Invariants
Level: | $30$ | $\SL_2$-level: | $30$ | Newform level: | $900$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot30^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $1$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 30D1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.72.1.28 |
Level structure
$\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}2&5\\11&1\end{bmatrix}$, $\begin{bmatrix}19&10\\26&1\end{bmatrix}$, $\begin{bmatrix}21&5\\22&27\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 30-isogeny field degree: | $12$ |
Cyclic 30-torsion field degree: | $96$ |
Full 30-torsion field degree: | $1920$ |
Jacobian
Conductor: | $2^{2}\cdot3^{2}\cdot5^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 900.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x z - x w + y^{2} + z^{2} - w^{2} $ |
$=$ | $4 x^{2} + x z + 2 z^{2} - 2 z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} - 5 x^{3} z + 7 x^{2} y^{2} + x y^{2} z - 5 x z^{3} + 4 y^{4} - 8 y^{2} z^{2} + 5 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{3^3}{2^{30}}\cdot\frac{1131753626219xz^{17}-58968692067448xz^{16}w+354826487055824xz^{15}w^{2}-372710010639520xz^{14}w^{3}-1648907491712080xz^{13}w^{4}+4770044482656128xz^{12}w^{5}-5080414398528256xz^{11}w^{6}+2513566719392768xz^{10}w^{7}-218967994812160xz^{9}w^{8}+47968122664960xz^{8}w^{9}-363887215935488xz^{7}w^{10}+201462973382656xz^{6}w^{11}+177847134625792xz^{5}w^{12}-219035139768320xz^{4}w^{13}+96968780677120xz^{3}w^{14}-7397044649984xz^{2}w^{15}-5431293181952xzw^{16}+2056497135616xw^{17}+3275981939334z^{18}-26408214047654z^{17}w-74592337587749z^{16}w^{2}+998294866236664z^{15}w^{3}-2783355210356900z^{14}w^{4}+3115884086876128z^{13}w^{5}-917441127870128z^{12}w^{6}-1143940833605888z^{11}w^{7}+687724582227328z^{10}w^{8}+1084096583718400z^{9}w^{9}-1699363565809408z^{8}w^{10}+1041265394413568z^{7}w^{11}-260367488714752z^{6}w^{12}-9602687746048z^{5}w^{13}+24327965696000z^{4}w^{14}-7269044322304z^{3}w^{15}+10831036350464z^{2}w^{16}-6483497844736zw^{17}+2140672557056w^{18}}{(z-w)^{15}(29xz^{2}+92xzw+44xw^{2}-6z^{3}+46z^{2}w+61zw^{2}+4w^{3})}$ |
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 |
---|---|---|---|---|---|---|
15.36.0.b.1 | $15$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
30.36.0.f.2 | $30$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
30.36.1.p.1 | $30$ | $2$ | $2$ | $1$ | $1$ | 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 |
---|---|---|---|---|---|---|
30.144.9.g.2 | $30$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
30.144.9.z.2 | $30$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
30.144.9.bb.1 | $30$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
30.144.9.bm.2 | $30$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
30.216.9.e.2 | $30$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
30.360.21.j.1 | $30$ | $5$ | $5$ | $21$ | $4$ | $1^{8}\cdot2^{6}$ |
60.144.9.bo.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.144.9.fe.2 | $60$ | $2$ | $2$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
60.144.9.ge.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.144.9.in.2 | $60$ | $2$ | $2$ | $9$ | $2$ | $1^{4}\cdot2^{2}$ |
60.288.13.sj.2 | $60$ | $4$ | $4$ | $13$ | $5$ | $1^{6}\cdot2^{3}$ |
90.216.13.ch.1 | $90$ | $3$ | $3$ | $13$ | $?$ | not computed |
120.144.9.iyw.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.jat.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.rbp.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.rcr.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.rvo.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.rxl.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.smt.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.soj.2 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
150.360.21.d.2 | $150$ | $5$ | $5$ | $21$ | $?$ | not computed |
210.144.9.jy.2 | $210$ | $2$ | $2$ | $9$ | $?$ | not computed |
210.144.9.jz.1 | $210$ | $2$ | $2$ | $9$ | $?$ | not computed |
210.144.9.ko.1 | $210$ | $2$ | $2$ | $9$ | $?$ | not computed |
210.144.9.kp.2 | $210$ | $2$ | $2$ | $9$ | $?$ | not computed |
330.144.9.jy.2 | $330$ | $2$ | $2$ | $9$ | $?$ | not computed |
330.144.9.jz.2 | $330$ | $2$ | $2$ | $9$ | $?$ | not computed |
330.144.9.ko.1 | $330$ | $2$ | $2$ | $9$ | $?$ | not computed |
330.144.9.kp.2 | $330$ | $2$ | $2$ | $9$ | $?$ | not computed |