Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-12$) |
Other labels
| Cummins and Pauli (CP) label: | 8C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.1.77 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&21\\20&7\end{bmatrix}$, $\begin{bmatrix}9&11\\20&23\end{bmatrix}$, $\begin{bmatrix}15&10\\10&1\end{bmatrix}$, $\begin{bmatrix}21&17\\20&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 396x - 3024 $ |
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 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^4\cdot3^3\,\frac{175560x^{2}y^{14}-556395456x^{2}y^{13}z+505196473440x^{2}y^{12}z^{2}-239401131835392x^{2}y^{11}z^{3}+73655864261038080x^{2}y^{10}z^{4}-16366237189101950976x^{2}y^{9}z^{5}+2786046152332269634560x^{2}y^{8}z^{6}-375974315534088561623040x^{2}y^{7}z^{7}+40980710472635385076285440x^{2}y^{6}z^{8}-3634077437742470650147897344x^{2}y^{5}z^{9}+261288899247841744929068482560x^{2}y^{4}z^{10}-14982751899013337608315094433792x^{2}y^{3}z^{11}+660254651834581043241719546511360x^{2}y^{2}z^{12}-20679322829724430460261074206720000x^{2}yz^{13}+367756518750574297327329750810624000x^{2}z^{14}-6600xy^{15}+46019904xy^{14}z-57454003200xy^{13}z^{2}+33285219482880xy^{12}z^{3}-11964937295139840xy^{11}z^{4}+3043797172820386560xy^{10}z^{5}-588147451409165377536xy^{9}z^{6}+89948788737677084160000xy^{8}z^{7}-11153566798074971454799872xy^{7}z^{8}+1135574516652350501778554880xy^{6}z^{9}-95247605896599063331904421888xy^{5}z^{10}+6543063800051694015866903986176xy^{4}z^{11}-361584621462616500786751460081664xy^{3}z^{12}+15476256654392133572379778193817600xy^{2}z^{13}-475015682656201394550033663551078400xyz^{14}+8447574190121576622693157586337792000xz^{15}+125y^{16}-3208352y^{15}z+5901309568y^{14}z^{2}-4106473461504y^{13}z^{3}+1630450635565824y^{12}z^{4}-438690223470292992y^{11}z^{5}+87458122642904193024y^{10}z^{6}-13600684555358879342592y^{9}z^{7}+1701454393885269047721984y^{8}z^{8}-174338902785492250301693952y^{7}z^{9}+14760989870321412519369375744y^{6}z^{10}-1033856786046527226983200849920y^{5}z^{11}+59518109146446249778559501991936y^{4}z^{12}-2772834330126243414495160317247488y^{3}z^{13}+101154583874421068260853741959249920y^{2}z^{14}-2722365704394097879619563917882163200yz^{15}+48413951581376221249480074209525760000z^{16}}{192x^{2}y^{14}+9857376x^{2}y^{12}z^{2}-305307159552x^{2}y^{10}z^{4}+8224724326652928x^{2}y^{8}z^{6}-216843378748778151936x^{2}y^{6}z^{8}+3502279521129033968320512x^{2}y^{4}z^{10}-4982147463103710042802618368x^{2}y^{2}z^{12}-521077749599939200149586496716800x^{2}z^{14}-11664xy^{14}z+92503296xy^{12}z^{3}+624308228352xy^{10}z^{5}+37009237511245824xy^{8}z^{7}-3663237318248133033984xy^{6}z^{9}+97725657192186967763189760xy^{4}z^{11}-553451978166037942553721962496xy^{2}z^{13}-11969449144020658288122926160936960xz^{15}-y^{16}+58752y^{14}z^{2}-11535789312y^{12}z^{4}+350945846747136y^{10}z^{6}-7047011998619320320y^{8}z^{8}+64640671043863830331392y^{6}z^{10}+884891230638279484088254464y^{4}z^{12}-20824462458244947093628460728320y^{2}z^{14}-68598193785856659374315996725641216z^{16}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.12.0.u.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.0.bt.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.12.1.by.1 | $24$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 24.48.1.br.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.ck.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.fl.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.fo.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.ld.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.lh.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.ls.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.48.1.lw.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.72.5.mj.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.96.5.ev.1 | $24$ | $4$ | $4$ | $5$ | $3$ | $1^{4}$ |
| 120.48.1.bjb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bjf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bkh.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bkl.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.btv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.btz.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bvb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.48.1.bvf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.120.9.sf.1 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.144.9.odd.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.240.17.fat.1 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.48.1.biz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bjd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bkf.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bkj.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.btt.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.btx.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.buz.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.48.1.bvd.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.13.md.1 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.48.1.biz.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bjd.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bkf.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bkj.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.btt.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.btx.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.buz.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.48.1.bvd.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.288.21.kh.1 | $264$ | $12$ | $12$ | $21$ | $?$ | not computed |
| 312.48.1.bjb.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bjf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bkh.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bkl.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.btv.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.btz.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bvb.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.48.1.bvf.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |