↑

Properties

Label 4.12.0.f.1
Level $4$
Index $12$
Genus $0$
Analytic rank $0$
Cusps $3$
$\Q$-cusps $3$

Related objects

Downloads

Learn more

Invariants

Level: $4$ $\SL_2$-level: $4$
Index: $12$ $\PSL_2$-index:$12$
Genus: $0 = 1 + \frac{ 12 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$
Cusps: $3$ (all of which are rational) Cusp widths $4^{3}$ Cusp orbits $1^{3}$
Elliptic points: $2$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $3$
Rational CM points: yes $\quad(D =$ $-4,-7$)

Other labels

Cummins and Pauli (CP) label: 4F0
Sutherland and Zywina (SZ) label: 4F0-4a
Rouse and Zureick-Brown (RZB) label: X23
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 4.12.0.11

Level structure

$\GL_2(\Z/4\Z)$-generators: $\begin{bmatrix}0&3\\3&0\end{bmatrix}$, $\begin{bmatrix}1&0\\0&3\end{bmatrix}$
$\GL_2(\Z/4\Z)$-subgroup: $D_4$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 4-isogeny field degree: $2$
Cyclic 4-torsion field degree: $4$
Full 4-torsion field degree: $8$

Models

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

Rational points

This modular curve has infinitely many rational points, including 187 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{x^{12}(x^{2}-12y^{2})^{3}(x^{2}+8xy+20y^{2})^{3}}{y^{4}x^{12}(x+2y)^{4}(x+4y)^{4}}$

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
$X_{\mathrm{sp}}(4)$ $4$ $2$ $2$ $0$
4.24.0.c.1 $4$ $2$ $2$ $0$
8.24.0.n.1 $8$ $2$ $2$ $0$
8.24.0.v.1 $8$ $2$ $2$ $0$
8.24.0.bf.1 $8$ $2$ $2$ $0$
8.24.0.bh.1 $8$ $2$ $2$ $0$
8.24.0.bn.1 $8$ $2$ $2$ $0$
8.24.0.bt.1 $8$ $2$ $2$ $0$
8.24.1.r.1 $8$ $2$ $2$ $1$
8.24.1.x.1 $8$ $2$ $2$ $1$
8.24.1.bb.1 $8$ $2$ $2$ $1$
8.24.1.bd.1 $8$ $2$ $2$ $1$
12.24.0.l.1 $12$ $2$ $2$ $0$
12.24.0.n.1 $12$ $2$ $2$ $0$
12.36.1.bt.1 $12$ $3$ $3$ $1$
12.48.2.d.1 $12$ $4$ $4$ $2$
20.24.0.h.1 $20$ $2$ $2$ $0$
20.24.0.j.1 $20$ $2$ $2$ $0$
20.60.4.v.1 $20$ $5$ $5$ $4$
20.72.3.bp.1 $20$ $6$ $6$ $3$
20.120.7.cr.1 $20$ $10$ $10$ $7$
24.24.0.cq.1 $24$ $2$ $2$ $0$
24.24.0.da.1 $24$ $2$ $2$ $0$
24.24.0.dr.1 $24$ $2$ $2$ $0$
24.24.0.dt.1 $24$ $2$ $2$ $0$
24.24.0.dx.1 $24$ $2$ $2$ $0$
24.24.0.ed.1 $24$ $2$ $2$ $0$
24.24.1.dd.1 $24$ $2$ $2$ $1$
24.24.1.dj.1 $24$ $2$ $2$ $1$
24.24.1.dn.1 $24$ $2$ $2$ $1$
24.24.1.dp.1 $24$ $2$ $2$ $1$
28.24.0.g.1 $28$ $2$ $2$ $0$
28.24.0.i.1 $28$ $2$ $2$ $0$
28.96.6.n.1 $28$ $8$ $8$ $6$
28.252.15.t.1 $28$ $21$ $21$ $15$
28.336.21.cp.1 $28$ $28$ $28$ $21$
36.324.22.ct.1 $36$ $27$ $27$ $22$
40.24.0.co.1 $40$ $2$ $2$ $0$
40.24.0.cy.1 $40$ $2$ $2$ $0$
40.24.0.dp.1 $40$ $2$ $2$ $0$
40.24.0.dr.1 $40$ $2$ $2$ $0$
40.24.0.dv.1 $40$ $2$ $2$ $0$
40.24.0.eb.1 $40$ $2$ $2$ $0$
40.24.1.dd.1 $40$ $2$ $2$ $1$
40.24.1.dj.1 $40$ $2$ $2$ $1$
40.24.1.dn.1 $40$ $2$ $2$ $1$
40.24.1.dp.1 $40$ $2$ $2$ $1$
44.24.0.f.1 $44$ $2$ $2$ $0$
44.24.0.h.1 $44$ $2$ $2$ $0$
44.144.10.d.1 $44$ $12$ $12$ $10$
44.660.45.r.1 $44$ $55$ $55$ $45$
44.660.47.b.1 $44$ $55$ $55$ $47$
44.792.55.cn.1 $44$ $66$ $66$ $55$
52.24.0.i.1 $52$ $2$ $2$ $0$
52.24.0.k.1 $52$ $2$ $2$ $0$
52.168.11.bp.1 $52$ $14$ $14$ $11$
52.936.67.cn.1 $52$ $78$ $78$ $67$
52.1092.78.cr.1 $52$ $91$ $91$ $78$
52.1092.80.b.1 $52$ $91$ $91$ $80$
56.24.0.ci.1 $56$ $2$ $2$ $0$
56.24.0.cs.1 $56$ $2$ $2$ $0$
56.24.0.dj.1 $56$ $2$ $2$ $0$
56.24.0.dl.1 $56$ $2$ $2$ $0$
56.24.0.dp.1 $56$ $2$ $2$ $0$
56.24.0.dv.1 $56$ $2$ $2$ $0$
56.24.1.cf.1 $56$ $2$ $2$ $1$
56.24.1.cl.1 $56$ $2$ $2$ $1$
56.24.1.cp.1 $56$ $2$ $2$ $1$
56.24.1.cr.1 $56$ $2$ $2$ $1$
60.24.0.z.1 $60$ $2$ $2$ $0$
60.24.0.bd.1 $60$ $2$ $2$ $0$
68.24.0.h.1 $68$ $2$ $2$ $0$
68.24.0.j.1 $68$ $2$ $2$ $0$
68.216.15.bp.1 $68$ $18$ $18$ $15$
68.1632.121.cn.1 $68$ $136$ $136$ $121$
68.1836.136.bx.1 $68$ $153$ $153$ $136$
76.24.0.g.1 $76$ $2$ $2$ $0$
76.24.0.i.1 $76$ $2$ $2$ $0$
76.240.18.l.1 $76$ $20$ $20$ $18$
84.24.0.x.1 $84$ $2$ $2$ $0$
84.24.0.bb.1 $84$ $2$ $2$ $0$
88.24.0.ci.1 $88$ $2$ $2$ $0$
88.24.0.cs.1 $88$ $2$ $2$ $0$
88.24.0.dj.1 $88$ $2$ $2$ $0$
88.24.0.dl.1 $88$ $2$ $2$ $0$
88.24.0.dp.1 $88$ $2$ $2$ $0$
88.24.0.dv.1 $88$ $2$ $2$ $0$
88.24.1.cf.1 $88$ $2$ $2$ $1$
88.24.1.cl.1 $88$ $2$ $2$ $1$
88.24.1.cp.1 $88$ $2$ $2$ $1$
88.24.1.cr.1 $88$ $2$ $2$ $1$
92.24.0.f.1 $92$ $2$ $2$ $0$
92.24.0.h.1 $92$ $2$ $2$ $0$
92.288.22.d.1 $92$ $24$ $24$ $22$
104.24.0.co.1 $104$ $2$ $2$ $0$
104.24.0.cy.1 $104$ $2$ $2$ $0$
104.24.0.dp.1 $104$ $2$ $2$ $0$
104.24.0.dr.1 $104$ $2$ $2$ $0$
104.24.0.dv.1 $104$ $2$ $2$ $0$
104.24.0.eb.1 $104$ $2$ $2$ $0$
104.24.1.cf.1 $104$ $2$ $2$ $1$
104.24.1.cl.1 $104$ $2$ $2$ $1$
104.24.1.cp.1 $104$ $2$ $2$ $1$
104.24.1.cr.1 $104$ $2$ $2$ $1$
116.24.0.h.1 $116$ $2$ $2$ $0$
116.24.0.j.1 $116$ $2$ $2$ $0$
120.24.0.hs.1 $120$ $2$ $2$ $0$
120.24.0.im.1 $120$ $2$ $2$ $0$
120.24.0.kb.1 $120$ $2$ $2$ $0$
120.24.0.kd.1 $120$ $2$ $2$ $0$
120.24.0.kh.1 $120$ $2$ $2$ $0$
120.24.0.kn.1 $120$ $2$ $2$ $0$
120.24.1.mn.1 $120$ $2$ $2$ $1$
120.24.1.mt.1 $120$ $2$ $2$ $1$
120.24.1.mx.1 $120$ $2$ $2$ $1$
120.24.1.mz.1 $120$ $2$ $2$ $1$
124.24.0.g.1 $124$ $2$ $2$ $0$
124.24.0.i.1 $124$ $2$ $2$ $0$
132.24.0.x.1 $132$ $2$ $2$ $0$
132.24.0.bb.1 $132$ $2$ $2$ $0$
136.24.0.co.1 $136$ $2$ $2$ $0$
136.24.0.cy.1 $136$ $2$ $2$ $0$
136.24.0.dp.1 $136$ $2$ $2$ $0$
136.24.0.dr.1 $136$ $2$ $2$ $0$
136.24.0.dv.1 $136$ $2$ $2$ $0$
136.24.0.eb.1 $136$ $2$ $2$ $0$
136.24.1.cf.1 $136$ $2$ $2$ $1$
136.24.1.cl.1 $136$ $2$ $2$ $1$
136.24.1.cp.1 $136$ $2$ $2$ $1$
136.24.1.cr.1 $136$ $2$ $2$ $1$
140.24.0.t.1 $140$ $2$ $2$ $0$
140.24.0.x.1 $140$ $2$ $2$ $0$
148.24.0.i.1 $148$ $2$ $2$ $0$
148.24.0.k.1 $148$ $2$ $2$ $0$
152.24.0.ci.1 $152$ $2$ $2$ $0$
152.24.0.cs.1 $152$ $2$ $2$ $0$
152.24.0.dj.1 $152$ $2$ $2$ $0$
152.24.0.dl.1 $152$ $2$ $2$ $0$
152.24.0.dp.1 $152$ $2$ $2$ $0$
152.24.0.dv.1 $152$ $2$ $2$ $0$
152.24.1.cf.1 $152$ $2$ $2$ $1$
152.24.1.cl.1 $152$ $2$ $2$ $1$
152.24.1.cp.1 $152$ $2$ $2$ $1$
152.24.1.cr.1 $152$ $2$ $2$ $1$
156.24.0.z.1 $156$ $2$ $2$ $0$
156.24.0.bd.1 $156$ $2$ $2$ $0$
164.24.0.h.1 $164$ $2$ $2$ $0$
164.24.0.j.1 $164$ $2$ $2$ $0$
168.24.0.hm.1 $168$ $2$ $2$ $0$
168.24.0.ig.1 $168$ $2$ $2$ $0$
168.24.0.jv.1 $168$ $2$ $2$ $0$
168.24.0.jx.1 $168$ $2$ $2$ $0$
168.24.0.kb.1 $168$ $2$ $2$ $0$
168.24.0.kh.1 $168$ $2$ $2$ $0$
168.24.1.jt.1 $168$ $2$ $2$ $1$
168.24.1.jz.1 $168$ $2$ $2$ $1$
168.24.1.kd.1 $168$ $2$ $2$ $1$
168.24.1.kf.1 $168$ $2$ $2$ $1$
172.24.0.g.1 $172$ $2$ $2$ $0$
172.24.0.i.1 $172$ $2$ $2$ $0$
184.24.0.ci.1 $184$ $2$ $2$ $0$
184.24.0.cs.1 $184$ $2$ $2$ $0$
184.24.0.dj.1 $184$ $2$ $2$ $0$
184.24.0.dl.1 $184$ $2$ $2$ $0$
184.24.0.dp.1 $184$ $2$ $2$ $0$
184.24.0.dv.1 $184$ $2$ $2$ $0$
184.24.1.cf.1 $184$ $2$ $2$ $1$
184.24.1.cl.1 $184$ $2$ $2$ $1$
184.24.1.cp.1 $184$ $2$ $2$ $1$
184.24.1.cr.1 $184$ $2$ $2$ $1$
188.24.0.f.1 $188$ $2$ $2$ $0$
188.24.0.h.1 $188$ $2$ $2$ $0$
204.24.0.z.1 $204$ $2$ $2$ $0$
204.24.0.bd.1 $204$ $2$ $2$ $0$
212.24.0.h.1 $212$ $2$ $2$ $0$
212.24.0.j.1 $212$ $2$ $2$ $0$
220.24.0.t.1 $220$ $2$ $2$ $0$
220.24.0.x.1 $220$ $2$ $2$ $0$
228.24.0.x.1 $228$ $2$ $2$ $0$
228.24.0.bb.1 $228$ $2$ $2$ $0$
232.24.0.co.1 $232$ $2$ $2$ $0$
232.24.0.cy.1 $232$ $2$ $2$ $0$
232.24.0.dp.1 $232$ $2$ $2$ $0$
232.24.0.dr.1 $232$ $2$ $2$ $0$
232.24.0.dv.1 $232$ $2$ $2$ $0$
232.24.0.eb.1 $232$ $2$ $2$ $0$
232.24.1.cf.1 $232$ $2$ $2$ $1$
232.24.1.cl.1 $232$ $2$ $2$ $1$
232.24.1.cp.1 $232$ $2$ $2$ $1$
232.24.1.cr.1 $232$ $2$ $2$ $1$
236.24.0.f.1 $236$ $2$ $2$ $0$
236.24.0.h.1 $236$ $2$ $2$ $0$
244.24.0.i.1 $244$ $2$ $2$ $0$
244.24.0.k.1 $244$ $2$ $2$ $0$
248.24.0.ci.1 $248$ $2$ $2$ $0$
248.24.0.cs.1 $248$ $2$ $2$ $0$
248.24.0.dj.1 $248$ $2$ $2$ $0$
248.24.0.dl.1 $248$ $2$ $2$ $0$
248.24.0.dp.1 $248$ $2$ $2$ $0$
248.24.0.dv.1 $248$ $2$ $2$ $0$
248.24.1.cf.1 $248$ $2$ $2$ $1$
248.24.1.cl.1 $248$ $2$ $2$ $1$
248.24.1.cp.1 $248$ $2$ $2$ $1$
248.24.1.cr.1 $248$ $2$ $2$ $1$
260.24.0.z.1 $260$ $2$ $2$ $0$
260.24.0.bd.1 $260$ $2$ $2$ $0$
264.24.0.hm.1 $264$ $2$ $2$ $0$
264.24.0.ig.1 $264$ $2$ $2$ $0$
264.24.0.jv.1 $264$ $2$ $2$ $0$
264.24.0.jx.1 $264$ $2$ $2$ $0$
264.24.0.kb.1 $264$ $2$ $2$ $0$
264.24.0.kh.1 $264$ $2$ $2$ $0$
264.24.1.jt.1 $264$ $2$ $2$ $1$
264.24.1.jz.1 $264$ $2$ $2$ $1$
264.24.1.kd.1 $264$ $2$ $2$ $1$
264.24.1.kf.1 $264$ $2$ $2$ $1$
268.24.0.g.1 $268$ $2$ $2$ $0$
268.24.0.i.1 $268$ $2$ $2$ $0$
276.24.0.x.1 $276$ $2$ $2$ $0$
276.24.0.bb.1 $276$ $2$ $2$ $0$
280.24.0.hk.1 $280$ $2$ $2$ $0$
280.24.0.ie.1 $280$ $2$ $2$ $0$
280.24.0.jt.1 $280$ $2$ $2$ $0$
280.24.0.jv.1 $280$ $2$ $2$ $0$
280.24.0.jz.1 $280$ $2$ $2$ $0$
280.24.0.kf.1 $280$ $2$ $2$ $0$
280.24.1.jp.1 $280$ $2$ $2$ $1$
280.24.1.jv.1 $280$ $2$ $2$ $1$
280.24.1.jz.1 $280$ $2$ $2$ $1$
280.24.1.kb.1 $280$ $2$ $2$ $1$
284.24.0.f.1 $284$ $2$ $2$ $0$
284.24.0.h.1 $284$ $2$ $2$ $0$
292.24.0.i.1 $292$ $2$ $2$ $0$
292.24.0.k.1 $292$ $2$ $2$ $0$
296.24.0.co.1 $296$ $2$ $2$ $0$
296.24.0.cy.1 $296$ $2$ $2$ $0$
296.24.0.dp.1 $296$ $2$ $2$ $0$
296.24.0.dr.1 $296$ $2$ $2$ $0$
296.24.0.dv.1 $296$ $2$ $2$ $0$
296.24.0.eb.1 $296$ $2$ $2$ $0$
296.24.1.cf.1 $296$ $2$ $2$ $1$
296.24.1.cl.1 $296$ $2$ $2$ $1$
296.24.1.cp.1 $296$ $2$ $2$ $1$
296.24.1.cr.1 $296$ $2$ $2$ $1$
308.24.0.r.1 $308$ $2$ $2$ $0$
308.24.0.v.1 $308$ $2$ $2$ $0$
312.24.0.hs.1 $312$ $2$ $2$ $0$
312.24.0.im.1 $312$ $2$ $2$ $0$
312.24.0.kb.1 $312$ $2$ $2$ $0$
312.24.0.kd.1 $312$ $2$ $2$ $0$
312.24.0.kh.1 $312$ $2$ $2$ $0$
312.24.0.kn.1 $312$ $2$ $2$ $0$
312.24.1.jt.1 $312$ $2$ $2$ $1$
312.24.1.jz.1 $312$ $2$ $2$ $1$
312.24.1.kd.1 $312$ $2$ $2$ $1$
312.24.1.kf.1 $312$ $2$ $2$ $1$
316.24.0.g.1 $316$ $2$ $2$ $0$
316.24.0.i.1 $316$ $2$ $2$ $0$
328.24.0.co.1 $328$ $2$ $2$ $0$
328.24.0.cy.1 $328$ $2$ $2$ $0$
328.24.0.dp.1 $328$ $2$ $2$ $0$
328.24.0.dr.1 $328$ $2$ $2$ $0$
328.24.0.dv.1 $328$ $2$ $2$ $0$
328.24.0.eb.1 $328$ $2$ $2$ $0$
328.24.1.cf.1 $328$ $2$ $2$ $1$
328.24.1.cl.1 $328$ $2$ $2$ $1$
328.24.1.cp.1 $328$ $2$ $2$ $1$
328.24.1.cr.1 $328$ $2$ $2$ $1$
332.24.0.f.1 $332$ $2$ $2$ $0$
332.24.0.h.1 $332$ $2$ $2$ $0$