Invariants
Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $36$ | ||
Index: | $144$ | $\PSL_2$-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 | $3^{8}\cdot12^{4}$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $6$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12S1 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}1&60\\73&23\end{bmatrix}$, $\begin{bmatrix}31&24\\18&49\end{bmatrix}$, $\begin{bmatrix}53&36\\24&23\end{bmatrix}$, $\begin{bmatrix}65&72\\69&5\end{bmatrix}$, $\begin{bmatrix}67&36\\61&5\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.72.1.f.1 for the level structure with $-I$) |
Cyclic 84-isogeny field degree: | $8$ |
Cyclic 84-torsion field degree: | $192$ |
Full 84-torsion field degree: | $64512$ |
Jacobian
Conductor: | $?$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 36.2.a.a |
Models
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} + 1 $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{y(y-z)^{3}(y+3z)^{9}(y^{2}+6yz-3z^{2})^{3}(y^{6}+234y^{5}z+747y^{4}z^{2}+540y^{3}z^{3}-729y^{2}z^{4}-486yz^{5}-243z^{6})^{3}}{zy^{4}(y-3z)^{12}(y-z)^{4}(y+z)^{4}(y+3z)^{12}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
$X_0(4)$ | $4$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
21.24.0-3.a.1.1 | $21$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
42.72.0-6.a.1.1 | $42$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
84.48.0-12.g.1.10 | $84$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
84.48.0-12.g.1.12 | $84$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
84.72.0-6.a.1.3 | $84$ | $2$ | $2$ | $0$ | $?$ | 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 |
---|---|---|---|---|---|---|
84.288.3-12.c.1.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
84.288.3-12.c.1.7 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
84.288.3-12.c.2.2 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
84.288.3-12.c.2.5 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
84.288.3-84.c.1.3 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
84.288.3-84.c.1.6 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
84.288.3-84.c.2.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
84.288.3-84.c.2.8 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
84.288.5-12.b.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-12.n.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-12.u.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-12.v.1.2 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-84.ca.1.4 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-84.cb.1.4 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-84.ci.1.8 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
84.288.5-84.cj.1.8 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.3-24.c.1.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-24.c.1.19 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-24.c.2.4 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-24.c.2.17 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-168.c.1.3 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-168.c.1.22 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-168.c.2.7 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-168.c.2.18 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-24.f.1.4 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-24.f.1.21 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-168.f.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-168.f.1.28 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-24.g.1.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-24.g.1.23 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-168.g.1.2 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.3-168.g.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.5-24.ba.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.dq.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.ei.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.ej.1.8 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.eq.1.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.er.1.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.eu.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.ev.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.fc.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.fd.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.gq.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-24.gx.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.oa.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.oh.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.pc.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.pd.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.pg.1.14 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.ph.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.pk.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.pl.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.po.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.pp.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.rk.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.5-168.rr.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
168.288.7-24.xq.1.24 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.7-24.xq.1.25 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.7-24.xr.1.20 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.7-24.xr.1.29 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.7-168.czm.1.4 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.7-168.czm.1.64 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.7-168.czn.1.12 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.288.7-168.czn.1.56 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
252.432.7-36.k.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-36.k.1.8 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-36.n.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.n.1.5 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.n.1.20 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.p.1.2 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.p.1.20 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-36.r.1.1 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.r.1.2 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.7-252.r.1.20 | $252$ | $3$ | $3$ | $7$ | $?$ | not computed |
252.432.10-36.g.1.1 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
252.432.10-36.g.1.11 | $252$ | $3$ | $3$ | $10$ | $?$ | not computed |
252.432.13-36.f.1.1 | $252$ | $3$ | $3$ | $13$ | $?$ | not computed |