Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| 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 | $1^{2}\cdot2^{5}\cdot4$ | ||
| 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: | 24.192.1.1864 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&15\\8&1\end{bmatrix}$, $\begin{bmatrix}11&3\\4&5\end{bmatrix}$, $\begin{bmatrix}23&9\\20&7\end{bmatrix}$, $\begin{bmatrix}23&21\\4&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $(C_2\times D_4):D_{12}$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.dn.3 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 6x + 7 $ |
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}{3^4}\cdot\frac{24x^{2}y^{30}-52488x^{2}y^{28}z^{2}-104188680x^{2}y^{26}z^{4}+174037243464x^{2}y^{24}z^{6}-7694878790952x^{2}y^{22}z^{8}-42031973758391640x^{2}y^{20}z^{10}+11968123213963454616x^{2}y^{18}z^{12}-1354507675385850668232x^{2}y^{16}z^{14}+77128273107660430799496x^{2}y^{14}z^{16}-2192538737195931165138936x^{2}y^{12}z^{18}+16697782733149017651798504x^{2}y^{10}z^{20}+775329781878936387168131448x^{2}y^{8}z^{22}-25120685992665739849298852472x^{2}y^{6}z^{24}+315578620344753639423646330968x^{2}y^{4}z^{26}-1780428640337427264920550486648x^{2}y^{2}z^{28}+3433683811429574365005347993352x^{2}z^{30}-24xy^{30}z+1102248xy^{28}z^{3}-1581935832xy^{26}z^{5}-642836229912xy^{24}z^{7}+1284686992007784xy^{22}z^{9}-370691087882695704xy^{20}z^{11}+36584339574044814120xy^{18}z^{13}-658409918723874558936xy^{16}z^{15}-133442669740081567371912xy^{14}z^{17}+11545893449282646945088056xy^{12}z^{19}-441135030064199161660387272xy^{10}z^{21}+9331956021806706234328999416xy^{8}z^{23}-109903003563441477626887743432xy^{6}z^{25}+631157248567674496316182524984xy^{4}z^{27}-890214331985964458663610037896xy^{2}z^{29}-3433683811429574365005347993352xz^{31}-y^{32}-2640y^{30}z^{2}+4601448y^{28}z^{4}+13624940016y^{26}z^{6}-19081376114220y^{24}z^{8}+5010678654394608y^{22}z^{10}+100519052172885432y^{20}z^{12}-149580741435855529680y^{18}z^{14}+19495855879014591897354y^{16}z^{16}-1230938762501510089861104y^{14}z^{18}+44811881583424646244745752y^{12}z^{20}-977830175181685094411571504y^{10}z^{22}+12336358613640270140362341492y^{8}z^{24}-75362055856259454989516982960y^{6}z^{26}+18840496552486790754293213256y^{4}z^{28}+2034775649731492551812065213968y^{2}z^{30}-6867367702625591806883205850065z^{32}}{z^{2}y^{8}(y^{2}-27z^{2})(x^{2}y^{18}+1701x^{2}y^{16}z^{2}-139968x^{2}y^{14}z^{4}-1102248x^{2}y^{12}z^{6}+36137988x^{2}y^{10}z^{8}+1894055724x^{2}y^{8}z^{10}+8523250758x^{2}y^{6}z^{12}-523017660150x^{2}y^{4}z^{14}+3671583974253x^{2}y^{2}z^{16}-7625597484987x^{2}z^{18}-16xy^{18}z+4374xy^{16}z^{3}+559872xy^{14}z^{5}+11967264xy^{12}z^{7}+136048896xy^{10}z^{9}-2539756539xy^{8}z^{11}-52689186504xy^{6}z^{13}+899590375458xy^{4}z^{15}-4518872583696xy^{2}z^{17}+7625597484987xz^{19}-80y^{18}z^{2}-56376y^{16}z^{4}-225261y^{14}z^{6}-18994095y^{12}z^{8}-287509581y^{10}z^{10}+11063007297y^{8}z^{12}+186349255209y^{6}z^{14}-4508412230493y^{4}z^{16}+27395665038657y^{2}z^{18}-53379182394909z^{20})}$ |
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.2.3 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-12.c.2.23 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bt.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.0-24.bt.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.96.1-24.iu.1.12 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.96.1-24.iu.1.18 | $24$ | $2$ | $2$ | $1$ | $0$ | 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.cx.2.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.dm.3.7 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.eh.2.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.em.3.7 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.ev.4.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.ez.4.3 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 24.384.5-24.fr.4.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.384.5-24.fy.4.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
| 24.576.9-24.t.1.2 | $24$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 120.384.5-120.bal.4.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ban.3.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bbb.2.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bbd.3.11 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bcx.4.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bcz.4.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdn.3.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.bdp.4.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bal.1.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ban.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bbb.3.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bbd.2.14 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bcx.3.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bcz.3.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdn.3.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bdp.3.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bal.2.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ban.4.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bbb.4.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bbd.4.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bcx.4.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bcz.4.11 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdn.4.3 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.bdp.3.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bal.3.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ban.2.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bbb.4.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bbd.2.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bcx.4.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bcz.2.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdn.2.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.bdp.2.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |