↑

Properties

Label 10.72.1.a.1
Level $10$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $6$

Related objects

Downloads

Learn more

Invariants

Level: $10$ $\SL_2$-level: $10$ Newform level: $20$
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 $2^{6}\cdot10^{6}$ 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: 10K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 10.72.1.2

Level structure

$\GL_2(\Z/10\Z)$-generators: $\begin{bmatrix}1&6\\0&3\end{bmatrix}$, $\begin{bmatrix}9&8\\0&7\end{bmatrix}$
$\GL_2(\Z/10\Z)$-subgroup: $C_2\times F_5$
Contains $-I$: yes
Quadratic refinements: 10.144.1-10.a.1.1, 10.144.1-10.a.1.2, 20.144.1-10.a.1.1, 20.144.1-10.a.1.2, 20.144.1-10.a.1.3, 20.144.1-10.a.1.4, 20.144.1-10.a.1.5, 20.144.1-10.a.1.6, 20.144.1-10.a.1.7, 20.144.1-10.a.1.8, 20.144.1-10.a.1.9, 20.144.1-10.a.1.10, 30.144.1-10.a.1.1, 30.144.1-10.a.1.2, 40.144.1-10.a.1.1, 40.144.1-10.a.1.2, 40.144.1-10.a.1.3, 40.144.1-10.a.1.4, 40.144.1-10.a.1.5, 40.144.1-10.a.1.6, 40.144.1-10.a.1.7, 40.144.1-10.a.1.8, 40.144.1-10.a.1.9, 40.144.1-10.a.1.10, 40.144.1-10.a.1.11, 40.144.1-10.a.1.12, 60.144.1-10.a.1.1, 60.144.1-10.a.1.2, 60.144.1-10.a.1.3, 60.144.1-10.a.1.4, 60.144.1-10.a.1.5, 60.144.1-10.a.1.6, 60.144.1-10.a.1.7, 60.144.1-10.a.1.8, 60.144.1-10.a.1.9, 60.144.1-10.a.1.10, 70.144.1-10.a.1.1, 70.144.1-10.a.1.2, 110.144.1-10.a.1.1, 110.144.1-10.a.1.2, 120.144.1-10.a.1.1, 120.144.1-10.a.1.2, 120.144.1-10.a.1.3, 120.144.1-10.a.1.4, 120.144.1-10.a.1.5, 120.144.1-10.a.1.6, 120.144.1-10.a.1.7, 120.144.1-10.a.1.8, 120.144.1-10.a.1.9, 120.144.1-10.a.1.10, 120.144.1-10.a.1.11, 120.144.1-10.a.1.12, 130.144.1-10.a.1.1, 130.144.1-10.a.1.2, 140.144.1-10.a.1.1, 140.144.1-10.a.1.2, 140.144.1-10.a.1.3, 140.144.1-10.a.1.4, 140.144.1-10.a.1.5, 140.144.1-10.a.1.6, 140.144.1-10.a.1.7, 140.144.1-10.a.1.8, 140.144.1-10.a.1.9, 140.144.1-10.a.1.10, 170.144.1-10.a.1.1, 170.144.1-10.a.1.2, 190.144.1-10.a.1.1, 190.144.1-10.a.1.2, 210.144.1-10.a.1.1, 210.144.1-10.a.1.2, 220.144.1-10.a.1.1, 220.144.1-10.a.1.2, 220.144.1-10.a.1.3, 220.144.1-10.a.1.4, 220.144.1-10.a.1.5, 220.144.1-10.a.1.6, 220.144.1-10.a.1.7, 220.144.1-10.a.1.8, 220.144.1-10.a.1.9, 220.144.1-10.a.1.10, 230.144.1-10.a.1.1, 230.144.1-10.a.1.2, 260.144.1-10.a.1.1, 260.144.1-10.a.1.2, 260.144.1-10.a.1.3, 260.144.1-10.a.1.4, 260.144.1-10.a.1.5, 260.144.1-10.a.1.6, 260.144.1-10.a.1.7, 260.144.1-10.a.1.8, 260.144.1-10.a.1.9, 260.144.1-10.a.1.10, 280.144.1-10.a.1.1, 280.144.1-10.a.1.2, 280.144.1-10.a.1.3, 280.144.1-10.a.1.4, 280.144.1-10.a.1.5, 280.144.1-10.a.1.6, 280.144.1-10.a.1.7, 280.144.1-10.a.1.8, 280.144.1-10.a.1.9, 280.144.1-10.a.1.10, 280.144.1-10.a.1.11, 280.144.1-10.a.1.12, 290.144.1-10.a.1.1, 290.144.1-10.a.1.2, 310.144.1-10.a.1.1, 310.144.1-10.a.1.2, 330.144.1-10.a.1.1, 330.144.1-10.a.1.2
Cyclic 10-isogeny field degree: $1$
Cyclic 10-torsion field degree: $2$
Full 10-torsion field degree: $40$

Jacobian

Conductor: $2^{2}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 20.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} - x $
 Copy content Toggle raw display

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
$(1:1:1)$, $(-1:1:1)$, $(1:-1:1)$, $(-1:-1:1)$, $(0:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{24x^{2}y^{22}-7096x^{2}y^{20}z^{2}+53000x^{2}y^{18}z^{4}+2377368x^{2}y^{16}z^{6}+3874160x^{2}y^{14}z^{8}-104874928x^{2}y^{12}z^{10}-195497328x^{2}y^{10}z^{12}+1164680880x^{2}y^{8}z^{14}-797319688x^{2}y^{6}z^{16}-163723800x^{2}y^{4}z^{18}-10026264x^{2}y^{2}z^{20}-199624x^{2}z^{22}-240xy^{22}z+18768xy^{20}z^{3}+130160xy^{18}z^{5}-3113040xy^{16}z^{7}-23610720xy^{14}z^{9}+63022880xy^{12}z^{11}+367140576xy^{10}z^{13}-954859680xy^{8}z^{15}+446608720xy^{6}z^{17}+98397840xy^{4}z^{19}+6141360xy^{2}z^{21}+123376xz^{23}-y^{24}+1532y^{22}z^{2}-41122y^{20}z^{4}-765940y^{18}z^{6}+1911185y^{16}z^{8}+52595192y^{14}z^{10}+39257636y^{12}z^{12}-628137480y^{10}z^{14}+510470545y^{8}z^{16}+102253580y^{6}z^{18}+6217566y^{4}z^{20}+123388y^{2}z^{22}-z^{24}}{z^{3}y^{2}(y-z)^{5}(y+z)^{5}(130x^{2}y^{6}z-4214x^{2}y^{4}z^{3}+17030x^{2}y^{2}z^{5}-15250x^{2}z^{7}+xy^{8}-500xy^{6}z^{2}+6070xy^{4}z^{4}-14740xy^{2}z^{6}+9425xz^{8}-17y^{8}z+1555y^{6}z^{3}-8915y^{4}z^{5}+9425y^{2}z^{7})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X(2)$ $2$ $12$ $12$ $0$ $0$ full Jacobian
$X_{\pm1}(5)$ $5$ $6$ $6$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
10.24.1.a.1 $10$ $3$ $3$ $1$ $0$ dimension zero
$X_{\pm1}(10)$ $10$ $2$ $2$ $0$ $0$ full Jacobian
10.36.0.b.2 $10$ $2$ $2$ $0$ $0$ full Jacobian
10.36.1.a.1 $10$ $2$ $2$ $1$ $0$ 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
10.360.13.a.1 $10$ $5$ $5$ $13$ $0$ $1^{6}\cdot2^{3}$
20.144.3.a.1 $20$ $2$ $2$ $3$ $0$ $2$
20.144.3.b.2 $20$ $2$ $2$ $3$ $0$ $2$
20.144.5.a.1 $20$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
20.144.5.b.1 $20$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
20.144.5.e.1 $20$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
20.144.5.f.1 $20$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
20.144.7.f.2 $20$ $2$ $2$ $7$ $0$ $2\cdot4$
20.144.7.g.1 $20$ $2$ $2$ $7$ $0$ $2\cdot4$
30.216.13.a.2 $30$ $3$ $3$ $13$ $0$ $1^{6}\cdot2^{3}$
30.288.13.a.1 $30$ $4$ $4$ $13$ $0$ $1^{6}\cdot2^{3}$
40.144.3.a.2 $40$ $2$ $2$ $3$ $0$ $2$
40.144.3.b.1 $40$ $2$ $2$ $3$ $0$ $2$
40.144.5.a.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.d.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.144.5.m.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.144.5.p.1 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.144.7.k.2 $40$ $2$ $2$ $7$ $0$ $2\cdot4$
40.144.7.l.1 $40$ $2$ $2$ $7$ $0$ $2\cdot4$
50.360.13.a.1 $50$ $5$ $5$ $13$ $0$ $1^{6}\cdot2^{3}$
60.144.3.b.2 $60$ $2$ $2$ $3$ $0$ $2$
60.144.3.c.1 $60$ $2$ $2$ $3$ $0$ $2$
60.144.5.y.1 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.144.5.z.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.bk.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.5.bl.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.144.7.eh.1 $60$ $2$ $2$ $7$ $0$ $2\cdot4$
60.144.7.ei.2 $60$ $2$ $2$ $7$ $0$ $2\cdot4$
70.576.37.a.2 $70$ $8$ $8$ $37$ $0$ $1^{12}\cdot2^{8}\cdot4^{2}$
70.1512.109.a.1 $70$ $21$ $21$ $109$ $14$ $1^{12}\cdot2^{28}\cdot4^{10}$
70.2016.145.a.1 $70$ $28$ $28$ $145$ $14$ $1^{24}\cdot2^{36}\cdot4^{12}$
120.144.3.b.2 $120$ $2$ $2$ $3$ $?$ not computed
120.144.3.c.1 $120$ $2$ $2$ $3$ $?$ not computed
120.144.5.cu.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cx.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ee.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.eh.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.7.yh.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.yi.1 $120$ $2$ $2$ $7$ $?$ not computed
140.144.3.a.2 $140$ $2$ $2$ $3$ $?$ not computed
140.144.3.b.1 $140$ $2$ $2$ $3$ $?$ not computed
140.144.5.a.1 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.b.1 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.e.1 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.f.1 $140$ $2$ $2$ $5$ $?$ not computed
140.144.7.a.1 $140$ $2$ $2$ $7$ $?$ not computed
140.144.7.b.2 $140$ $2$ $2$ $7$ $?$ not computed
220.144.3.a.1 $220$ $2$ $2$ $3$ $?$ not computed
220.144.3.b.1 $220$ $2$ $2$ $3$ $?$ not computed
220.144.5.a.1 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.b.1 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.e.1 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.f.1 $220$ $2$ $2$ $5$ $?$ not computed
220.144.7.a.1 $220$ $2$ $2$ $7$ $?$ not computed
220.144.7.b.1 $220$ $2$ $2$ $7$ $?$ not computed
260.144.3.a.1 $260$ $2$ $2$ $3$ $?$ not computed
260.144.3.b.2 $260$ $2$ $2$ $3$ $?$ not computed
260.144.5.a.1 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.b.1 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.e.1 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.f.1 $260$ $2$ $2$ $5$ $?$ not computed
260.144.7.a.2 $260$ $2$ $2$ $7$ $?$ not computed
260.144.7.b.1 $260$ $2$ $2$ $7$ $?$ not computed
280.144.3.a.2 $280$ $2$ $2$ $3$ $?$ not computed
280.144.3.b.1 $280$ $2$ $2$ $3$ $?$ not computed
280.144.5.a.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.d.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.m.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.p.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.7.a.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.b.1 $280$ $2$ $2$ $7$ $?$ not computed