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: | $6$ (none of which are rational) | Cusp widths | $10^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3$) |
Other labels
| Cummins and Pauli (CP) label: | 10C2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 10.60.2.4 |
Level structure
| $\GL_2(\Z/10\Z)$-generators: | $\begin{bmatrix}4&3\\7&1\end{bmatrix}$, $\begin{bmatrix}6&9\\3&4\end{bmatrix}$ |
| $\GL_2(\Z/10\Z)$-subgroup: | $C_{24}:C_2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 10-isogeny field degree: | $6$ |
| Cyclic 10-torsion field degree: | $24$ |
| Full 10-torsion field degree: | $48$ |
Jacobian
| Conductor: | $2^{4}\cdot5^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{2}$ |
| Newforms: | 20.2.a.a, 100.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} + x y - y^{2} + z w - w^{2} - w t - t^{2} $ |
| $=$ | $x z + 2 x t + y z - y w$ | |
| $=$ | $2 x^{2} + 2 x y - 2 y^{2} + z^{2} - 2 z w + z t + 2 w^{2} + w t + 2 t^{2}$ | |
| $=$ | $x z - 4 x w - x t - y w - 2 y t$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{6} + 48 x^{5} z + 11 x^{4} y^{2} + 40 x^{4} z^{2} + 7 x^{3} y^{2} z + 9 x^{2} y^{2} z^{2} + \cdots - z^{6} $ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 2x^{5} + 5x^{4} + 5x^{3} - 5x^{2} + 18x - 11 $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Plane model | Weierstrass model | Embedded model | |
|---|---|---|---|---|---|---|---|
| 27.a3 | $-3$ | $0$ | $0.000$ | $(0:-1:1)$, $(0:1:1)$ | $(1:1:0)$, $(1:-1:0)$ | $(0:-1:-2:-2:1)$, $(0:1:-2:-2:1)$ | |
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ | $=$ | $\displaystyle x^{2}y+xy^{2}+\frac{1}{4}y^{3}$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -\frac{11}{4}x^{8}t-\frac{29}{4}x^{7}yt-\frac{79}{8}x^{6}y^{2}t-\frac{37}{4}x^{5}y^{3}t-\frac{395}{64}x^{4}y^{4}t-\frac{183}{64}x^{3}y^{5}t-\frac{57}{64}x^{2}y^{6}t-\frac{11}{64}xy^{7}t-\frac{1}{64}y^{8}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x^{3}-x^{2}y-\frac{1}{4}xy^{2}$ |
Maps to other modular curves
$j$-invariant map of degree 60 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 5^2\,\frac{34375xy^{9}-185625xy^{7}t^{2}+88250xy^{5}t^{4}-205700xy^{3}t^{6}-957225xyt^{8}-21875y^{10}+121250y^{8}t^{2}-85875y^{6}t^{4}+139150y^{4}t^{6}+556850y^{2}t^{8}+66816zw^{9}-1271040zw^{8}t+4636832zw^{7}t^{2}-938208zw^{6}t^{3}-5917871zw^{5}t^{4}-3101201zw^{4}t^{5}+2660187zw^{3}t^{6}+1296191zw^{2}t^{7}-280082zwt^{8}+151497zt^{9}-120832w^{10}+1753088w^{9}t-3625984w^{8}t^{2}-3155008w^{7}t^{3}-1790312w^{6}t^{4}+6190044w^{5}t^{5}+6987918w^{4}t^{6}+3891909w^{3}t^{7}-123238w^{2}t^{8}+315120wt^{9}+551046t^{10}}{5zw^{9}-80zw^{8}t+185zw^{7}t^{2}+160zw^{6}t^{3}-225zw^{5}t^{4}-185zw^{4}t^{5}+10zw^{3}t^{6}+30zw^{2}t^{7}+5zwt^{8}-9w^{10}+105w^{9}t-80w^{8}t^{2}-250w^{7}t^{3}-260w^{6}t^{4}-44w^{5}t^{5}+215w^{4}t^{6}+160w^{3}t^{7}+25w^{2}t^{8}-5wt^{9}-t^{10}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 5.30.0.b.1 | $5$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 10.20.0.a.1 | $10$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}^+(10)$ | $10$ | $3$ | $3$ | $0$ | $0$ | 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.d.1 | $10$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 10.120.5.h.1 | $10$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 10.180.6.b.1 | $10$ | $3$ | $3$ | $6$ | $0$ | $1^{4}$ |
| 20.120.5.t.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 20.120.5.bn.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{3}$ |
| 20.240.13.df.1 | $20$ | $4$ | $4$ | $13$ | $5$ | $1^{11}$ |
| 30.120.5.l.1 | $30$ | $2$ | $2$ | $5$ | $2$ | $1^{3}$ |
| 30.120.5.x.1 | $30$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 30.180.10.o.1 | $30$ | $3$ | $3$ | $10$ | $3$ | $1^{6}\cdot2$ |
| 30.240.15.bb.1 | $30$ | $4$ | $4$ | $15$ | $2$ | $1^{13}$ |
| 40.120.5.ck.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 40.120.5.ct.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{3}$ |
| 40.120.5.fh.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{3}$ |
| 40.120.5.fn.1 | $40$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 50.300.14.k.1 | $50$ | $5$ | $5$ | $14$ | $12$ | $2^{2}\cdot8$ |
| 50.300.22.f.1 | $50$ | $5$ | $5$ | $22$ | $10$ | $2^{2}\cdot8^{2}$ |
| 50.300.22.h.1 | $50$ | $5$ | $5$ | $22$ | $20$ | $2^{2}\cdot8^{2}$ |
| 60.120.5.ch.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{3}$ |
| 60.120.5.et.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 70.120.5.bc.1 | $70$ | $2$ | $2$ | $5$ | $2$ | $1^{3}$ |
| 70.120.5.bf.1 | $70$ | $2$ | $2$ | $5$ | $1$ | $1^{3}$ |
| 70.480.35.bs.1 | $70$ | $8$ | $8$ | $35$ | $10$ | $1^{23}\cdot2^{5}$ |
| 70.1260.92.h.1 | $70$ | $21$ | $21$ | $92$ | $53$ | $1^{8}\cdot2^{17}\cdot3^{4}\cdot4^{9}$ |
| 70.1680.125.cg.1 | $70$ | $28$ | $28$ | $125$ | $63$ | $1^{31}\cdot2^{22}\cdot3^{4}\cdot4^{9}$ |
| 110.120.5.u.1 | $110$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 110.120.5.x.1 | $110$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.5.ib.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.5.ih.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.5.qu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.5.qx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 130.120.5.bc.1 | $130$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 130.120.5.bf.1 | $130$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.120.5.eg.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.120.5.ex.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 170.120.5.u.1 | $170$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 170.120.5.x.1 | $170$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 190.120.5.bc.1 | $190$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 190.120.5.bf.1 | $190$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.120.5.cs.1 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 210.120.5.db.1 | $210$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.120.5.dy.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.120.5.ep.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 230.120.5.u.1 | $230$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 230.120.5.x.1 | $230$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.120.5.eg.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.120.5.ex.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.5.ou.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.5.ox.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.5.rc.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.5.rf.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 290.120.5.u.1 | $290$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 290.120.5.x.1 | $290$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 310.120.5.bc.1 | $310$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 310.120.5.bf.1 | $310$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.120.5.ck.1 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 330.120.5.ct.1 | $330$ | $2$ | $2$ | $5$ | $?$ | not computed |