Invariants
Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
Index: | $24$ | $\PSL_2$-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 | $2^{2}\cdot4\cdot16$ | Cusp orbits | $1^{4}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16A1 |
Rouse and Zureick-Brown (RZB) label: | X159 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.24.1.1 |
Level structure
Jacobian
Conductor: | $2^{6}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} - 4x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Weierstrass model |
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$(0:0:1)$, $(-2:0:1)$, $(0:1:0)$, $(2:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^2\,\frac{10848x^{2}y^{4}z^{2}-4193280x^{2}z^{6}-176xy^{6}z+1037568xy^{2}z^{5}+y^{8}-264960y^{4}z^{4}+16777216z^{8}}{zy^{2}(8x^{2}y^{2}z+xy^{4}+64xz^{4}+16y^{2}z^{3})}$ |
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 |
---|---|---|---|---|---|---|
$X_0(8)$ | $8$ | $2$ | $2$ | $0$ | $0$ | 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 |
---|---|---|---|---|---|---|
16.48.1.a.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.48.1.d.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.48.1.g.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.48.1.i.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.48.1.q.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.48.1.q.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.48.1.r.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.48.1.r.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.48.1.s.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.48.1.s.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.48.1.t.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.48.1.t.2 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.q.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.r.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.u.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.v.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.bg.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.bg.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.bh.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.bh.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.bi.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.bi.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.bj.1 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.48.1.bj.2 | $48$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
48.72.5.a.1 | $48$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
48.96.5.mh.1 | $48$ | $4$ | $4$ | $5$ | $1$ | $1^{4}$ |
80.48.1.q.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.48.1.r.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.48.1.u.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.48.1.v.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.48.1.bg.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.48.1.bg.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.48.1.bh.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.48.1.bh.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.48.1.bi.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.48.1.bi.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.48.1.bj.1 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.48.1.bj.2 | $80$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
80.120.9.a.1 | $80$ | $5$ | $5$ | $9$ | $?$ | not computed |
80.144.9.i.1 | $80$ | $6$ | $6$ | $9$ | $?$ | not computed |
80.240.17.cm.1 | $80$ | $10$ | $10$ | $17$ | $?$ | not computed |
112.48.1.q.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.48.1.r.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.48.1.u.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.48.1.v.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.48.1.bg.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.48.1.bg.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.48.1.bh.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.48.1.bh.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.48.1.bi.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.48.1.bi.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.48.1.bj.1 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.48.1.bj.2 | $112$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
112.192.13.g.1 | $112$ | $8$ | $8$ | $13$ | $?$ | not computed |
176.48.1.q.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.48.1.r.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.48.1.u.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.48.1.v.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.48.1.bg.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.48.1.bg.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.48.1.bh.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.48.1.bh.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.48.1.bi.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.48.1.bi.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.48.1.bj.1 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.48.1.bj.2 | $176$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
176.288.21.g.1 | $176$ | $12$ | $12$ | $21$ | $?$ | not computed |
208.48.1.q.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
208.48.1.r.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
208.48.1.u.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
208.48.1.v.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
208.48.1.bg.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
208.48.1.bg.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
208.48.1.bh.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
208.48.1.bh.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
208.48.1.bi.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
208.48.1.bi.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
208.48.1.bj.1 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
208.48.1.bj.2 | $208$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.q.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.r.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.u.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.v.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.bg.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.bg.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.bh.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.bh.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.bi.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.bi.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.bj.1 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
240.48.1.bj.2 | $240$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.q.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.r.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.u.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.v.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.bg.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.bg.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.bh.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.bh.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.bi.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.bi.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.bj.1 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
272.48.1.bj.2 | $272$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.q.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.r.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.u.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.v.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.bg.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.bg.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.bh.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.bh.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.bi.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.bi.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.bj.1 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
304.48.1.bj.2 | $304$ | $2$ | $2$ | $1$ | $?$ | dimension zero |