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Properties

Label 16.96.1.l.2
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: X476
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.96.1.193

Level structure

$\GL_2(\Z/16\Z)$-generators: $\begin{bmatrix}3&3\\8&5\end{bmatrix}$, $\begin{bmatrix}7&5\\0&5\end{bmatrix}$, $\begin{bmatrix}9&15\\0&3\end{bmatrix}$, $\begin{bmatrix}15&6\\0&7\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup: $(C_2^2\times D_8):C_4$
Contains $-I$: yes
Quadratic refinements: 16.192.1-16.l.2.1, 16.192.1-16.l.2.2, 16.192.1-16.l.2.3, 16.192.1-16.l.2.4, 16.192.1-16.l.2.5, 16.192.1-16.l.2.6, 32.192.1-16.l.2.1, 32.192.1-16.l.2.2, 32.192.1-16.l.2.3, 32.192.1-16.l.2.4, 48.192.1-16.l.2.1, 48.192.1-16.l.2.2, 48.192.1-16.l.2.3, 48.192.1-16.l.2.4, 48.192.1-16.l.2.5, 48.192.1-16.l.2.6, 80.192.1-16.l.2.1, 80.192.1-16.l.2.2, 80.192.1-16.l.2.3, 80.192.1-16.l.2.4, 80.192.1-16.l.2.5, 80.192.1-16.l.2.6, 96.192.1-16.l.2.1, 96.192.1-16.l.2.2, 96.192.1-16.l.2.3, 96.192.1-16.l.2.4, 112.192.1-16.l.2.1, 112.192.1-16.l.2.2, 112.192.1-16.l.2.3, 112.192.1-16.l.2.4, 112.192.1-16.l.2.5, 112.192.1-16.l.2.6, 160.192.1-16.l.2.1, 160.192.1-16.l.2.2, 160.192.1-16.l.2.3, 160.192.1-16.l.2.4, 176.192.1-16.l.2.1, 176.192.1-16.l.2.2, 176.192.1-16.l.2.3, 176.192.1-16.l.2.4, 176.192.1-16.l.2.5, 176.192.1-16.l.2.6, 208.192.1-16.l.2.1, 208.192.1-16.l.2.2, 208.192.1-16.l.2.3, 208.192.1-16.l.2.4, 208.192.1-16.l.2.5, 208.192.1-16.l.2.6, 224.192.1-16.l.2.1, 224.192.1-16.l.2.2, 224.192.1-16.l.2.3, 224.192.1-16.l.2.4, 240.192.1-16.l.2.1, 240.192.1-16.l.2.2, 240.192.1-16.l.2.3, 240.192.1-16.l.2.4, 240.192.1-16.l.2.5, 240.192.1-16.l.2.6, 272.192.1-16.l.2.1, 272.192.1-16.l.2.2, 272.192.1-16.l.2.3, 272.192.1-16.l.2.4, 272.192.1-16.l.2.5, 272.192.1-16.l.2.6, 304.192.1-16.l.2.1, 304.192.1-16.l.2.2, 304.192.1-16.l.2.3, 304.192.1-16.l.2.4, 304.192.1-16.l.2.5, 304.192.1-16.l.2.6
Cyclic 16-isogeny field degree: $2$
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
$(-2:0:1)$, $(2:0:1)$, $(0:1:0)$, $(0:0:1)$

Maps to other modular curves

$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^2}\cdot\frac{2948928x^{2}y^{28}z^{2}-5496804096000x^{2}y^{24}z^{6}+2737106818891776x^{2}y^{20}z^{10}-1084996567380787200x^{2}y^{16}z^{14}+21727457013865119744x^{2}y^{12}z^{18}-125059246854493962240x^{2}y^{8}z^{22}+814738116182016x^{2}y^{4}z^{26}+1180591550348667125760x^{2}z^{30}-2912xy^{30}z+117817422336xy^{26}z^{5}-1150381403799552xy^{22}z^{9}+36054650382712832xy^{18}z^{13}-8549468382947180544xy^{14}z^{17}+180640200984562237440xy^{10}z^{21}-1314330718661456691200xy^{6}z^{25}+3246626974565067128832xy^{2}z^{29}+y^{32}-1153187328y^{28}z^{4}+133278010032128y^{24}z^{8}+32350121223520256y^{20}z^{12}-2747184021195718656y^{16}z^{16}+48289154466773467136y^{12}z^{20}-318193114881116340224y^{8}z^{24}+737866499597870825472y^{4}z^{28}+281474976710656z^{32}}{zy^{4}(68x^{2}y^{24}z-961024x^{2}y^{20}z^{5}+1264680960x^{2}y^{16}z^{9}+14843748810752x^{2}y^{12}z^{13}+4103377327751168x^{2}y^{8}z^{17}+132856158942658560x^{2}y^{4}z^{21}+288230101273804800x^{2}z^{25}+xy^{26}-1024xy^{22}z^{4}-249815040xy^{18}z^{8}+1408672727040xy^{14}z^{12}+968669928620032xy^{10}z^{16}+74872352394969088xy^{6}z^{20}+648518415060828160xy^{2}z^{24}+1472y^{24}z^{3}-21671936y^{20}z^{7}+72803614720y^{16}z^{11}+106651168276480y^{12}z^{15}+12384896827785216y^{8}z^{19}+144115291155070976y^{4}z^{23}+1099511627776z^{27})}$

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.n.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.j.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.y.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.0.z.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
16.48.1.g.1 $16$ $2$ $2$ $1$ $0$ dimension zero
16.48.1.q.2 $16$ $2$ $2$ $1$ $0$ 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.bu.1 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
16.192.5.bv.1 $16$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
32.192.5.o.2 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.s.1 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.w.1 $32$ $2$ $2$ $5$ $0$ $2^{2}$
32.192.5.bd.2 $32$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.gp.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.gq.2 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
48.288.17.lw.1 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.oq.1 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
80.192.5.ln.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.lo.2 $80$ $2$ $2$ $5$ $?$ not computed
96.192.5.bg.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.bo.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.bs.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.cp.2 $96$ $2$ $2$ $5$ $?$ not computed
112.192.5.gp.1 $112$ $2$ $2$ $5$ $?$ not computed
112.192.5.gq.2 $112$ $2$ $2$ $5$ $?$ not computed
160.192.5.cg.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.cs.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.cw.2 $160$ $2$ $2$ $5$ $?$ not computed
160.192.5.ef.2 $160$ $2$ $2$ $5$ $?$ not computed
176.192.5.gp.1 $176$ $2$ $2$ $5$ $?$ not computed
176.192.5.gq.2 $176$ $2$ $2$ $5$ $?$ not computed
208.192.5.ln.2 $208$ $2$ $2$ $5$ $?$ not computed
208.192.5.lo.2 $208$ $2$ $2$ $5$ $?$ not computed
224.192.5.bg.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.bo.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.bs.2 $224$ $2$ $2$ $5$ $?$ not computed
224.192.5.cp.2 $224$ $2$ $2$ $5$ $?$ not computed
240.192.5.brd.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bre.2 $240$ $2$ $2$ $5$ $?$ not computed
272.192.5.ln.1 $272$ $2$ $2$ $5$ $?$ not computed
272.192.5.lo.2 $272$ $2$ $2$ $5$ $?$ not computed
304.192.5.gp.2 $304$ $2$ $2$ $5$ $?$ not computed
304.192.5.gq.2 $304$ $2$ $2$ $5$ $?$ not computed