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Properties

Label 16.96.1.g.1
Level $16$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $4$

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Invariants

Level: $16$ $\SL_2$-level: $16$ Newform level: $64$
Index: $96$ $\PSL_2$-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 and Zureick-Brown (RZB) label: X477
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.96.1.263

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}5&11\\0&9\end{bmatrix}$, $\begin{bmatrix}7&14\\0&15\end{bmatrix}$, $\begin{bmatrix}15&3\\0&1\end{bmatrix}$, $\begin{bmatrix}15&6\\0&7\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup: $\OD_{32}:C_2^3$
Contains $-I$: yes
Quadratic refinements: 16.192.1-16.g.1.1, 16.192.1-16.g.1.2, 16.192.1-16.g.1.3, 16.192.1-16.g.1.4, 16.192.1-16.g.1.5, 16.192.1-16.g.1.6, 32.192.1-16.g.1.1, 32.192.1-16.g.1.2, 32.192.1-16.g.1.3, 32.192.1-16.g.1.4, 48.192.1-16.g.1.1, 48.192.1-16.g.1.2, 48.192.1-16.g.1.3, 48.192.1-16.g.1.4, 48.192.1-16.g.1.5, 48.192.1-16.g.1.6, 80.192.1-16.g.1.1, 80.192.1-16.g.1.2, 80.192.1-16.g.1.3, 80.192.1-16.g.1.4, 80.192.1-16.g.1.5, 80.192.1-16.g.1.6, 96.192.1-16.g.1.1, 96.192.1-16.g.1.2, 96.192.1-16.g.1.3, 96.192.1-16.g.1.4, 112.192.1-16.g.1.1, 112.192.1-16.g.1.2, 112.192.1-16.g.1.3, 112.192.1-16.g.1.4, 112.192.1-16.g.1.5, 112.192.1-16.g.1.6, 160.192.1-16.g.1.1, 160.192.1-16.g.1.2, 160.192.1-16.g.1.3, 160.192.1-16.g.1.4, 176.192.1-16.g.1.1, 176.192.1-16.g.1.2, 176.192.1-16.g.1.3, 176.192.1-16.g.1.4, 176.192.1-16.g.1.5, 176.192.1-16.g.1.6, 208.192.1-16.g.1.1, 208.192.1-16.g.1.2, 208.192.1-16.g.1.3, 208.192.1-16.g.1.4, 208.192.1-16.g.1.5, 208.192.1-16.g.1.6, 224.192.1-16.g.1.1, 224.192.1-16.g.1.2, 224.192.1-16.g.1.3, 224.192.1-16.g.1.4, 240.192.1-16.g.1.1, 240.192.1-16.g.1.2, 240.192.1-16.g.1.3, 240.192.1-16.g.1.4, 240.192.1-16.g.1.5, 240.192.1-16.g.1.6, 272.192.1-16.g.1.1, 272.192.1-16.g.1.2, 272.192.1-16.g.1.3, 272.192.1-16.g.1.4, 272.192.1-16.g.1.5, 272.192.1-16.g.1.6, 304.192.1-16.g.1.1, 304.192.1-16.g.1.2, 304.192.1-16.g.1.3, 304.192.1-16.g.1.4, 304.192.1-16.g.1.5, 304.192.1-16.g.1.6
Cyclic 16-isogeny field degree: $1$
Cyclic 16-torsion field degree: $8$
Full 16-torsion field degree: $256$

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 $
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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

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.0.m.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.e.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.u.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.v.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.1.d.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.q.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.r.2 $16$ $2$ $2$ $1$ $0$ 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
16.192.5.bj.1 $16$ $2$ $2$ $5$ $0$ $2^{2}$
16.192.5.bk.1 $16$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.f.2 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.h.1 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.i.1 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.j.1 $32$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.fp.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.fq.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.288.17.fk.2 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.jj.2 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
80.192.5.kf.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.kg.1 $80$ $2$ $2$ $5$ $?$ not computed
96.192.5.n.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.r.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.s.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.t.1 $96$ $2$ $2$ $5$ $?$ not computed
112.192.5.fp.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.fq.1 $112$ $2$ $2$ $5$ $?$ not computed
160.192.5.r.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.v.1 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.w.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.bb.1 $160$ $2$ $2$ $5$ $?$ not computed
176.192.5.fp.2 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.fq.1 $176$ $2$ $2$ $5$ $?$ not computed
208.192.5.kf.1 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.kg.1 $208$ $2$ $2$ $5$ $?$ not computed
224.192.5.n.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.r.1 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.s.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.t.1 $224$ $2$ $2$ $5$ $?$ not computed
240.192.5.bod.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.boe.1 $240$ $2$ $2$ $5$ $?$ not computed
272.192.5.kf.1 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.kg.1 $272$ $2$ $2$ $5$ $?$ not computed
304.192.5.fp.1 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.fq.1 $304$ $2$ $2$ $5$ $?$ not computed