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Properties

Label 24.36.1.go.1
Level $24$
Index $36$
Genus $1$
Analytic rank $0$
Cusps $2$
$\Q$-cusps $2$

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Invariants

Level: $24$ $\SL_2$-level: $24$ Newform level: $32$
Index: $36$ $\PSL_2$-index:$36$
Genus: $1 = 1 + \frac{ 36 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$
Cusps: $2$ (all of which are rational) Cusp widths $12\cdot24$ Cusp orbits $1^{2}$
Elliptic points: $8$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label: 24D1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.36.1.80

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}3&5\\8&9\end{bmatrix}$, $\begin{bmatrix}5&23\\20&7\end{bmatrix}$, $\begin{bmatrix}7&0\\0&19\end{bmatrix}$, $\begin{bmatrix}9&19\\14&15\end{bmatrix}$, $\begin{bmatrix}15&8\\8&3\end{bmatrix}$, $\begin{bmatrix}21&10\\4&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 24-isogeny field degree: $16$
Cyclic 24-torsion field degree: $128$
Full 24-torsion field degree: $2048$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x $
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Rational points

This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:1:0)$, $(0:0:1)$

Maps to other modular curves

$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^6\,\frac{960x^{2}y^{8}z^{2}+1056x^{2}y^{4}z^{6}+28x^{2}z^{10}+384xy^{10}z+2048xy^{6}z^{5}+296xy^{2}z^{9}+64y^{12}+1232y^{8}z^{4}+268y^{4}z^{8}-z^{12}}{z^{8}(xz+y^{2})^{2}}$

Modular covers

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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 Kernel decomposition
12.18.0.l.1 $12$ $2$ $2$ $0$ $0$ 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
24.72.1.dt.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.1.du.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.1.dx.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.1.dy.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.1.ej.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.1.ek.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.1.en.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.1.eo.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.72.3.bco.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bcp.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bcq.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bcr.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bcw.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.bcx.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bcy.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bcz.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bde.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.bdf.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bdg.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bdh.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bdm.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.bdn.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bdo.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bdp.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bdu.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bdv.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bdw.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.bdx.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bec.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bed.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bee.1 $24$ $2$ $2$ $3$ $2$ $1^{2}$
24.72.3.bef.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.bek.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bel.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bem.1 $24$ $2$ $2$ $3$ $2$ $1^{2}$
24.72.3.ben.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.bes.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.bet.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.72.3.beu.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.3.bev.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.72.5.j.1 $24$ $2$ $2$ $5$ $0$ $1^{4}$
24.72.5.x.1 $24$ $2$ $2$ $5$ $2$ $1^{4}$
24.72.5.cb.1 $24$ $2$ $2$ $5$ $0$ $1^{4}$
24.72.5.ch.1 $24$ $2$ $2$ $5$ $2$ $1^{4}$
24.72.5.eb.1 $24$ $2$ $2$ $5$ $0$ $1^{4}$
24.72.5.ec.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$
24.72.5.ef.1 $24$ $2$ $2$ $5$ $0$ $1^{4}$
24.72.5.eg.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$
24.72.5.in.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$
24.72.5.io.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$
24.72.5.ir.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$
24.72.5.is.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$
24.72.5.jd.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$
24.72.5.je.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$
24.72.5.jh.1 $24$ $2$ $2$ $5$ $2$ $1^{4}$
24.72.5.ji.1 $24$ $2$ $2$ $5$ $1$ $1^{4}$
72.108.7.gh.1 $72$ $3$ $3$ $7$ $?$ not computed
72.324.19.go.1 $72$ $9$ $9$ $19$ $?$ not computed
120.72.1.us.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.1.uu.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.1.uw.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.1.uy.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.1.vy.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.1.wa.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.1.wc.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.1.we.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.3.hfw.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hfx.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hfy.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hfz.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hge.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hgf.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hgg.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hgh.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hgm.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hgn.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hgo.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hgp.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hgu.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hgv.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hgw.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hgx.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hhc.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hhd.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hhe.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hhf.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hhk.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hhl.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hhm.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hhn.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hhs.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hht.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hhu.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hhv.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hia.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hib.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hic.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.hid.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.5.byi.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.byk.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.bym.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.byo.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.byy.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.bza.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.bzc.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.bze.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.cae.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.cag.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.cai.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.cak.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.cau.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.caw.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.cay.1 $120$ $2$ $2$ $5$ $?$ not computed
120.72.5.cba.1 $120$ $2$ $2$ $5$ $?$ not computed
120.180.13.bwk.1 $120$ $5$ $5$ $13$ $?$ not computed
120.216.13.chc.1 $120$ $6$ $6$ $13$ $?$ not computed
168.72.1.iu.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.72.1.iw.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.72.1.iy.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.72.1.ja.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.72.1.ka.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.72.1.kc.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.72.1.ke.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.72.1.kg.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.72.3.gis.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.git.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.giu.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.giv.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gja.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjb.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjc.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjd.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gji.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjj.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjk.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjl.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjq.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjr.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjs.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjt.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjy.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gjz.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gka.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gkb.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gkg.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gkh.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gki.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gkj.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gko.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gkp.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gkq.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gkr.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gkw.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gkx.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gky.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.3.gkz.1 $168$ $2$ $2$ $3$ $?$ not computed
168.72.5.bjc.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.bje.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.bjg.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.bji.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.bjs.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.bju.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.bjw.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.bjy.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.bky.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.bla.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.blc.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.ble.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.blo.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.blq.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.bls.1 $168$ $2$ $2$ $5$ $?$ not computed
168.72.5.blu.1 $168$ $2$ $2$ $5$ $?$ not computed
168.288.23.bq.1 $168$ $8$ $8$ $23$ $?$ not computed
264.72.1.iq.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.72.1.is.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.72.1.iu.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.72.1.iw.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.72.1.jw.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.72.1.jy.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.72.1.ka.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.72.1.kc.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.72.3.gis.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.git.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.giu.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.giv.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gja.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjb.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjc.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjd.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gji.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjj.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjk.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjl.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjq.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjr.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjs.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjt.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjy.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gjz.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gka.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gkb.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gkg.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gkh.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gki.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gkj.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gko.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gkp.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gkq.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gkr.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gkw.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gkx.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gky.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.3.gkz.1 $264$ $2$ $2$ $3$ $?$ not computed
264.72.5.bjc.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.bje.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.bjg.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.bji.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.bjs.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.bju.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.bjw.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.bjy.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.bky.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.bla.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.blc.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.ble.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.blo.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.blq.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.bls.1 $264$ $2$ $2$ $5$ $?$ not computed
264.72.5.blu.1 $264$ $2$ $2$ $5$ $?$ not computed
312.72.1.iu.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.72.1.iw.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.72.1.iy.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.72.1.ja.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.72.1.ka.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.72.1.kc.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.72.1.ke.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.72.1.kg.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.72.3.gis.1 $312$ $2$ $2$ $3$ $?$ not computed
312.72.3.git.1 $312$ $2$ $2$ $3$ $?$ not computed
312.72.3.giu.1 $312$ $2$ $2$ $3$ $?$ not computed
312.72.3.giv.1 $312$ $2$ $2$ $3$ $?$ not computed
312.72.3.gja.1 $312$ $2$ $2$ $3$ $?$ not computed
312.72.3.gjb.1 $312$ $2$ $2$ $3$ $?$ not computed
312.72.3.gjc.1 $312$ $2$ $2$ $3$ $?$ not computed
312.72.3.gjd.1 $312$ $2$ $2$ $3$ $?$ not computed
312.72.3.gji.1 $312$ $2$ $2$ $3$ $?$ not computed
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