Invariants
Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (none of which are rational) | Cusp widths | $12^{12}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $4$ | ||||||
$\overline{\Q}$-gonality: | $4$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12B7 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.144.7.27 |
Level structure
Jacobian
Conductor: | $2^{22}\cdot3^{13}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{7}$ |
Newforms: | 24.2.a.a, 36.2.a.a$^{2}$, 72.2.a.a, 144.2.a.a$^{2}$, 144.2.a.b |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
$ 0 $ | $=$ | $ x u + z v $ |
$=$ | $y v + z u$ | |
$=$ | $x y - z^{2}$ | |
$=$ | $w u + w v + t v$ | |
$=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 729 x^{12} - 486 x^{10} y^{2} + 486 x^{10} z^{2} + 486 x^{8} y^{4} - 648 x^{8} y^{2} z^{2} + \cdots + 16 y^{4} z^{8} $ |
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:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}u$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 12.72.4.q.1 :
$\displaystyle X$ | $=$ | $\displaystyle -x-y$ |
$\displaystyle Y$ | $=$ | $\displaystyle z$ |
$\displaystyle Z$ | $=$ | $\displaystyle -t$ |
$\displaystyle W$ | $=$ | $\displaystyle u-v$ |
Equation of the image curve:
$0$ | $=$ | $ 6X^{2}-Z^{2}+W^{2} $ |
$=$ | $ 3X^{3}+24Y^{3}-XZ^{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 |
---|---|---|---|---|---|---|
12.72.2.i.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{5}$ |
12.72.2.j.1 | $12$ | $2$ | $2$ | $2$ | $0$ | $1^{5}$ |
12.72.3.bi.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
12.72.3.ea.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
12.72.3.eb.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
12.72.4.q.1 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
12.72.4.s.1 | $12$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
24.288.17.clo.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
24.288.17.clw.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
24.288.17.csk.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
24.288.17.csv.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
24.288.17.dfr.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
24.288.17.dge.1 | $24$ | $2$ | $2$ | $17$ | $5$ | $1^{10}$ |
24.288.17.dii.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
24.288.17.diq.1 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{10}$ |
36.432.31.bj.1 | $36$ | $3$ | $3$ | $31$ | $6$ | $1^{24}$ |
36.1296.91.cq.1 | $36$ | $9$ | $9$ | $91$ | $32$ | $1^{48}\cdot2^{18}$ |
60.720.55.nw.1 | $60$ | $5$ | $5$ | $55$ | $18$ | $1^{48}$ |
60.864.61.bsr.1 | $60$ | $6$ | $6$ | $61$ | $8$ | $1^{54}$ |
60.1440.109.ciy.1 | $60$ | $10$ | $10$ | $109$ | $38$ | $1^{102}$ |
120.288.17.tcq.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
120.288.17.tdg.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
120.288.17.tho.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
120.288.17.tie.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
120.288.17.weq.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
120.288.17.wfg.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
120.288.17.wjo.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
120.288.17.wke.1 | $120$ | $2$ | $2$ | $17$ | $?$ | not computed |
168.288.17.rdy.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
168.288.17.reo.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
168.288.17.riw.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
168.288.17.rjm.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
168.288.17.tnm.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
168.288.17.toc.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
168.288.17.tsk.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
168.288.17.tta.1 | $168$ | $2$ | $2$ | $17$ | $?$ | not computed |
264.288.17.ret.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
264.288.17.rfj.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
264.288.17.rjr.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
264.288.17.rkh.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
264.288.17.toj.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
264.288.17.toz.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
264.288.17.tth.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
264.288.17.ttx.1 | $264$ | $2$ | $2$ | $17$ | $?$ | not computed |
312.288.17.rec.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
312.288.17.res.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
312.288.17.rja.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
312.288.17.rjq.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
312.288.17.tnq.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
312.288.17.tog.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
312.288.17.tso.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |
312.288.17.tte.1 | $312$ | $2$ | $2$ | $17$ | $?$ | not computed |