{20}z^{12}-25127910012616704y^{18}z^{14}-1106918490607976448y^{16}z^{16}-22428912064516849664y^{14}z^{18}-260818897549316325376y^{12}z^{20}-1845310773072972742656y^{10}z^{22}-7873096944563353812992y^{8}z^{24}-18529834003593351397376y^{6}z^{26}-16612956898535439073280y^{4}z^{28}+13631147237947570388992y^{2}z^{30}+27262293490732722290688z^{32}}{zy^{4}(y^{2}+8z^{2})^{2}(112x^{2}y^{20}z-97792x^{2}y^{18}z^{3}-16081152x^{2}y^{16}z^{5}+52953088x^{2}y^{14}z^{7}+23182639104x^{2}y^{12}z^{9}+278334013440x^{2}y^{10}z^{11}+558037467136x^{2}y^{8}z^{13}+1073741824x^{2}y^{6}z^{15}-4294967296x^{2}y^{4}z^{17}-8589934592x^{2}y^{2}z^{19}-4294967296x^{2}z^{21}-xy^{22}+1640xy^{20}z^{2}+490944xy^{18}z^{4}-37654528xy^{16}z^{6}-5075755008xy^{14}z^{8}-98772910080xy^{12}z^{10}-417796194304xy^{10}z^{12}+369098752xy^{8}z^{14}-822083584xy^{6}z^{16}-2013265920xy^{4}z^{18}-1073741824xy^{2}z^{20}-13y^{22}z-8152y^{20}z^{3}+2790336y^{18}z^{5}+381992192y^{16}z^{7}+9569214464y^{14}z^{9}+40454062080y^{12}z^{11}-139940331520y^{10}z^{13}-554690412544y^{8}z^{15}+1929379840y^{6}z^{17}-15166603264y^{4}z^{19}-26843545600y^{2}z^{21}-12884901888z^{23})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.0-12.a.2.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-12.a.2.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.o.2.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.o.2.18 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.bx.1.12 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.bx.1.16 | $24$ | $2$ | $2$ | $1$ | $0$ | 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.e.2.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.f.4.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.g.2.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.h.3.10 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.bw.2.5 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.bx.4.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.bz.4.2 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.ca.3.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.576.9-24.d.2.10 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 120.384.5-120.iz.3.16 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ja.3.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.jc.3.14 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.jd.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.jq.4.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.jr.4.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.jt.4.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ju.3.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.iz.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ja.2.8 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.jc.4.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.jd.4.8 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.jq.1.12 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.jr.3.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.jt.2.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ju.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.iz.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ja.4.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.jc.3.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.jd.3.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.jq.1.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.jr.4.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.jt.2.4 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ju.2.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.iz.3.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ja.3.16 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.jc.3.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.jd.3.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.jq.3.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.jr.4.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.jt.3.12 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ju.4.12 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |