Invariants
| Level: | $10$ | $\SL_2$-level: | $10$ | Newform level: | $100$ | ||
| Index: | $60$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $2 = 1 + \frac{ 60 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $10^{6}$ | Cusp orbits | $1^{2}\cdot4$ | ||
| Elliptic points: | $4$ 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: | 10C2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 10.60.2.3 |
Level structure
| $\GL_2(\Z/10\Z)$-generators: | $\begin{bmatrix}1&1\\0&3\end{bmatrix}$, $\begin{bmatrix}2&3\\5&8\end{bmatrix}$ |
| $\GL_2(\Z/10\Z)$-subgroup: | $C_{12}:C_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 10-isogeny field degree: | $3$ |
| Cyclic 10-torsion field degree: | $12$ |
| Full 10-torsion field degree: | $48$ |
Jacobian
| Conductor: | $2^{4}\cdot5^{4}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $2$ |
| Newforms: | 100.2.c.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{3} + x^{2} y + x y^{2} + x y z + y^{3} + 2 y^{2} z + y z^{2} - y w^{2} - z w^{2} $ |
| $=$ | $2 x^{3} - x^{2} z + x y^{2} + 2 x y z + x z^{2} + 3 x w^{2} - y^{2} z - y z^{2} + z w^{2}$ | |
| $=$ | $2 x^{3} - 2 x^{2} y - x^{2} z + 2 x y^{2} + 3 x y z - 3 x w^{2} - y^{3} - y^{2} z + y w^{2}$ | |
| $=$ | $2 x^{2} y - 2 x^{2} z + x y z + x z^{2} + y^{3} - 2 y z^{2} + 3 y w^{2} + 3 z w^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} + 5 x^{4} z - 11 x^{3} y^{2} + 10 x^{3} z^{2} - 18 x^{2} y^{2} z + 5 x^{2} z^{3} + 54 x y^{4} + \cdots + z^{5} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{5} + 5x^{3} + 5x - 11 $ |
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 |
|---|
| $(0:0:0:1)$, $(1/2:-1:1:0)$ |
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ | $=$ | $\displaystyle \frac{1}{9}y^{3}+\frac{1}{6}y^{2}z-\frac{1}{9}yz^{2}-yw^{2}-\frac{1}{6}z^{3}-zw^{2}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{5}{5832}y^{8}w-\frac{35}{5832}y^{7}zw-\frac{5}{5832}y^{6}z^{2}w+\frac{5}{324}y^{6}w^{3}+\frac{155}{1944}y^{5}z^{3}w+\frac{25}{162}y^{5}zw^{3}+\frac{145}{648}y^{4}z^{4}w+\frac{175}{324}y^{4}z^{2}w^{3}+\frac{505}{1944}y^{3}z^{5}w+\frac{65}{81}y^{3}z^{3}w^{3}+\frac{845}{5832}y^{2}z^{6}w+\frac{175}{324}y^{2}z^{4}w^{3}+\frac{215}{5832}yz^{7}w+\frac{25}{162}yz^{5}w^{3}+\frac{5}{1458}z^{8}w+\frac{5}{324}z^{6}w^{3}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{18}y^{3}+\frac{5}{18}y^{2}z+\frac{5}{18}yz^{2}+\frac{1}{18}z^{3}$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^{10}\cdot3}\cdot\frac{2921525xyz^{10}-15634750xyz^{8}w^{2}-61762200xyz^{6}w^{4}+125183000xyz^{4}w^{6}+34204400xyz^{2}w^{8}-24961248xyw^{10}+1369900xz^{11}+9453250xz^{9}w^{2}+16873800xz^{7}w^{4}-149819000xz^{5}w^{6}-178865600xz^{3}w^{8}+87705504xzw^{10}-1836950y^{2}z^{10}-15104000y^{2}z^{8}w^{2}+75312600y^{2}z^{6}w^{4}+39604000y^{2}z^{4}w^{6}-91791200y^{2}z^{2}w^{8}+2600832y^{2}w^{10}-1670850yz^{11}-2341500yz^{9}w^{2}+27100800yz^{7}w^{4}-105054000yz^{5}w^{6}+101162400yz^{3}w^{8}+78032832yzw^{10}+918475z^{12}+5594500z^{10}w^{2}-37234800z^{8}w^{4}-176792000z^{6}w^{6}+130003600z^{4}w^{8}+112987584z^{2}w^{10}-768000w^{12}}{w^{10}(3xy+6xz-2y^{2}-2yz+z^{2})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 10.12.0.a.1 | $10$ | $5$ | $5$ | $0$ | $0$ | full Jacobian |
| 10.12.0.a.2 | $10$ | $5$ | $5$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 10.120.5.c.1 | $10$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 10.120.5.e.1 | $10$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 10.180.6.a.1 | $10$ | $3$ | $3$ | $6$ | $0$ | $1^{2}\cdot2$ |
| 20.120.5.q.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 20.120.5.w.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 20.240.13.cn.1 | $20$ | $4$ | $4$ | $13$ | $2$ | $1^{7}\cdot2^{2}$ |
| 30.120.5.i.1 | $30$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 30.120.5.o.1 | $30$ | $2$ | $2$ | $5$ | $2$ | $1^{3}$ |
| 30.180.10.h.1 | $30$ | $3$ | $3$ | $10$ | $1$ | $1^{2}\cdot2^{3}$ |
| 30.240.15.v.1 | $30$ | $4$ | $4$ | $15$ | $0$ | $1^{5}\cdot2^{4}$ |
| 40.120.5.by.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{3}$ |
| 40.120.5.ch.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 40.120.5.cz.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 40.120.5.df.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{3}$ |
| 50.300.14.a.1 | $50$ | $5$ | $5$ | $14$ | $0$ | $4\cdot8$ |
| 50.300.14.c.1 | $50$ | $5$ | $5$ | $14$ | $0$ | $4\cdot8$ |
| 50.300.14.e.1 | $50$ | $5$ | $5$ | $14$ | $0$ | $4\cdot8$ |
| 50.300.14.g.1 | $50$ | $5$ | $5$ | $14$ | $0$ | $4\cdot8$ |
| 50.300.14.i.1 | $50$ | $5$ | $5$ | $14$ | $4$ | $2^{2}\cdot8$ |
| 50.300.20.a.1 | $50$ | $5$ | $5$ | $20$ | $2$ | $2^{3}\cdot4^{3}$ |
| 50.300.20.a.2 | $50$ | $5$ | $5$ | $20$ | $2$ | $2^{3}\cdot4^{3}$ |
| 60.120.5.bq.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 60.120.5.cu.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{3}$ |
| 70.120.5.y.1 | $70$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 70.120.5.z.1 | $70$ | $2$ | $2$ | $5$ | $2$ | $1^{3}$ |
| 70.480.35.bp.1 | $70$ | $8$ | $8$ | $35$ | $2$ | $1^{5}\cdot2^{14}$ |
| 70.1260.92.e.1 | $70$ | $21$ | $21$ | $92$ | $17$ | $1^{2}\cdot2^{14}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |
| 70.1680.125.cd.1 | $70$ | $28$ | $28$ | $125$ | $19$ | $1^{7}\cdot2^{28}\cdot3^{2}\cdot4^{10}\cdot6\cdot8$ |
| 110.120.5.q.1 | $110$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 110.120.5.r.1 | $110$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.5.ft.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.5.fz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.5.jw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.5.jz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 130.120.5.y.1 | $130$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 130.120.5.z.1 | $130$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.120.5.dm.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.120.5.dp.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 170.120.5.q.1 | $170$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 170.120.5.r.1 | $170$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 190.120.5.y.1 | $190$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 190.120.5.z.1 | $190$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.120.5.cg.1 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.120.5.cj.1 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.120.5.de.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.120.5.dh.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 230.120.5.q.1 | $230$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 230.120.5.r.1 | $230$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.120.5.dm.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.120.5.dp.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.5.ma.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.5.md.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.5.mm.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.5.mp.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 290.120.5.q.1 | $290$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 290.120.5.r.1 | $290$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 310.120.5.y.1 | $310$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 310.120.5.z.1 | $310$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.120.5.by.1 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.120.5.cb.1 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |