↑

Properties

Label 8.12.0.q.1
Level $8$
Index $12$
Genus $0$
Analytic rank $0$
Cusps $3$
$\Q$-cusps $1$

Related objects

Downloads

Learn more

Invariants

Level: $8$ $\SL_2$-level: $8$
Index: $12$ $\PSL_2$-index:$12$
Genus: $0 = 1 + \frac{ 12 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$
Cusps: $3$ (of which $1$ is rational) Cusp widths $2^{2}\cdot8$ Cusp orbits $1\cdot2$
Elliptic points: $2$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $1$
Rational CM points: yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label: 8D0
Sutherland and Zywina (SZ) label: 8D0-8c
Rouse and Zureick-Brown (RZB) label: X35
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 8.12.0.32

Level structure

$\GL_2(\Z/8\Z)$-generators: $\begin{bmatrix}3&3\\6&1\end{bmatrix}$, $\begin{bmatrix}3&4\\2&5\end{bmatrix}$, $\begin{bmatrix}5&5\\0&7\end{bmatrix}$, $\begin{bmatrix}7&5\\4&5\end{bmatrix}$
$\GL_2(\Z/8\Z)$-subgroup: $C_2^3.C_2^4$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 8-isogeny field degree: $4$
Cyclic 8-torsion field degree: $16$
Full 8-torsion field degree: $128$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points, including 73 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 12 to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{x^{12}(x^{2}+4y^{2})^{3}(x^{2}+12y^{2})^{3}}{y^{8}x^{12}(x^{2}+8y^{2})^{2}}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
4.6.0.e.1 $4$ $2$ $2$ $0$ $0$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
8.24.0.f.1 $8$ $2$ $2$ $0$
8.24.0.l.1 $8$ $2$ $2$ $0$
8.24.0.p.1 $8$ $2$ $2$ $0$
8.24.0.s.1 $8$ $2$ $2$ $0$
8.24.0.bc.1 $8$ $2$ $2$ $0$
8.24.0.bd.1 $8$ $2$ $2$ $0$
8.24.0.bg.1 $8$ $2$ $2$ $0$
8.24.0.bh.1 $8$ $2$ $2$ $0$
8.24.1.q.1 $8$ $2$ $2$ $1$
8.24.1.r.1 $8$ $2$ $2$ $1$
8.24.1.u.1 $8$ $2$ $2$ $1$
8.24.1.v.1 $8$ $2$ $2$ $1$
24.24.0.ce.1 $24$ $2$ $2$ $0$
24.24.0.cg.1 $24$ $2$ $2$ $0$
24.24.0.ci.1 $24$ $2$ $2$ $0$
24.24.0.ck.1 $24$ $2$ $2$ $0$
24.24.0.dg.1 $24$ $2$ $2$ $0$
24.24.0.dh.1 $24$ $2$ $2$ $0$
24.24.0.dk.1 $24$ $2$ $2$ $0$
24.24.0.dl.1 $24$ $2$ $2$ $0$
24.24.1.cu.1 $24$ $2$ $2$ $1$
24.24.1.cv.1 $24$ $2$ $2$ $1$
24.24.1.cy.1 $24$ $2$ $2$ $1$
24.24.1.cz.1 $24$ $2$ $2$ $1$
24.36.1.fo.1 $24$ $3$ $3$ $1$
24.48.2.h.1 $24$ $4$ $4$ $2$
40.24.0.cc.1 $40$ $2$ $2$ $0$
40.24.0.ce.1 $40$ $2$ $2$ $0$
40.24.0.cg.1 $40$ $2$ $2$ $0$
40.24.0.ci.1 $40$ $2$ $2$ $0$
40.24.0.de.1 $40$ $2$ $2$ $0$
40.24.0.df.1 $40$ $2$ $2$ $0$
40.24.0.di.1 $40$ $2$ $2$ $0$
40.24.0.dj.1 $40$ $2$ $2$ $0$
40.24.1.cu.1 $40$ $2$ $2$ $1$
40.24.1.cv.1 $40$ $2$ $2$ $1$
40.24.1.cy.1 $40$ $2$ $2$ $1$
40.24.1.cz.1 $40$ $2$ $2$ $1$
40.60.4.ci.1 $40$ $5$ $5$ $4$
40.72.3.fc.1 $40$ $6$ $6$ $3$
40.120.7.ie.1 $40$ $10$ $10$ $7$
56.24.0.bw.1 $56$ $2$ $2$ $0$
56.24.0.by.1 $56$ $2$ $2$ $0$
56.24.0.ca.1 $56$ $2$ $2$ $0$
56.24.0.cc.1 $56$ $2$ $2$ $0$
56.24.0.cy.1 $56$ $2$ $2$ $0$
56.24.0.cz.1 $56$ $2$ $2$ $0$
56.24.0.dc.1 $56$ $2$ $2$ $0$
56.24.0.dd.1 $56$ $2$ $2$ $0$
56.24.1.bw.1 $56$ $2$ $2$ $1$
56.24.1.bx.1 $56$ $2$ $2$ $1$
56.24.1.ca.1 $56$ $2$ $2$ $1$
56.24.1.cb.1 $56$ $2$ $2$ $1$
56.96.6.o.1 $56$ $8$ $8$ $6$
56.252.15.by.1 $56$ $21$ $21$ $15$
56.336.21.ic.1 $56$ $28$ $28$ $21$
72.324.22.io.1 $72$ $27$ $27$ $22$
88.24.0.bw.1 $88$ $2$ $2$ $0$
88.24.0.by.1 $88$ $2$ $2$ $0$
88.24.0.ca.1 $88$ $2$ $2$ $0$
88.24.0.cc.1 $88$ $2$ $2$ $0$
88.24.0.cy.1 $88$ $2$ $2$ $0$
88.24.0.cz.1 $88$ $2$ $2$ $0$
88.24.0.dc.1 $88$ $2$ $2$ $0$
88.24.0.dd.1 $88$ $2$ $2$ $0$
88.24.1.bw.1 $88$ $2$ $2$ $1$
88.24.1.bx.1 $88$ $2$ $2$ $1$
88.24.1.ca.1 $88$ $2$ $2$ $1$
88.24.1.cb.1 $88$ $2$ $2$ $1$
88.144.10.g.1 $88$ $12$ $12$ $10$
104.24.0.cc.1 $104$ $2$ $2$ $0$
104.24.0.ce.1 $104$ $2$ $2$ $0$
104.24.0.cg.1 $104$ $2$ $2$ $0$
104.24.0.ci.1 $104$ $2$ $2$ $0$
104.24.0.de.1 $104$ $2$ $2$ $0$
104.24.0.df.1 $104$ $2$ $2$ $0$
104.24.0.di.1 $104$ $2$ $2$ $0$
104.24.0.dj.1 $104$ $2$ $2$ $0$
104.24.1.bw.1 $104$ $2$ $2$ $1$
104.24.1.bx.1 $104$ $2$ $2$ $1$
104.24.1.ca.1 $104$ $2$ $2$ $1$
104.24.1.cb.1 $104$ $2$ $2$ $1$
104.168.11.fc.1 $104$ $14$ $14$ $11$
120.24.0.gw.1 $120$ $2$ $2$ $0$
120.24.0.gy.1 $120$ $2$ $2$ $0$
120.24.0.ha.1 $120$ $2$ $2$ $0$
120.24.0.hc.1 $120$ $2$ $2$ $0$
120.24.0.ja.1 $120$ $2$ $2$ $0$
120.24.0.jb.1 $120$ $2$ $2$ $0$
120.24.0.je.1 $120$ $2$ $2$ $0$
120.24.0.jf.1 $120$ $2$ $2$ $0$
120.24.1.lo.1 $120$ $2$ $2$ $1$
120.24.1.lp.1 $120$ $2$ $2$ $1$
120.24.1.ls.1 $120$ $2$ $2$ $1$
120.24.1.lt.1 $120$ $2$ $2$ $1$
136.24.0.cc.1 $136$ $2$ $2$ $0$
136.24.0.ce.1 $136$ $2$ $2$ $0$
136.24.0.cg.1 $136$ $2$ $2$ $0$
136.24.0.ci.1 $136$ $2$ $2$ $0$
136.24.0.de.1 $136$ $2$ $2$ $0$
136.24.0.df.1 $136$ $2$ $2$ $0$
136.24.0.di.1 $136$ $2$ $2$ $0$
136.24.0.dj.1 $136$ $2$ $2$ $0$
136.24.1.bw.1 $136$ $2$ $2$ $1$
136.24.1.bx.1 $136$ $2$ $2$ $1$
136.24.1.ca.1 $136$ $2$ $2$ $1$
136.24.1.cb.1 $136$ $2$ $2$ $1$
136.216.15.fc.1 $136$ $18$ $18$ $15$
152.24.0.bw.1 $152$ $2$ $2$ $0$
152.24.0.by.1 $152$ $2$ $2$ $0$
152.24.0.ca.1 $152$ $2$ $2$ $0$
152.24.0.cc.1 $152$ $2$ $2$ $0$
152.24.0.cy.1 $152$ $2$ $2$ $0$
152.24.0.cz.1 $152$ $2$ $2$ $0$
152.24.0.dc.1 $152$ $2$ $2$ $0$
152.24.0.dd.1 $152$ $2$ $2$ $0$
152.24.1.bw.1 $152$ $2$ $2$ $1$
152.24.1.bx.1 $152$ $2$ $2$ $1$
152.24.1.ca.1 $152$ $2$ $2$ $1$
152.24.1.cb.1 $152$ $2$ $2$ $1$
152.240.18.o.1 $152$ $20$ $20$ $18$
168.24.0.gq.1 $168$ $2$ $2$ $0$
168.24.0.gs.1 $168$ $2$ $2$ $0$
168.24.0.gu.1 $168$ $2$ $2$ $0$
168.24.0.gw.1 $168$ $2$ $2$ $0$
168.24.0.iu.1 $168$ $2$ $2$ $0$
168.24.0.iv.1 $168$ $2$ $2$ $0$
168.24.0.iy.1 $168$ $2$ $2$ $0$
168.24.0.iz.1 $168$ $2$ $2$ $0$
168.24.1.iu.1 $168$ $2$ $2$ $1$
168.24.1.iv.1 $168$ $2$ $2$ $1$
168.24.1.iy.1 $168$ $2$ $2$ $1$
168.24.1.iz.1 $168$ $2$ $2$ $1$
184.24.0.bw.1 $184$ $2$ $2$ $0$
184.24.0.by.1 $184$ $2$ $2$ $0$
184.24.0.ca.1 $184$ $2$ $2$ $0$
184.24.0.cc.1 $184$ $2$ $2$ $0$
184.24.0.cy.1 $184$ $2$ $2$ $0$
184.24.0.cz.1 $184$ $2$ $2$ $0$
184.24.0.dc.1 $184$ $2$ $2$ $0$
184.24.0.dd.1 $184$ $2$ $2$ $0$
184.24.1.bw.1 $184$ $2$ $2$ $1$
184.24.1.bx.1 $184$ $2$ $2$ $1$
184.24.1.ca.1 $184$ $2$ $2$ $1$
184.24.1.cb.1 $184$ $2$ $2$ $1$
184.288.22.g.1 $184$ $24$ $24$ $22$
232.24.0.cc.1 $232$ $2$ $2$ $0$
232.24.0.ce.1 $232$ $2$ $2$ $0$
232.24.0.cg.1 $232$ $2$ $2$ $0$
232.24.0.ci.1 $232$ $2$ $2$ $0$
232.24.0.de.1 $232$ $2$ $2$ $0$
232.24.0.df.1 $232$ $2$ $2$ $0$
232.24.0.di.1 $232$ $2$ $2$ $0$
232.24.0.dj.1 $232$ $2$ $2$ $0$
232.24.1.bw.1 $232$ $2$ $2$ $1$
232.24.1.bx.1 $232$ $2$ $2$ $1$
232.24.1.ca.1 $232$ $2$ $2$ $1$
232.24.1.cb.1 $232$ $2$ $2$ $1$
248.24.0.bw.1 $248$ $2$ $2$ $0$
248.24.0.by.1 $248$ $2$ $2$ $0$
248.24.0.ca.1 $248$ $2$ $2$ $0$
248.24.0.cc.1 $248$ $2$ $2$ $0$
248.24.0.cy.1 $248$ $2$ $2$ $0$
248.24.0.cz.1 $248$ $2$ $2$ $0$
248.24.0.dc.1 $248$ $2$ $2$ $0$
248.24.0.dd.1 $248$ $2$ $2$ $0$
248.24.1.bw.1 $248$ $2$ $2$ $1$
248.24.1.bx.1 $248$ $2$ $2$ $1$
248.24.1.ca.1 $248$ $2$ $2$ $1$
248.24.1.cb.1 $248$ $2$ $2$ $1$
264.24.0.gq.1 $264$ $2$ $2$ $0$
264.24.0.gs.1 $264$ $2$ $2$ $0$
264.24.0.gu.1 $264$ $2$ $2$ $0$
264.24.0.gw.1 $264$ $2$ $2$ $0$
264.24.0.iu.1 $264$ $2$ $2$ $0$
264.24.0.iv.1 $264$ $2$ $2$ $0$
264.24.0.iy.1 $264$ $2$ $2$ $0$
264.24.0.iz.1 $264$ $2$ $2$ $0$
264.24.1.iu.1 $264$ $2$ $2$ $1$
264.24.1.iv.1 $264$ $2$ $2$ $1$
264.24.1.iy.1 $264$ $2$ $2$ $1$
264.24.1.iz.1 $264$ $2$ $2$ $1$
280.24.0.go.1 $280$ $2$ $2$ $0$
280.24.0.gq.1 $280$ $2$ $2$ $0$
280.24.0.gs.1 $280$ $2$ $2$ $0$
280.24.0.gu.1 $280$ $2$ $2$ $0$
280.24.0.is.1 $280$ $2$ $2$ $0$
280.24.0.it.1 $280$ $2$ $2$ $0$
280.24.0.iw.1 $280$ $2$ $2$ $0$
280.24.0.ix.1 $280$ $2$ $2$ $0$
280.24.1.iq.1 $280$ $2$ $2$ $1$
280.24.1.ir.1 $280$ $2$ $2$ $1$
280.24.1.iu.1 $280$ $2$ $2$ $1$
280.24.1.iv.1 $280$ $2$ $2$ $1$
296.24.0.cc.1 $296$ $2$ $2$ $0$
296.24.0.ce.1 $296$ $2$ $2$ $0$
296.24.0.cg.1 $296$ $2$ $2$ $0$
296.24.0.ci.1 $296$ $2$ $2$ $0$
296.24.0.de.1 $296$ $2$ $2$ $0$
296.24.0.df.1 $296$ $2$ $2$ $0$
296.24.0.di.1 $296$ $2$ $2$ $0$
296.24.0.dj.1 $296$ $2$ $2$ $0$
296.24.1.bw.1 $296$ $2$ $2$ $1$
296.24.1.bx.1 $296$ $2$ $2$ $1$
296.24.1.ca.1 $296$ $2$ $2$ $1$
296.24.1.cb.1 $296$ $2$ $2$ $1$
312.24.0.gw.1 $312$ $2$ $2$ $0$
312.24.0.gy.1 $312$ $2$ $2$ $0$
312.24.0.ha.1 $312$ $2$ $2$ $0$
312.24.0.hc.1 $312$ $2$ $2$ $0$
312.24.0.ja.1 $312$ $2$ $2$ $0$
312.24.0.jb.1 $312$ $2$ $2$ $0$
312.24.0.je.1 $312$ $2$ $2$ $0$
312.24.0.jf.1 $312$ $2$ $2$ $0$
312.24.1.iu.1 $312$ $2$ $2$ $1$
312.24.1.iv.1 $312$ $2$ $2$ $1$
312.24.1.iy.1 $312$ $2$ $2$ $1$
312.24.1.iz.1 $312$ $2$ $2$ $1$
328.24.0.cc.1 $328$ $2$ $2$ $0$
328.24.0.ce.1 $328$ $2$ $2$ $0$
328.24.0.cg.1 $328$ $2$ $2$ $0$
328.24.0.ci.1 $328$ $2$ $2$ $0$
328.24.0.de.1 $328$ $2$ $2$ $0$
328.24.0.df.1 $328$ $2$ $2$ $0$
328.24.0.di.1 $328$ $2$ $2$ $0$
328.24.0.dj.1 $328$ $2$ $2$ $0$
328.24.1.bw.1 $328$ $2$ $2$ $1$
328.24.1.bx.1 $328$ $2$ $2$ $1$
328.24.1.ca.1 $328$ $2$ $2$ $1$
328.24.1.cb.1 $328$ $2$ $2$ $1$