Invariants
| Level: | $30$ | $\SL_2$-level: | $30$ | Newform level: | $150$ | ||
| Index: | $120$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $5\cdot10\cdot15\cdot30$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30D4 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.120.4.5 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}1&17\\18&29\end{bmatrix}$, $\begin{bmatrix}25&14\\12&25\end{bmatrix}$, $\begin{bmatrix}29&3\\18&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.60.4.b.1 for the level structure with $-I$) |
| Cyclic 30-isogeny field degree: | $6$ |
| Cyclic 30-torsion field degree: | $48$ |
| Full 30-torsion field degree: | $1152$ |
Jacobian
| Conductor: | $2^{2}\cdot3^{2}\cdot5^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}$ |
| Newforms: | 50.2.a.b$^{2}$, 75.2.a.a$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x^{2} - 4 x y + x z - x w + 3 y z + 2 w^{2} $ |
| $=$ | $x^{3} - x^{2} y + x^{2} z - 2 x^{2} w - 2 x y z - x y w - x z w + x w^{2} - y^{2} z + y z^{2} + z w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{6} - 12 x^{5} y + 7 x^{5} z - 20 x^{4} y^{2} + 7 x^{4} y z + 3 x^{4} z^{2} - 8 x^{3} y^{3} + \cdots + y^{2} z^{4} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:1:0)$, $(1/2:-3/4:1:0)$, $(0:1:0:0)$, $(-1:0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2^2}\cdot\frac{50030411238xyz^{8}+7180833708xyz^{7}w+335696608332xyz^{6}w^{2}+45549588600xyz^{5}w^{3}+794055601440xyz^{4}w^{4}+102047531328xyz^{3}w^{5}+780417218112xyz^{2}w^{6}+74511393664xyzw^{7}+260400325120xyw^{8}-537477120xz^{9}+3090493440xz^{8}w+1119744xz^{7}w^{2}+16030835388xz^{6}w^{3}+24397625736xz^{5}w^{4}+27111804192xz^{4}w^{5}+67811781312xz^{3}w^{6}+24242703552xz^{2}w^{7}+51887783040xzw^{8}+14680979968xw^{9}+107495424y^{10}+268738560y^{8}w^{2}-268738560y^{7}w^{3}+705438720y^{6}w^{4}-1330255872y^{5}w^{5}+2720977920y^{4}w^{6}-5576325120y^{3}w^{7}+18373210074y^{2}z^{8}-13271121324y^{2}z^{7}w+110771297280y^{2}z^{6}w^{2}-64549593360y^{2}z^{5}w^{3}+225014988960y^{2}z^{4}w^{4}-96224884992y^{2}z^{3}w^{5}+160468468608y^{2}z^{2}w^{6}-41091594496y^{2}zw^{7}+11522165760y^{2}w^{8}-11576598003yz^{9}-20951301807yz^{8}w-82975160052yz^{7}w^{2}-120742340904yz^{6}w^{3}-205114847400yz^{5}w^{4}-239500712160yz^{4}w^{5}-219119339424yz^{3}w^{6}-169651136000yz^{2}w^{7}-93356862336yzw^{8}-24035304960yw^{9}+6718464z^{10}-11106305523z^{8}w^{2}-13405782204z^{7}w^{3}-78005109240z^{6}w^{4}-72983015928z^{5}w^{5}-198526744800z^{4}w^{6}-130457767392z^{3}w^{7}-223656500352z^{2}w^{8}-74556275840zw^{9}-100217231872w^{10}}{109061316xyz^{8}+339521544xyz^{7}w+835226964xyz^{6}w^{2}+1328332608xyz^{5}w^{3}+1603071540xyz^{4}w^{4}+1408129686xyz^{3}w^{5}+881728080xyz^{2}w^{6}+358665940xyzw^{7}+73497424xyw^{8}+20802744xz^{7}w^{2}+50732568xz^{6}w^{3}+112149360xz^{5}w^{4}+152191224xz^{4}w^{5}+155653704xz^{3}w^{6}+111653352xz^{2}w^{7}+52533420xzw^{8}+12793600xw^{9}+41605488y^{2}z^{8}+84231576y^{2}z^{7}w+181573488y^{2}z^{6}w^{2}+217433160y^{2}z^{5}w^{3}+202919040y^{2}z^{4}w^{4}+128221290y^{2}z^{3}w^{5}+51898188y^{2}z^{2}w^{6}+10367888y^{2}zw^{7}-23326542yz^{9}-127389834yz^{8}w-323549154yz^{7}w^{2}-599016546yz^{6}w^{3}-770135310yz^{5}w^{4}-730235133yz^{4}w^{5}-484989573yz^{3}w^{6}-208698026yz^{2}w^{7}-45151992yzw^{8}-23326542z^{8}w^{2}-104063292z^{7}w^{3}-266138946z^{6}w^{4}-467022348z^{5}w^{5}-593161650z^{4}w^{6}-546611427z^{3}w^{7}-355768446z^{2}w^{8}-149579048zw^{9}-31564768w^{10}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 30.60.4.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 4X^{6}-12X^{5}Y+7X^{5}Z-20X^{4}Y^{2}+7X^{4}YZ+3X^{4}Z^{2}-8X^{3}Y^{3}+2X^{3}Y^{2}Z+7X^{3}YZ^{2}+X^{3}Z^{3}+13X^{2}Y^{2}Z^{2}+8X^{2}YZ^{3}+X^{2}Z^{4}+12XY^{3}Z^{2}+10XY^{2}Z^{3}+2XYZ^{4}+4Y^{4}Z^{2}+4Y^{3}Z^{3}+Y^{2}Z^{4} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 6.24.0-6.a.1.3 | $6$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.24.0-6.a.1.3 | $6$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 30.40.1-15.a.1.1 | $30$ | $3$ | $3$ | $1$ | $0$ | $1^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 30.240.8-30.a.1.4 | $30$ | $2$ | $2$ | $8$ | $0$ | $1^{4}$ |
| 30.240.8-30.c.1.4 | $30$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 30.240.8-30.f.1.2 | $30$ | $2$ | $2$ | $8$ | $0$ | $1^{4}$ |
| 30.240.8-30.h.1.2 | $30$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 30.360.10-30.a.1.9 | $30$ | $3$ | $3$ | $10$ | $0$ | $1^{6}$ |
| 30.360.12-30.a.1.2 | $30$ | $3$ | $3$ | $12$ | $2$ | $1^{8}$ |
| 30.480.13-30.c.1.4 | $30$ | $4$ | $4$ | $13$ | $0$ | $1^{9}$ |
| 60.240.8-60.k.1.12 | $60$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 60.240.8-60.n.1.1 | $60$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 60.240.8-60.o.1.1 | $60$ | $2$ | $2$ | $8$ | $0$ | $1^{4}$ |
| 60.240.8-60.s.1.10 | $60$ | $2$ | $2$ | $8$ | $4$ | $1^{4}$ |
| 60.240.8-60.t.1.1 | $60$ | $2$ | $2$ | $8$ | $4$ | $1^{4}$ |
| 60.240.8-60.u.1.1 | $60$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 60.240.8-60.bi.1.6 | $60$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 60.240.8-60.bj.1.1 | $60$ | $2$ | $2$ | $8$ | $3$ | $1^{4}$ |
| 60.240.8-60.bk.1.1 | $60$ | $2$ | $2$ | $8$ | $0$ | $1^{4}$ |
| 60.240.8-60.bm.1.10 | $60$ | $2$ | $2$ | $8$ | $0$ | $1^{4}$ |
| 60.240.8-60.bn.1.1 | $60$ | $2$ | $2$ | $8$ | $0$ | $1^{4}$ |
| 60.240.8-60.bo.1.1 | $60$ | $2$ | $2$ | $8$ | $1$ | $1^{4}$ |
| 60.240.9-60.dm.1.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 60.240.9-60.dn.1.1 | $60$ | $2$ | $2$ | $9$ | $4$ | $1^{5}$ |
| 60.240.9-60.do.1.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 60.240.9-60.dp.1.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 60.240.9-60.dq.1.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 60.240.9-60.dr.1.1 | $60$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
| 60.240.9-60.ds.1.1 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
| 60.240.9-60.dt.1.1 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
| 90.360.12-90.a.1.4 | $90$ | $3$ | $3$ | $12$ | $?$ | not computed |
| 90.360.14-90.c.1.3 | $90$ | $3$ | $3$ | $14$ | $?$ | not computed |
| 90.360.14-90.d.1.5 | $90$ | $3$ | $3$ | $14$ | $?$ | not computed |
| 120.240.8-120.bq.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.cd.1.21 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.eg.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.eh.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gm.1.25 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gn.1.17 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.go.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.gp.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.iy.1.17 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.iz.1.17 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.ja.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.jb.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.je.1.17 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.jf.1.17 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.jg.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.8-120.jh.1.1 | $120$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 120.240.9-120.xg.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.9-120.xh.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.9-120.xi.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.9-120.xj.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.9-120.xk.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.9-120.xl.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.9-120.xm.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.240.9-120.xn.1.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 210.240.8-210.b.1.4 | $210$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 210.240.8-210.c.1.7 | $210$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 210.240.8-210.d.1.4 | $210$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 210.240.8-210.e.1.3 | $210$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 330.240.8-330.a.1.8 | $330$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 330.240.8-330.b.1.7 | $330$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 330.240.8-330.c.1.4 | $330$ | $2$ | $2$ | $8$ | $?$ | not computed |
| 330.240.8-330.d.1.4 | $330$ | $2$ | $2$ | $8$ | $?$ | not computed |