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Properties

Label 10.72.1.a.2
Level $10$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $6$

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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: 10.72.1.3

Level structure

$\GL_2(\Z/10\Z)$-generators: $\begin{bmatrix}7&6\\0&9\end{bmatrix}$, $\begin{bmatrix}9&4\\0&9\end{bmatrix}$
$\GL_2(\Z/10\Z)$-subgroup: $C_2\times F_5$
Contains $-I$: yes
Quadratic refinements: 10.144.1-10.a.2.1, 10.144.1-10.a.2.2, 20.144.1-10.a.2.1, 20.144.1-10.a.2.2, 20.144.1-10.a.2.3, 20.144.1-10.a.2.4, 20.144.1-10.a.2.5, 20.144.1-10.a.2.6, 20.144.1-10.a.2.7, 20.144.1-10.a.2.8, 20.144.1-10.a.2.9, 20.144.1-10.a.2.10, 30.144.1-10.a.2.1, 30.144.1-10.a.2.2, 40.144.1-10.a.2.1, 40.144.1-10.a.2.2, 40.144.1-10.a.2.3, 40.144.1-10.a.2.4, 40.144.1-10.a.2.5, 40.144.1-10.a.2.6, 40.144.1-10.a.2.7, 40.144.1-10.a.2.8, 40.144.1-10.a.2.9, 40.144.1-10.a.2.10, 40.144.1-10.a.2.11, 40.144.1-10.a.2.12, 60.144.1-10.a.2.1, 60.144.1-10.a.2.2, 60.144.1-10.a.2.3, 60.144.1-10.a.2.4, 60.144.1-10.a.2.5, 60.144.1-10.a.2.6, 60.144.1-10.a.2.7, 60.144.1-10.a.2.8, 60.144.1-10.a.2.9, 60.144.1-10.a.2.10, 70.144.1-10.a.2.1, 70.144.1-10.a.2.2, 110.144.1-10.a.2.1, 110.144.1-10.a.2.2, 120.144.1-10.a.2.1, 120.144.1-10.a.2.2, 120.144.1-10.a.2.3, 120.144.1-10.a.2.4, 120.144.1-10.a.2.5, 120.144.1-10.a.2.6, 120.144.1-10.a.2.7, 120.144.1-10.a.2.8, 120.144.1-10.a.2.9, 120.144.1-10.a.2.10, 120.144.1-10.a.2.11, 120.144.1-10.a.2.12, 130.144.1-10.a.2.1, 130.144.1-10.a.2.2, 140.144.1-10.a.2.1, 140.144.1-10.a.2.2, 140.144.1-10.a.2.3, 140.144.1-10.a.2.4, 140.144.1-10.a.2.5, 140.144.1-10.a.2.6, 140.144.1-10.a.2.7, 140.144.1-10.a.2.8, 140.144.1-10.a.2.9, 140.144.1-10.a.2.10, 170.144.1-10.a.2.1, 170.144.1-10.a.2.2, 190.144.1-10.a.2.1, 190.144.1-10.a.2.2, 210.144.1-10.a.2.1, 210.144.1-10.a.2.2, 220.144.1-10.a.2.1, 220.144.1-10.a.2.2, 220.144.1-10.a.2.3, 220.144.1-10.a.2.4, 220.144.1-10.a.2.5, 220.144.1-10.a.2.6, 220.144.1-10.a.2.7, 220.144.1-10.a.2.8, 220.144.1-10.a.2.9, 220.144.1-10.a.2.10, 230.144.1-10.a.2.1, 230.144.1-10.a.2.2, 260.144.1-10.a.2.1, 260.144.1-10.a.2.2, 260.144.1-10.a.2.3, 260.144.1-10.a.2.4, 260.144.1-10.a.2.5, 260.144.1-10.a.2.6, 260.144.1-10.a.2.7, 260.144.1-10.a.2.8, 260.144.1-10.a.2.9, 260.144.1-10.a.2.10, 280.144.1-10.a.2.1, 280.144.1-10.a.2.2, 280.144.1-10.a.2.3, 280.144.1-10.a.2.4, 280.144.1-10.a.2.5, 280.144.1-10.a.2.6, 280.144.1-10.a.2.7, 280.144.1-10.a.2.8, 280.144.1-10.a.2.9, 280.144.1-10.a.2.10, 280.144.1-10.a.2.11, 280.144.1-10.a.2.12, 290.144.1-10.a.2.1, 290.144.1-10.a.2.2, 310.144.1-10.a.2.1, 310.144.1-10.a.2.2, 330.144.1-10.a.2.1, 330.144.1-10.a.2.2
Cyclic 10-isogeny field degree: $1$
Cyclic 10-torsion field degree: $4$
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 $
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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
$(-1:-1:1)$, $(1:-1:1)$, $(0:0:1)$, $(0:1:0)$, $(-1:1:1)$, $(1:1:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{696x^{2}y^{22}+92771176x^{2}y^{20}z^{2}+1616055400x^{2}y^{18}z^{4}+20871094392x^{2}y^{16}z^{6}-1712205696080x^{2}y^{14}z^{8}+20307647267728x^{2}y^{12}z^{10}-106629694737072x^{2}y^{10}z^{12}+301749581547120x^{2}y^{8}z^{14}-491801772313832x^{2}y^{6}z^{16}+462176513502600x^{2}y^{4}z^{18}-232849121094456x^{2}y^{2}z^{20}+48736572265624x^{2}z^{22}+163680xy^{22}z+59351712xy^{20}z^{3}-3817353440xy^{18}z^{5}-130966052640xy^{16}z^{7}+3513248656320xy^{14}z^{9}-29523815860160xy^{12}z^{11}+123435016355904xy^{10}z^{13}-293310218571840xy^{8}z^{15}+414794907542240xy^{6}z^{17}-346032715012320xy^{4}z^{19}+157379150389920xy^{2}z^{21}-30120849609376xz^{23}+y^{24}+13345828y^{22}z^{2}-442352798y^{20}z^{4}+2851667860y^{18}z^{6}+512512808815y^{16}z^{8}-7981334027192y^{14}z^{10}+48469620081244y^{12}z^{12}-151336760681400y^{10}z^{14}+265143010744175y^{8}z^{16}-263452133768300y^{6}z^{18}+138763427904354y^{4}z^{20}-30120849608668y^{2}z^{22}+z^{24}}{y^{2}(y-z)(y+z)(x^{2}y^{18}-12665x^{2}y^{16}z^{2}+2964610x^{2}y^{14}z^{4}-53311862x^{2}y^{12}z^{6}+165735924x^{2}y^{10}z^{8}-126904328x^{2}y^{8}z^{10}-6610x^{2}y^{6}z^{12}+630x^{2}y^{4}z^{14}-37x^{2}y^{2}z^{16}+x^{2}z^{18}-46xy^{18}z+100705xy^{16}z^{3}-8308512xy^{14}z^{5}+68138076xy^{12}z^{7}-137056156xy^{10}z^{9}+78181158xy^{8}z^{11}-7280xy^{6}z^{13}+668xy^{4}z^{15}-38xy^{2}z^{17}+xz^{19}+975y^{18}z^{2}-622214y^{16}z^{4}+21646119y^{14}z^{6}-88842581y^{12}z^{8}+78338841y^{10}z^{10}-40725y^{8}z^{12}+5901y^{6}z^{14}-591y^{4}z^{16}+36y^{2}z^{18}-z^{20})}$

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