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Properties

Label 36.72.1.c.1
Level $36$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $6$

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$X_0(36)$ is isomorphic to the elliptic curve $y^2=x^3+1$.

Invariants

Level: $36$ $\SL_2$-level: $36$ Newform level: $36$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $6$ are rational) Cusp widths $1^{6}\cdot4^{3}\cdot9^{2}\cdot36$ Cusp orbits $1^{6}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 36C1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 36.72.1.17

Level structure

$\GL_2(\Z/36\Z)$-generators: $\begin{bmatrix}5&31\\0&11\end{bmatrix}$, $\begin{bmatrix}11&19\\0&31\end{bmatrix}$, $\begin{bmatrix}13&8\\0&5\end{bmatrix}$, $\begin{bmatrix}13&30\\0&7\end{bmatrix}$, $\begin{bmatrix}25&33\\0&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 36.144.1-36.c.1.1, 36.144.1-36.c.1.2, 36.144.1-36.c.1.3, 36.144.1-36.c.1.4, 36.144.1-36.c.1.5, 36.144.1-36.c.1.6, 36.144.1-36.c.1.7, 36.144.1-36.c.1.8, 36.144.1-36.c.1.9, 36.144.1-36.c.1.10, 36.144.1-36.c.1.11, 36.144.1-36.c.1.12, 72.144.1-36.c.1.1, 72.144.1-36.c.1.2, 72.144.1-36.c.1.3, 72.144.1-36.c.1.4, 72.144.1-36.c.1.5, 72.144.1-36.c.1.6, 72.144.1-36.c.1.7, 72.144.1-36.c.1.8, 72.144.1-36.c.1.9, 72.144.1-36.c.1.10, 72.144.1-36.c.1.11, 72.144.1-36.c.1.12, 72.144.1-36.c.1.13, 72.144.1-36.c.1.14, 72.144.1-36.c.1.15, 72.144.1-36.c.1.16, 72.144.1-36.c.1.17, 72.144.1-36.c.1.18, 72.144.1-36.c.1.19, 72.144.1-36.c.1.20, 72.144.1-36.c.1.21, 72.144.1-36.c.1.22, 72.144.1-36.c.1.23, 72.144.1-36.c.1.24, 72.144.1-36.c.1.25, 72.144.1-36.c.1.26, 72.144.1-36.c.1.27, 72.144.1-36.c.1.28, 180.144.1-36.c.1.1, 180.144.1-36.c.1.2, 180.144.1-36.c.1.3, 180.144.1-36.c.1.4, 180.144.1-36.c.1.5, 180.144.1-36.c.1.6, 180.144.1-36.c.1.7, 180.144.1-36.c.1.8, 180.144.1-36.c.1.9, 180.144.1-36.c.1.10, 180.144.1-36.c.1.11, 180.144.1-36.c.1.12, 252.144.1-36.c.1.1, 252.144.1-36.c.1.2, 252.144.1-36.c.1.3, 252.144.1-36.c.1.4, 252.144.1-36.c.1.5, 252.144.1-36.c.1.6, 252.144.1-36.c.1.7, 252.144.1-36.c.1.8, 252.144.1-36.c.1.9, 252.144.1-36.c.1.10, 252.144.1-36.c.1.11, 252.144.1-36.c.1.12
Cyclic 36-isogeny field degree: $1$
Cyclic 36-torsion field degree: $12$
Full 36-torsion field degree: $5184$

Jacobian

Conductor: $2^{2}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 36.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + 1 $
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Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:-1:1)$, $(2:-3:1)$, $(0:1:0)$, $(2:3:1)$, $(0:1:1)$, $(-1:0:1)$

Maps to other modular curves

$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{y(y-z)^{9}(y+3z)^{3}(y^{2}+6yz-3z^{2})^{3}(y^{6}-6y^{5}z+27y^{4}z^{2}+60y^{3}z^{3}-249y^{2}z^{4}+234yz^{5}-3z^{6})^{3}}{z^{3}y^{2}(y-3z)^{4}(y-z)^{12}(y+z)^{12}(y+3z)^{4}}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(4)$ $4$ $12$ $12$ $0$ $0$ full Jacobian
$X_0(9)$ $9$ $6$ $6$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_0(12)$ $12$ $3$ $3$ $0$ $0$ full Jacobian
$X_0(18)$ $18$ $2$ $2$ $0$ $0$ full Jacobian

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
36.144.3.c.1 $36$ $2$ $2$ $3$ $0$ $2$
36.144.3.c.2 $36$ $2$ $2$ $3$ $0$ $2$
36.144.3.c.3 $36$ $2$ $2$ $3$ $0$ $2$
36.144.3.c.4 $36$ $2$ $2$ $3$ $0$ $2$
36.144.5.b.1 $36$ $2$ $2$ $5$ $0$ $1^{4}$
36.144.5.h.1 $36$ $2$ $2$ $5$ $0$ $1^{4}$
36.144.5.k.1 $36$ $2$ $2$ $5$ $0$ $1^{4}$
36.144.5.l.1 $36$ $2$ $2$ $5$ $0$ $1^{4}$
36.216.7.u.1 $36$ $3$ $3$ $7$ $0$ $2^{3}$
36.216.7.u.2 $36$ $3$ $3$ $7$ $0$ $2^{3}$
36.216.7.w.1 $36$ $3$ $3$ $7$ $0$ $1^{6}$
36.216.10.g.1 $36$ $3$ $3$ $10$ $0$ $1^{9}$
72.144.3.c.1 $72$ $2$ $2$ $3$ $?$ not computed
72.144.3.c.2 $72$ $2$ $2$ $3$ $?$ not computed
72.144.3.c.3 $72$ $2$ $2$ $3$ $?$ not computed
72.144.3.c.4 $72$ $2$ $2$ $3$ $?$ not computed
72.144.3.f.1 $72$ $2$ $2$ $3$ $?$ not computed
72.144.3.f.2 $72$ $2$ $2$ $3$ $?$ not computed
72.144.3.g.1 $72$ $2$ $2$ $3$ $?$ not computed
72.144.3.g.2 $72$ $2$ $2$ $3$ $?$ not computed
72.144.5.f.1 $72$ $2$ $2$ $5$ $?$ not computed
72.144.5.w.1 $72$ $2$ $2$ $5$ $?$ not computed
72.144.5.be.1 $72$ $2$ $2$ $5$ $?$ not computed
72.144.5.bh.1 $72$ $2$ $2$ $5$ $?$ not computed
72.144.5.bk.1 $72$ $2$ $2$ $5$ $?$ not computed
$X_0(72)$ $72$ $2$ $2$ $5$ $?$ not computed
72.144.5.bm.1 $72$ $2$ $2$ $5$ $?$ not computed
72.144.5.bn.1 $72$ $2$ $2$ $5$ $?$ not computed
72.144.5.bo.1 $72$ $2$ $2$ $5$ $?$ not computed
72.144.5.bp.1 $72$ $2$ $2$ $5$ $?$ not computed
72.144.5.bq.1 $72$ $2$ $2$ $5$ $?$ not computed
72.144.5.br.1 $72$ $2$ $2$ $5$ $?$ not computed
72.144.7.e.1 $72$ $2$ $2$ $7$ $?$ not computed
72.144.7.e.2 $72$ $2$ $2$ $7$ $?$ not computed
72.144.7.f.1 $72$ $2$ $2$ $7$ $?$ not computed
72.144.7.f.2 $72$ $2$ $2$ $7$ $?$ not computed
108.216.7.c.1 $108$ $3$ $3$ $7$ $?$ not computed
$X_0(108)$ $108$ $3$ $3$ $10$ $?$ not computed
108.216.13.c.1 $108$ $3$ $3$ $13$ $?$ not computed
180.144.3.c.1 $180$ $2$ $2$ $3$ $?$ not computed
180.144.3.c.2 $180$ $2$ $2$ $3$ $?$ not computed
180.144.3.c.3 $180$ $2$ $2$ $3$ $?$ not computed
180.144.3.c.4 $180$ $2$ $2$ $3$ $?$ not computed
180.144.5.k.1 $180$ $2$ $2$ $5$ $?$ not computed
180.144.5.l.1 $180$ $2$ $2$ $5$ $?$ not computed
180.144.5.o.1 $180$ $2$ $2$ $5$ $?$ not computed
180.144.5.p.1 $180$ $2$ $2$ $5$ $?$ not computed
252.144.3.c.1 $252$ $2$ $2$ $3$ $?$ not computed
252.144.3.c.2 $252$ $2$ $2$ $3$ $?$ not computed
252.144.3.c.3 $252$ $2$ $2$ $3$ $?$ not computed
252.144.3.c.4 $252$ $2$ $2$ $3$ $?$ not computed
252.144.5.k.1 $252$ $2$ $2$ $5$ $?$ not computed
252.144.5.l.1 $252$ $2$ $2$ $5$ $?$ not computed
252.144.5.o.1 $252$ $2$ $2$ $5$ $?$ not computed
252.144.5.p.1 $252$ $2$ $2$ $5$ $?$ not computed
252.216.7.t.1 $252$ $3$ $3$ $7$ $?$ not computed
252.216.7.t.2 $252$ $3$ $3$ $7$ $?$ not computed
252.216.7.v.1 $252$ $3$ $3$ $7$ $?$ not computed
252.216.7.v.2 $252$ $3$ $3$ $7$ $?$ not computed
252.216.7.x.1 $252$ $3$ $3$ $7$ $?$ not computed
252.216.7.x.2 $252$ $3$ $3$ $7$ $?$ not computed