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: | 24.192.1.1607 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&18\\18&5\end{bmatrix}$, $\begin{bmatrix}7&10\\6&13\end{bmatrix}$, $\begin{bmatrix}13&18\\0&13\end{bmatrix}$, $\begin{bmatrix}23&0\\12&19\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $S_3\times C_2^3:D_4$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.cp.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: | $2^{6}\cdot3^{2}$ |
| 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}{2^6\cdot3^6}\cdot\frac{48x^{2}y^{30}+93312x^{2}y^{28}z^{2}-33592320x^{2}y^{26}z^{4}-3851937275904x^{2}y^{24}z^{6}+1266051435134976x^{2}y^{22}z^{8}-12862342823262289920x^{2}y^{20}z^{10}-24244030489958874611712x^{2}y^{18}z^{12}+12008584070097916775104512x^{2}y^{16}z^{14}-1143666561233807533878018048x^{2}y^{14}z^{16}-68949369250643486876689760256x^{2}y^{12}z^{18}+5983416926391023541696470188032x^{2}y^{10}z^{20}-59046643829758819156652351029248x^{2}y^{8}z^{22}+15212703877594496914906822974898176x^{2}y^{6}z^{24}-129290407179696595904137318285443072x^{2}y^{4}z^{26}-29405065906854270953324094403897196544x^{2}y^{2}z^{28}+454034544748725490855096936058918535168x^{2}z^{30}-96xy^{30}z+5412096xy^{28}z^{3}+12241041408xy^{26}z^{5}+27939243147264xy^{24}z^{7}+93709121252622336xy^{22}z^{9}-227442078550645014528xy^{20}z^{11}-158119931506129958338560xy^{18}z^{13}+53359158646669320589934592xy^{16}z^{15}-2326626964717237267184222208xy^{14}z^{17}+247171155540851057896080801792xy^{12}z^{19}-38872583963244525196446145708032xy^{10}z^{21}-1321470352616209426794821883789312xy^{8}z^{23}+159535601304806363096888247941332992xy^{6}z^{25}-499837355796444904264630128756129792xy^{4}z^{27}-29612957181922368705547124136524906496xy^{2}z^{29}-908069089497450981710193872117837070336xz^{31}-y^{32}-21120y^{30}z^{2}-37698048y^{28}z^{4}-202575126528y^{26}z^{6}+334319905751040y^{24}z^{8}+825236085435531264y^{22}z^{10}-2104740368088985239552y^{20}z^{12}+161890600828535706746880y^{18}z^{14}-18771979010690013154246656y^{16}z^{16}-10800148763885844592739745792y^{14}z^{18}+2545849726065150540621501431808y^{12}z^{20}-93512103286048304970564693393408y^{10}z^{22}-4954788374433291516147167970459648y^{8}z^{24}+270340248814453057422323091001835520y^{6}z^{26}-3042655636953455373585206765804322816y^{4}z^{28}+136561468689177101238061308810557915136y^{2}z^{30}-3654728615697158484080862699595140956160z^{32}}{z^{4}y^{4}(y^{2}-216z^{2})^{6}(24x^{2}y^{10}-104976x^{2}y^{8}z^{2}-110854656x^{2}y^{6}z^{4}+48977602560x^{2}y^{4}z^{6}-228509902503936x^{2}z^{10}+456xy^{10}z+1539648xy^{8}z^{3}-1169012736xy^{6}z^{5}-104485552128xy^{4}z^{7}+38084983750656xy^{2}z^{9}-914039610015744xz^{11}-y^{12}-9552y^{10}z^{2}+13110336y^{8}z^{4}+3184551936y^{6}z^{6}-1286478360576y^{4}z^{8}+76169967501312y^{2}z^{10}-914039610015744z^{12})}$ |
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.1.9 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-12.a.1.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.o.2.12 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.o.2.26 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.bz.1.3 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bz.1.13 | $24$ | $2$ | $2$ | $1$ | $1$ | 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.bq.1.5 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 24.384.5-24.br.2.5 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.bs.1.6 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 24.384.5-24.bt.1.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.bv.3.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.bw.4.6 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.by.1.9 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.bz.1.2 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.576.9-24.e.2.3 | $24$ | $3$ | $3$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
| 120.384.5-120.kl.4.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.km.3.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ko.3.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kp.2.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ks.4.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kt.2.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kv.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.kw.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kl.3.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.km.3.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ko.2.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kp.2.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ks.4.8 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kt.2.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kv.2.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.kw.3.12 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kl.3.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.km.1.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ko.2.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kp.3.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ks.3.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kt.3.14 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kv.1.9 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.kw.3.9 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kl.3.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.km.2.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ko.2.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kp.3.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ks.1.12 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kt.4.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kv.1.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.kw.1.15 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |