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Properties

Label 10.120.5.c.1
Level $10$
Index $120$
Genus $5$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $2$

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Invariants

Level: $10$ $\SL_2$-level: $10$ Newform level: $100$
Index: $120$ $\PSL_2$-index:$120$
Genus: $5 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $2$ are rational) Cusp widths $10^{12}$ Cusp orbits $1^{2}\cdot2\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 10A5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 10.120.5.2

Level structure

$\GL_2(\Z/10\Z)$-generators: $\begin{bmatrix}1&4\\0&9\end{bmatrix}$, $\begin{bmatrix}2&1\\5&9\end{bmatrix}$
$\GL_2(\Z/10\Z)$-subgroup: $C_2\times C_{12}$
Contains $-I$: yes
Quadratic refinements: 10.240.5-10.c.1.1, 10.240.5-10.c.1.2, 20.240.5-10.c.1.1, 20.240.5-10.c.1.2, 20.240.5-10.c.1.3, 20.240.5-10.c.1.4, 30.240.5-10.c.1.1, 30.240.5-10.c.1.2, 40.240.5-10.c.1.1, 40.240.5-10.c.1.2, 40.240.5-10.c.1.3, 40.240.5-10.c.1.4, 40.240.5-10.c.1.5, 40.240.5-10.c.1.6, 60.240.5-10.c.1.1, 60.240.5-10.c.1.2, 60.240.5-10.c.1.3, 60.240.5-10.c.1.4, 70.240.5-10.c.1.1, 70.240.5-10.c.1.2, 110.240.5-10.c.1.1, 110.240.5-10.c.1.2, 120.240.5-10.c.1.1, 120.240.5-10.c.1.2, 120.240.5-10.c.1.3, 120.240.5-10.c.1.4, 120.240.5-10.c.1.5, 120.240.5-10.c.1.6, 130.240.5-10.c.1.1, 130.240.5-10.c.1.2, 140.240.5-10.c.1.1, 140.240.5-10.c.1.2, 140.240.5-10.c.1.3, 140.240.5-10.c.1.4, 170.240.5-10.c.1.1, 170.240.5-10.c.1.2, 190.240.5-10.c.1.1, 190.240.5-10.c.1.2, 210.240.5-10.c.1.1, 210.240.5-10.c.1.2, 220.240.5-10.c.1.1, 220.240.5-10.c.1.2, 220.240.5-10.c.1.3, 220.240.5-10.c.1.4, 230.240.5-10.c.1.1, 230.240.5-10.c.1.2, 260.240.5-10.c.1.1, 260.240.5-10.c.1.2, 260.240.5-10.c.1.3, 260.240.5-10.c.1.4, 280.240.5-10.c.1.1, 280.240.5-10.c.1.2, 280.240.5-10.c.1.3, 280.240.5-10.c.1.4, 280.240.5-10.c.1.5, 280.240.5-10.c.1.6, 290.240.5-10.c.1.1, 290.240.5-10.c.1.2, 310.240.5-10.c.1.1, 310.240.5-10.c.1.2, 330.240.5-10.c.1.1, 330.240.5-10.c.1.2
Cyclic 10-isogeny field degree: $3$
Cyclic 10-torsion field degree: $6$
Full 10-torsion field degree: $24$

Jacobian

Conductor: $2^{10}\cdot5^{8}$
Simple: no
Squarefree: no
Decomposition: $1^{3}\cdot2$
Newforms: 20.2.a.a$^{2}$, 100.2.a.a, 100.2.c.a

Models

Embedded model Embedded model in $\mathbb{P}^{6}$

$ 0 $ $=$ $ z^{2} v - z w v - z u v + w t v - t u v + u^{2} v $
$=$ $z^{2} u - z w u - z u^{2} + w t u - t u^{2} + u^{3}$
$=$ $z^{3} - z^{2} w - z^{2} u + z w t - z t u + z u^{2}$
$=$ $z^{2} t - z w t - z t u + w t^{2} - t^{2} u + t u^{2}$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{7} + x^{6} z + 110 x^{5} y^{2} + 9 x^{5} z^{2} - 75 x^{4} y^{2} z - 5 x^{4} z^{3} + 75 x^{3} y^{2} z^{2} + \cdots + 22 z^{7} $
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Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ -x^{11} - 11x^{6} + x $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model
$(0:0:0:0:0:0:1)$, $(2:1:1:1:1:1:0)$

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle v$
$\displaystyle Z$ $=$ $\displaystyle y$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle -\frac{2}{5}x-\frac{1}{5}y$
$\displaystyle Y$ $=$ $\displaystyle -\frac{22}{625}x^{5}v+\frac{3}{125}x^{4}yv-\frac{3}{125}x^{3}y^{2}v-\frac{2}{125}x^{2}y^{3}v-\frac{1}{625}y^{5}v$
$\displaystyle Z$ $=$ $\displaystyle -\frac{1}{5}x+\frac{2}{5}y$

Maps to other modular curves

$j$-invariant map of degree 120 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2}\cdot\frac{1686xy^{10}-2330xy^{8}v^{2}-52860xy^{6}v^{4}+60390xy^{4}v^{6}+3026590xy^{2}v^{8}-295130xu^{10}-3710942xu^{8}v^{2}-12167902xu^{6}v^{4}-198674xu^{4}v^{6}+12685304xu^{2}v^{8}-24364378xv^{10}+2728y^{11}-19740y^{9}v^{2}+55870y^{7}v^{4}-10430y^{5}v^{6}-1863880y^{3}v^{8}-868340yu^{10}-10887606yu^{8}v^{2}-42373626yu^{6}v^{4}-39665892yu^{4}v^{6}+30310562yu^{2}v^{8}+13515756yv^{10}+290510ztu^{9}+3952734ztu^{7}v^{2}+16609112ztu^{5}v^{4}+18568700ztu^{3}v^{6}-6573316ztuv^{8}+816225zu^{10}+10564015zu^{8}v^{2}+42491007zu^{6}v^{4}+46446938zu^{4}v^{6}-4525622zu^{2}v^{8}+20821050zv^{10}-525965wtu^{9}-6719781wtu^{7}v^{2}-27520095wtu^{5}v^{4}-34215218wtu^{3}v^{6}-692764wtuv^{8}+754110wu^{10}+11237174wu^{8}v^{2}+59502078wu^{6}v^{4}+130388944wu^{4}v^{6}+101965318wu^{2}v^{8}+12036250wv^{10}-463100t^{2}u^{9}-6058640t^{2}u^{7}v^{2}-27950096t^{2}u^{5}v^{4}-52023544t^{2}u^{3}v^{6}-29451984t^{2}uv^{8}+290010tu^{10}+5450034tu^{8}v^{2}+35217632tu^{6}v^{4}+93868380tu^{4}v^{6}+91306634tu^{2}v^{8}+6137800tv^{10}+290760u^{11}+5115534u^{9}v^{2}+31382132u^{7}v^{4}+74943160u^{5}v^{6}+37609124u^{3}v^{8}-49348570uv^{10}}{v^{10}(7x+11y-4z-3w+3t+4u)}$

Modular covers

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Cover information

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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
$X_{\mathrm{ns}}(2)$ $2$ $60$ $60$ $0$ $0$ full Jacobian
5.60.0.a.1 $5$ $2$ $2$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
5.60.0.a.1 $5$ $2$ $2$ $0$ $0$ full Jacobian
10.24.1.a.1 $10$ $5$ $5$ $1$ $0$ $1^{2}\cdot2$
10.24.1.a.2 $10$ $5$ $5$ $1$ $0$ $1^{2}\cdot2$
10.60.2.c.1 $10$ $2$ $2$ $2$ $0$ $1^{3}$
10.60.3.b.1 $10$ $2$ $2$ $3$ $0$ $2$

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 $10$ $3$ $3$ $13$ $0$ $1^{4}\cdot2^{2}$
20.240.15.x.1 $20$ $2$ $2$ $15$ $0$ $2^{3}\cdot4$
20.240.15.z.1 $20$ $2$ $2$ $15$ $0$ $2^{3}\cdot4$
20.480.29.bk.1 $20$ $4$ $4$ $29$ $3$ $1^{12}\cdot2^{6}$
30.360.25.q.1 $30$ $3$ $3$ $25$ $2$ $1^{8}\cdot2^{6}$
30.480.29.e.1 $30$ $4$ $4$ $29$ $1$ $1^{12}\cdot2^{6}$
40.240.15.jk.1 $40$ $2$ $2$ $15$ $0$ $2^{3}\cdot4$
40.240.15.jm.1 $40$ $2$ $2$ $15$ $0$ $2^{3}\cdot4$
50.600.37.a.1 $50$ $5$ $5$ $37$ $0$ $4\cdot8^{2}\cdot12$
50.600.37.b.1 $50$ $5$ $5$ $37$ $0$ $4\cdot8^{2}\cdot12$
50.600.37.c.1 $50$ $5$ $5$ $37$ $0$ $4\cdot8^{2}\cdot12$
50.600.37.d.1 $50$ $5$ $5$ $37$ $0$ $4\cdot8^{2}\cdot12$
50.600.37.e.1 $50$ $5$ $5$ $37$ $4$ $2^{2}\cdot6^{2}\cdot8^{2}$
50.600.41.a.1 $50$ $5$ $5$ $41$ $4$ $1^{2}\cdot2^{5}\cdot4^{6}$
50.600.41.a.2 $50$ $5$ $5$ $41$ $4$ $1^{2}\cdot2^{5}\cdot4^{6}$
60.240.15.ch.1 $60$ $2$ $2$ $15$ $0$ $2^{3}\cdot4$
60.240.15.cj.1 $60$ $2$ $2$ $15$ $0$ $2^{3}\cdot4$
70.360.13.d.1 $70$ $3$ $3$ $13$ $0$ $2^{2}\cdot4$
70.960.69.e.1 $70$ $8$ $8$ $69$ $7$ $1^{22}\cdot2^{17}\cdot4^{2}$
70.2520.193.p.1 $70$ $21$ $21$ $193$ $36$ $1^{12}\cdot2^{26}\cdot3^{4}\cdot4^{21}\cdot6^{2}\cdot8^{2}$
70.3360.257.m.1 $70$ $28$ $28$ $257$ $43$ $1^{34}\cdot2^{43}\cdot3^{4}\cdot4^{23}\cdot6^{2}\cdot8^{2}$
90.360.13.b.1 $90$ $3$ $3$ $13$ $?$ not computed
120.240.15.xs.1 $120$ $2$ $2$ $15$ $?$ not computed
120.240.15.xu.1 $120$ $2$ $2$ $15$ $?$ not computed
130.360.13.d.1 $130$ $3$ $3$ $13$ $?$ not computed
140.240.15.bs.1 $140$ $2$ $2$ $15$ $?$ not computed
140.240.15.bu.1 $140$ $2$ $2$ $15$ $?$ not computed
190.360.13.d.1 $190$ $3$ $3$ $13$ $?$ not computed
220.240.15.bs.1 $220$ $2$ $2$ $15$ $?$ not computed
220.240.15.bu.1 $220$ $2$ $2$ $15$ $?$ not computed
260.240.15.ci.1 $260$ $2$ $2$ $15$ $?$ not computed
260.240.15.ck.1 $260$ $2$ $2$ $15$ $?$ not computed
280.240.15.vs.1 $280$ $2$ $2$ $15$ $?$ not computed
280.240.15.vu.1 $280$ $2$ $2$ $15$ $?$ not computed
310.360.13.d.1 $310$ $3$ $3$ $13$ $?$ not computed