Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| 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: | $1$ | ||||||
| $\Q$-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: | 24.192.1.1734 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&6\\20&1\end{bmatrix}$, $\begin{bmatrix}13&9\\0&23\end{bmatrix}$, $\begin{bmatrix}19&0\\8&23\end{bmatrix}$, $\begin{bmatrix}19&3\\8&11\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $(C_2\times D_8):D_6$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.da.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 24x + 56 $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
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}{2^3\cdot3^3}\cdot\frac{48x^{2}y^{30}+5692032x^{2}y^{28}z^{2}+188620876800x^{2}y^{26}z^{4}+865293713842176x^{2}y^{24}z^{6}-18848487595015471104x^{2}y^{22}z^{8}-40618300162102009528320x^{2}y^{20}z^{10}+10067995772968125887152128x^{2}y^{18}z^{12}+15473670617226567981289439232x^{2}y^{16}z^{14}-794061797776012679744248086528x^{2}y^{14}z^{16}-445561858290011775724810200416256x^{2}y^{12}z^{18}+16553690172472716947879918926036992x^{2}y^{10}z^{20}+1282889542337633135313051365274550272x^{2}y^{8}z^{22}-22416156065344262934735628130838380544x^{2}y^{6}z^{24}-207500837287607204438376295614426120192x^{2}y^{4}z^{26}-29405065906854270953324094403897196544x^{2}y^{2}z^{28}+454034544748725490855096936058918535168x^{2}z^{30}-96xy^{30}z+44603136xy^{28}z^{3}+4318642096128xy^{26}z^{5}+95349934306934784xy^{24}z^{7}+264834223411475644416xy^{22}z^{9}-202168256505657135464448xy^{20}z^{11}-326010707038073564713451520xy^{18}z^{13}+28318858415885770964286308352xy^{16}z^{15}+23314170523492828896304627187712xy^{14}z^{17}-1254067647065260649666920870576128xy^{12}z^{19}-175640356042420846165764222172004352xy^{10}z^{21}+6280089793223554685578833408942931968xy^{8}z^{23}+109078772270106141130118994521481019392xy^{6}z^{25}-647561431005250698698444672932503355392xy^{4}z^{27}-3060667747674787596117320625848536006656xy^{2}z^{29}-908069089497450981710193872117837070336xz^{31}-y^{32}-176640y^{30}z^{2}-10764845568y^{28}z^{4}-235955358056448y^{26}z^{6}-903725526741073920y^{24}z^{8}+2161786961867009163264y^{22}z^{10}+3746998723354151425671168y^{20}z^{12}-546350747573344709011046400y^{18}z^{14}-490000357235068143306471899136y^{16}z^{16}+44766027472498693035127197401088y^{14}z^{18}+4906717146574462988126524211724288y^{12}z^{20}-456290103371475960071254237771726848y^{10}z^{22}+2046651966672012326530636387253747712y^{8}z^{24}+303155130874927691644962365807693660160y^{6}z^{26}-383483689011572341941729617474513534976y^{4}z^{28}-5925548112296553353585485694613464285184y^{2}z^{30}-3654728615697158484080862699595140956160z^{32}}{z^{2}y^{2}(y^{2}-216z^{2})^{6}(36x^{2}y^{14}-150336x^{2}y^{12}z^{2}+320806656x^{2}y^{10}z^{4}-593535983616x^{2}y^{8}z^{6}+129901662683136x^{2}y^{6}z^{8}-10438106657587200x^{2}y^{4}z^{10}+296148833645101056x^{2}y^{2}z^{12}+252xy^{14}z+1226016xy^{12}z^{3}-2238928128xy^{10}z^{5}-2081366710272xy^{8}z^{7}+826245624840192xy^{6}z^{9}-114858668772163584xy^{4}z^{11}+7403720841127526400xy^{2}z^{13}-191904444202025484288xz^{15}-y^{16}-11520y^{14}z^{2}+41824512y^{12}z^{4}-60089942016y^{10}z^{6}+1252375437312y^{8}z^{8}+2255216157130752y^{6}z^{10}-336140887693197312y^{4}z^{12}+18953525353286467584y^{2}z^{14}-383808888404050968576z^{16})}$ |
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.c.3.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-12.c.3.10 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bu.4.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bu.4.24 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.ik.1.2 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.ik.1.12 | $24$ | $2$ | $2$ | $1$ | $1$ | 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.ex.4.4 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.ey.3.2 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.ez.1.4 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.fa.2.8 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.fn.3.2 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 24.384.5-24.fo.4.4 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.fp.2.8 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 24.384.5-24.fq.1.4 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.576.9-24.bi.1.11 | $24$ | $3$ | $3$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 120.384.5-120.wj.4.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.wk.4.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.wl.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.wm.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.wz.4.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.xa.4.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.xb.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.xc.1.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.wj.4.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.wk.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.wl.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.wm.4.14 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.wz.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.xa.4.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.xb.4.14 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.xc.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.wj.4.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.wk.4.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.wl.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.wm.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.wz.4.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.xa.4.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.xb.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.xc.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.wj.4.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.wk.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.wl.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.wm.4.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.wz.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.xa.4.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.xb.4.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.xc.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |