Invariants
Level: | $16$ | $\SL_2$-level: | $16$ | Newform level: | $64$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $2^{8}\cdot4^{4}\cdot16^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 16M1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.192.1.126 |
Level structure
$\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&4\\0&9\end{bmatrix}$, $\begin{bmatrix}11&13\\0&7\end{bmatrix}$, $\begin{bmatrix}15&11\\0&9\end{bmatrix}$ |
$\GL_2(\Z/16\Z)$-subgroup: | $\OD_{32}:C_2^2$ |
Contains $-I$: | no $\quad$ (see 16.96.1.g.1 for the level structure with $-I$) |
Cyclic 16-isogeny field degree: | $1$ |
Cyclic 16-torsion field degree: | $8$ |
Full 16-torsion field degree: | $128$ |
Jacobian
Conductor: | $2^{6}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 64.2.a.a |
Models
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ x^{3} - 4x $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Weierstrass model |
---|
$(0:0:1)$, $(0:1:0)$, $(-2:0:1)$, $(2:0:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2}\cdot\frac{1440x^{2}y^{30}-6331507008x^{2}y^{28}z^{2}+20986578616320x^{2}y^{26}z^{4}-6329041585489920x^{2}y^{24}z^{6}+563262454239068160x^{2}y^{22}z^{8}-21730522821489524736x^{2}y^{20}z^{10}+433667781480929034240x^{2}y^{18}z^{12}-4948710315703507353600x^{2}y^{16}z^{14}+34119568108114642206720x^{2}y^{14}z^{16}-144342258184677526339584x^{2}y^{12}z^{18}+363883563600464691855360x^{2}y^{10}z^{20}-537412063354833296424960x^{2}y^{8}z^{22}+625791935148732375367680x^{2}y^{6}z^{24}-207525673731877572182016x^{2}y^{4}z^{26}+26563305132954778337280x^{2}y^{2}z^{28}-1180591550348667125760x^{2}z^{30}-717088xy^{30}z+144995788800xy^{28}z^{3}-170850445963776xy^{26}z^{5}+33012291743907840xy^{24}z^{7}-2252941884355903488xy^{22}z^{9}+72270563099856076800xy^{20}z^{11}-1262417726380543311872xy^{18}z^{13}+13009444261275327528960xy^{16}z^{15}-82471581097447460438016xy^{14}z^{17}+328112718679976580218880xy^{12}z^{19}-827393423544758965370880xy^{10}z^{21}+1230421470255462765035520xy^{8}z^{23}-465618928125134818508800xy^{6}z^{25}+66408280248651129815040xy^{4}z^{27}-3246626974565067128832xy^{2}z^{29}-y^{32}+135590400y^{30}z^{2}-2049829642752y^{28}z^{4}+1085901562920960y^{26}z^{6}-132101324886622208y^{24}z^{8}+6265238441101885440y^{22}z^{10}-145336112178427068416y^{20}z^{12}+1852447788216625397760y^{18}z^{14}-13849060543131040088064y^{16}z^{16}+62895491540280410112000y^{14}z^{18}-174704222098988126437376y^{12}z^{20}+268738072837776495083520y^{10}z^{22}-103471787185155610771456y^{8}z^{24}+14951175651152929751040y^{6}z^{26}-738658147969869545472y^{4}z^{28}+25332747903959040y^{2}z^{30}-281474976710656z^{32}}{y^{2}(x^{2}y^{28}+10432x^{2}y^{26}z^{2}-2946560x^{2}y^{24}z^{4}+176889856x^{2}y^{22}z^{6}+13410590720x^{2}y^{20}z^{8}-1268000489472x^{2}y^{18}z^{10}-6440731017216x^{2}y^{16}z^{12}+1244886640623616x^{2}y^{14}z^{14}+29798883857006592x^{2}y^{12}z^{16}+315183165318627328x^{2}y^{10}z^{18}+1888977344639533056x^{2}y^{8}z^{20}+6881491090931712000x^{2}y^{6}z^{22}+14915905954213003264x^{2}y^{4}z^{24}+13835044861142630400x^{2}y^{2}z^{26}+18446739675663040512x^{2}z^{28}+48xy^{28}z+52800xy^{26}z^{3}-18251776xy^{24}z^{5}+1640411136xy^{22}z^{7}-7301890048xy^{20}z^{9}-4322589999104xy^{18}z^{11}+35669988605952xy^{16}z^{13}+4497855024726016xy^{14}z^{15}+85902292536000512xy^{12}z^{17}+809944614852100096xy^{10}z^{19}+4476580366068482048xy^{8}z^{21}+15132098768054255616xy^{6}z^{23}+31128883922919751680xy^{4}z^{25}+46116861283785506816xy^{2}z^{27}+976y^{28}z^{2}-101376y^{26}z^{4}-33095168y^{24}z^{6}+6436618240y^{22}z^{8}-281169887232y^{20}z^{10}-6842473250816y^{18}z^{12}+310080609189888y^{16}z^{14}+10044258568372224y^{14}z^{16}+120871725133987840y^{12}z^{18}+783619497574531072y^{10}z^{20}+2994897639146782720y^{8}z^{22}+6917564761768984576y^{6}z^{24}+10376357313136033792y^{4}z^{26}+52776558133248y^{2}z^{28}+17592186044416z^{30})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.96.0-8.m.2.3 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
16.96.0-16.e.1.5 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
16.96.0-16.e.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
16.96.0-8.m.2.1 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
16.96.0-16.u.1.5 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
16.96.0-16.u.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
16.96.0-16.v.2.4 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
16.96.0-16.v.2.9 | $16$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
16.96.1-16.d.1.3 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.96.1-16.d.1.5 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.96.1-16.q.1.1 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.96.1-16.q.1.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.96.1-16.r.2.4 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
16.96.1-16.r.2.7 | $16$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
16.384.5-16.bj.1.4 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
16.384.5-16.bk.1.3 | $16$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
32.384.5-32.f.2.3 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
32.384.5-32.h.1.5 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
32.384.5-32.i.1.7 | $32$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
32.384.5-32.j.1.5 | $32$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.384.5-48.fp.1.8 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
48.384.5-48.fq.1.6 | $48$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
48.576.17-48.fk.2.21 | $48$ | $3$ | $3$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
48.768.17-48.jj.2.11 | $48$ | $4$ | $4$ | $17$ | $1$ | $1^{8}\cdot2^{4}$ |
80.384.5-80.kf.1.6 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.384.5-80.kg.1.8 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.384.5-96.n.2.6 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.384.5-96.r.1.12 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.384.5-96.s.1.16 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
96.384.5-96.t.1.11 | $96$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.384.5-112.fp.1.8 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
112.384.5-112.fq.1.6 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
160.384.5-160.r.2.6 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
160.384.5-160.v.1.7 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
160.384.5-160.w.2.15 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
160.384.5-160.bb.1.5 | $160$ | $2$ | $2$ | $5$ | $?$ | not computed |
176.384.5-176.fp.2.8 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
176.384.5-176.fq.1.4 | $176$ | $2$ | $2$ | $5$ | $?$ | not computed |
208.384.5-208.kf.1.6 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
208.384.5-208.kg.1.8 | $208$ | $2$ | $2$ | $5$ | $?$ | not computed |
224.384.5-224.n.2.6 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
224.384.5-224.r.1.8 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
224.384.5-224.s.2.16 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
224.384.5-224.t.1.6 | $224$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.bod.1.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.boe.1.12 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
272.384.5-272.kf.1.8 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
272.384.5-272.kg.1.5 | $272$ | $2$ | $2$ | $5$ | $?$ | not computed |
304.384.5-304.fp.1.8 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |
304.384.5-304.fq.1.6 | $304$ | $2$ | $2$ | $5$ | $?$ | not computed |