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Properties

Label 30.72.1.o.1
Level $30$
Index $72$
Genus $1$
Analytic rank $1$
Cusps $4$
$\Q$-cusps $2$

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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: $4$ (of which $2$ are rational) Cusp widths $6^{2}\cdot30^{2}$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $16$ 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: 30D1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 30.72.1.26

Level structure

$\GL_2(\Z/30\Z)$-generators: $\begin{bmatrix}13&29\\22&25\end{bmatrix}$, $\begin{bmatrix}22&25\\25&2\end{bmatrix}$, $\begin{bmatrix}27&10\\17&21\end{bmatrix}$
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

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 300x - 1375 $
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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 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^3\cdot5^3}\cdot\frac{90x^{2}y^{22}-149394375x^{2}y^{20}z^{2}+65374214062500x^{2}y^{18}z^{4}+2527458662109375x^{2}y^{16}z^{6}-2396194369913232421875000x^{2}y^{14}z^{8}+172229578437554658050537109375x^{2}y^{12}z^{10}+4221516895904176908988952636718750x^{2}y^{10}z^{12}-362583687693683164032304286956787109375x^{2}y^{8}z^{14}-10038202474158474836619831919670104980468750x^{2}y^{6}z^{16}-91599830541673754389461493492126464843750000000x^{2}y^{4}z^{18}-348473375772697612084924531644210219383239746093750x^{2}y^{2}z^{20}-473105273612741047896969087920151650905609130859375000x^{2}z^{22}-2475xy^{22}z+1875403125xy^{20}z^{3}+156874535156250xy^{18}z^{5}+4903996383984375000xy^{16}z^{7}-77827082860508972167968750xy^{14}z^{9}+8756696800482002352905273437500xy^{12}z^{11}-24760822844801095915031433105468750xy^{10}z^{13}-13756717392889430783281087875366210937500xy^{8}z^{15}-276403504893532849966454069316387176513671875xy^{6}z^{17}-2142530959690538845751930166967213153839111328125xy^{4}z^{19}-7296435607330206889298521402850747108459472656250000xy^{2}z^{21}-9116978581986525991077912915917113423347473144531250000xz^{23}-y^{24}+2493000y^{22}z^{2}-2145263484375y^{20}z^{4}+743356174218750000y^{18}z^{6}-79849681427805908203125y^{16}z^{8}+2070186403570917846679687500y^{14}z^{10}+232512634994890626033782958984375y^{12}z^{12}-7529725489637131119655609130859375000y^{10}z^{14}-358097057760510349286836087703704833984375y^{8}z^{16}-4314429656853098545980863392353057861328125000y^{6}z^{18}-21793971606559825012066108273342251777648925781250y^{4}z^{20}-47263347401695791988002082844730466604232788085937500y^{2}z^{22}-33757261069616287922374411951881484128534793853759765625z^{24}}{z^{2}y^{6}(60x^{2}y^{14}+759375x^{2}y^{12}z^{2}-1025156250x^{2}y^{10}z^{4}-28255869140625x^{2}y^{8}z^{6}+177347025604248046875x^{2}y^{4}z^{10}-221683782005310058593750x^{2}y^{2}z^{12}+74818276426792144775390625x^{2}z^{14}-750xy^{14}z+4556250xy^{12}z^{3}+82012500000xy^{10}z^{5}+80731054687500xy^{8}z^{7}-583858520507812500xy^{6}z^{9}+775893237018585205078125xy^{2}z^{13}-374091382133960723876953125xz^{15}-y^{16}-39000y^{14}z^{2}-463218750y^{12}z^{4}-102515625000y^{10}z^{6}+10192295654296875y^{8}z^{8}+2335434082031250000y^{6}z^{10}-56159891441345214843750y^{4}z^{12}+64288296781539916992187500y^{2}z^{14}-20575026017367839813232421875z^{16})}$

Modular covers

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Cover information

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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.a.2 $15$ $2$ $2$ $0$ $0$ full Jacobian
30.36.0.e.1 $30$ $2$ $2$ $0$ $0$ 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.h.1 $30$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
30.144.9.r.2 $30$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
30.144.9.bc.2 $30$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
30.144.9.bj.2 $30$ $2$ $2$ $9$ $2$ $1^{4}\cdot2^{2}$
30.216.9.d.1 $30$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
30.360.21.g.1 $30$ $5$ $5$ $21$ $4$ $1^{8}\cdot2^{6}$
60.144.9.bv.1 $60$ $2$ $2$ $9$ $2$ $1^{4}\cdot2^{2}$
60.144.9.eb.1 $60$ $2$ $2$ $9$ $1$ $1^{4}\cdot2^{2}$
60.144.9.gl.1 $60$ $2$ $2$ $9$ $2$ $1^{4}\cdot2^{2}$
60.144.9.ia.1 $60$ $2$ $2$ $9$ $2$ $1^{4}\cdot2^{2}$
60.288.13.sf.1 $60$ $4$ $4$ $13$ $5$ $1^{6}\cdot2^{3}$
90.216.13.cg.2 $90$ $3$ $3$ $13$ $?$ not computed
120.144.9.jba.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.jcx.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.npo.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.nqx.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.rxs.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.rzp.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.sjk.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.slh.1 $120$ $2$ $2$ $9$ $?$ not computed
150.360.21.c.1 $150$ $5$ $5$ $21$ $?$ not computed
210.144.9.jw.1 $210$ $2$ $2$ $9$ $?$ not computed
210.144.9.jx.2 $210$ $2$ $2$ $9$ $?$ not computed
210.144.9.km.2 $210$ $2$ $2$ $9$ $?$ not computed
210.144.9.kn.1 $210$ $2$ $2$ $9$ $?$ not computed
330.144.9.jw.1 $330$ $2$ $2$ $9$ $?$ not computed
330.144.9.jx.1 $330$ $2$ $2$ $9$ $?$ not computed
330.144.9.km.2 $330$ $2$ $2$ $9$ $?$ not computed
330.144.9.kn.1 $330$ $2$ $2$ $9$ $?$ not computed