Invariants
| Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2\cdot4\cdot6\cdot12$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12F1 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}24&7\\23&4\end{bmatrix}$, $\begin{bmatrix}50&9\\29&10\end{bmatrix}$, $\begin{bmatrix}70&83\\55&18\end{bmatrix}$, $\begin{bmatrix}73&18\\78&19\end{bmatrix}$, $\begin{bmatrix}75&22\\70&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.1.l.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $384$ |
| Full 84-torsion field degree: | $193536$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 48.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 24x + 36 $ |
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 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{4x^{2}y^{6}-303x^{2}y^{4}z^{2}+6128x^{2}y^{2}z^{4}-41041x^{2}z^{6}-30xy^{6}z+1518xy^{4}z^{3}-30793xy^{2}z^{5}+208572xz^{7}-y^{8}+82y^{6}z^{2}-2599y^{4}z^{4}+41089y^{2}z^{6}-257076z^{8}}{z^{4}y^{2}(8x^{2}-40xz-y^{2}+48z^{2})}$ |
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 |
|---|---|---|---|---|---|---|
| 4.6.0.e.1 | $4$ | $8$ | $4$ | $0$ | $0$ | full Jacobian |
| 21.8.0-3.a.1.2 | $21$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.24.0-6.a.1.4 | $42$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 84.24.0-6.a.1.8 | $84$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 84.96.1-12.a.1.12 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-12.h.1.10 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-12.o.1.6 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-12.p.1.6 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.bq.1.2 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.br.1.3 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.bu.1.9 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.1-84.bv.1.9 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 84.96.2-12.c.1.2 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-12.d.1.4 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-12.e.1.2 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-12.f.1.4 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-84.l.1.6 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-84.m.1.2 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-84.n.1.14 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.96.2-84.o.1.16 | $84$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 84.144.3-12.da.1.1 | $84$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 84.384.13-84.be.1.40 | $84$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 168.96.1-24.cu.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.et.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.jg.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.jj.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzo.1.22 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzr.1.20 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.caa.1.22 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cad.1.20 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.2-24.h.1.12 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-24.i.1.16 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-24.j.1.12 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-24.k.1.16 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-24.l.1.14 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-24.m.1.16 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-24.n.1.14 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-24.o.1.16 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-24.p.1.16 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-24.q.1.12 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-24.r.1.16 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-24.s.1.12 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.t.1.2 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.u.1.6 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.v.1.30 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.w.1.14 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.x.1.8 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.y.1.4 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.z.1.12 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.ba.1.24 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.bb.1.14 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.bc.1.30 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.bd.1.6 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.2-168.be.1.2 | $168$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 168.96.3-24.c.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-24.d.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-24.bu.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-24.bv.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.ca.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.cb.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.ce.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.cf.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.ci.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.cj.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-24.ck.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-24.cl.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.cm.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-168.cn.1.6 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-24.co.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3-24.cp.1.12 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 252.144.3-36.x.1.11 | $252$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 252.144.5-36.k.1.10 | $252$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 252.144.5-36.o.1.11 | $252$ | $3$ | $3$ | $5$ | $?$ | not computed |