Invariants
Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $12^{6}$ | Cusp orbits | $2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $3$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12A4 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.72.4.6 |
Level structure
Jacobian
Conductor: | $2^{10}\cdot3^{8}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{4}$ |
Newforms: | 36.2.a.a$^{3}$, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 3 }$
$ 0 $ | $=$ | $ 3 x^{2} + z^{2} + w^{2} $ |
$=$ | $ - x z w + 12 y^{3}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 3 x^{4} z^{2} + 9 x^{2} z^{4} + 16 y^{6} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle x$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^8\,\frac{(z^{2}-zw+w^{2})^{3}(z^{2}+zw+w^{2})^{3}}{w^{4}z^{4}(z^{2}+w^{2})^{2}}$ |
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 |
---|---|---|---|---|---|---|
12.24.0.b.1 | $12$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
12.36.1.bv.1 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{3}$ |
12.36.2.a.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
12.36.2.d.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
12.144.7.k.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
12.144.7.m.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
12.144.7.r.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
12.144.7.t.1 | $12$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
24.144.7.bq.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
24.144.7.cc.1 | $24$ | $2$ | $2$ | $7$ | $1$ | $1^{3}$ |
24.144.7.dh.1 | $24$ | $2$ | $2$ | $7$ | $2$ | $1^{3}$ |
24.144.7.dt.1 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
24.144.9.cj.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
24.144.9.ck.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
24.144.9.ga.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{5}$ |
24.144.9.gb.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
24.144.9.ie.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
24.144.9.if.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
24.144.9.iz.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{5}$ |
24.144.9.ja.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{5}$ |
36.216.16.d.1 | $36$ | $3$ | $3$ | $16$ | $3$ | $1^{12}$ |
36.648.46.i.1 | $36$ | $9$ | $9$ | $46$ | $15$ | $1^{36}\cdot2^{3}$ |
60.144.7.k.1 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
60.144.7.m.1 | $60$ | $2$ | $2$ | $7$ | $3$ | $1^{3}$ |
60.144.7.u.1 | $60$ | $2$ | $2$ | $7$ | $3$ | $1^{3}$ |
60.144.7.w.1 | $60$ | $2$ | $2$ | $7$ | $0$ | $1^{3}$ |
60.360.28.d.1 | $60$ | $5$ | $5$ | $28$ | $10$ | $1^{24}$ |
60.432.31.d.1 | $60$ | $6$ | $6$ | $31$ | $4$ | $1^{27}$ |
60.720.55.bh.1 | $60$ | $10$ | $10$ | $55$ | $21$ | $1^{51}$ |
84.144.7.k.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
84.144.7.m.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
84.144.7.r.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
84.144.7.t.1 | $84$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.144.7.bq.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.144.7.cc.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.144.7.dg.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.144.7.ds.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
120.144.9.ga.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.gb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.hk.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.hl.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.js.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.jt.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.ku.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
120.144.9.kv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
132.144.7.k.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
132.144.7.m.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
132.144.7.r.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
132.144.7.t.1 | $132$ | $2$ | $2$ | $7$ | $?$ | not computed |
156.144.7.k.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
156.144.7.m.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
156.144.7.r.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
156.144.7.t.1 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.144.7.bq.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.144.7.cc.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.144.7.cz.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.144.7.dl.1 | $168$ | $2$ | $2$ | $7$ | $?$ | not computed |
168.144.9.ga.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.gb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.hk.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.hl.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.jo.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.jp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.kq.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
168.144.9.kr.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
204.144.7.k.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
204.144.7.m.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
204.144.7.r.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
204.144.7.t.1 | $204$ | $2$ | $2$ | $7$ | $?$ | not computed |
228.144.7.k.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
228.144.7.m.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
228.144.7.r.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
228.144.7.t.1 | $228$ | $2$ | $2$ | $7$ | $?$ | not computed |
264.144.7.bq.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
264.144.7.cc.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
264.144.7.cz.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
264.144.7.dl.1 | $264$ | $2$ | $2$ | $7$ | $?$ | not computed |
264.144.9.ga.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.gb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.hk.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.hl.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.jo.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.jp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.kq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
264.144.9.kr.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
276.144.7.k.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
276.144.7.m.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
276.144.7.r.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
276.144.7.t.1 | $276$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.144.7.bq.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.144.7.cc.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.144.7.cz.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.144.7.dl.1 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.144.9.ga.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.gb.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.hk.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.hl.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.jo.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.jp.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.kq.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
312.144.9.kr.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |