Properties

Label 16.96.0-16.e.1.6
Level 1616
Index 9696
Genus 00
Analytic rank 00
Cusps 1010
Q\Q-cusps 22

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Invariants

Level: 1616 SL2\SL_2-level: 1616
Index: 9696 PSL2\PSL_2-index:4848
Genus: 0=1+481204031020 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}
Cusps: 1010 (of which 22 are rational) Cusp widths 281622^{8}\cdot16^{2} Cusp orbits 12241^{2}\cdot2^{4}
Elliptic points: 00 of order 22 and 00 of order 33
Q\Q-gonality: 11
Q\overline{\Q}-gonality: 11
Rational cusps: 22
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16G0
Rouse and Zureick-Brown (RZB) label: X217c
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 16.96.0.137

Level structure

GL2(Z/16Z)\GL_2(\Z/16\Z)-generators: [3081]\begin{bmatrix}3&0\\8&1\end{bmatrix}, [54813]\begin{bmatrix}5&4\\8&13\end{bmatrix}, [111385]\begin{bmatrix}11&13\\8&5\end{bmatrix}
GL2(Z/16Z)\GL_2(\Z/16\Z)-subgroup: C42.SD16C_4^2.\SD_{16}
Contains I-I: no \quad (see 16.48.0.e.1 for the level structure with I-I)
Cyclic 16-isogeny field degree: 22
Cyclic 16-torsion field degree: 88
Full 16-torsion field degree: 256256

Models

This modular curve is isomorphic to P1\mathbb{P}^1.

Rational points

This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.

Maps to other modular curves

jj-invariant map of degree 48 to the modular curve X(1)X(1) :

j\displaystyle j == 12(2xy)48(24832x8+33792x7y164096x6y2119040x5y3+166496x4y46720x3y522288x2y6+4752xy7191y8)3(48896x8304128x7y+356608x6y2+26880x5y3166496x4y4+29760x3y5+10256x2y6528xy797y8)3(2xy)50(2x+y)2(4x212xy+y2)2(4x2+4xy7y2)2(12x24xy+3y2)16(28x24xyy2)2\displaystyle \frac{1}{2}\cdot\frac{(2 x-y)^{48} (24832 x^{8}+33792 x^{7} y-164096 x^{6} y^{2}-119040 x^{5} y^{3}+166496 x^{4} y^{4}-6720 x^{3} y^{5}-22288 x^{2} y^{6}+4752 x y^{7}-191 y^{8})^{3} (48896 x^{8}-304128 x^{7} y+356608 x^{6} y^{2}+26880 x^{5} y^{3}-166496 x^{4} y^{4}+29760 x^{3} y^{5}+10256 x^{2} y^{6}-528 x y^{7}-97 y^{8})^{3}}{(2 x-y)^{50} (2 x+y)^{2} (4 x^{2}-12 x y+y^{2})^{2} (4 x^{2}+4 x y-7 y^{2})^{2} (12 x^{2}-4 x y+3 y^{2})^{16} (28 x^{2}-4 x y-y^{2})^{2}}

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
8.48.0-8.k.1.6 88 22 22 00 00
16.48.0-16.e.1.3 1616 22 22 00 00
16.48.0-16.e.1.6 1616 22 22 00 00
16.48.0-16.e.2.5 1616 22 22 00 00
16.48.0-16.e.2.11 1616 22 22 00 00
16.48.0-8.k.1.4 1616 22 22 00 00

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
16.192.1-16.g.1.2 1616 22 22 11
16.192.1-16.g.2.2 1616 22 22 11
16.192.1-16.i.1.2 1616 22 22 11
16.192.1-16.i.2.6 1616 22 22 11
48.192.1-48.u.1.2 4848 22 22 11
48.192.1-48.u.2.4 4848 22 22 11
48.192.1-48.y.1.2 4848 22 22 11
48.192.1-48.y.2.4 4848 22 22 11
48.288.8-48.q.1.25 4848 33 33 88
48.384.7-48.cl.1.12 4848 44 44 77
80.192.1-80.u.1.2 8080 22 22 11
80.192.1-80.u.2.4 8080 22 22 11
80.192.1-80.y.1.2 8080 22 22 11
80.192.1-80.y.2.4 8080 22 22 11
80.480.16-80.k.1.10 8080 55 55 1616
112.192.1-112.u.1.2 112112 22 22 11
112.192.1-112.u.2.4 112112 22 22 11
112.192.1-112.y.1.2 112112 22 22 11
112.192.1-112.y.2.4 112112 22 22 11
176.192.1-176.u.1.2 176176 22 22 11
176.192.1-176.u.2.4 176176 22 22 11
176.192.1-176.y.1.2 176176 22 22 11
176.192.1-176.y.2.4 176176 22 22 11
208.192.1-208.u.1.2 208208 22 22 11
208.192.1-208.u.2.4 208208 22 22 11
208.192.1-208.y.1.2 208208 22 22 11
208.192.1-208.y.2.4 208208 22 22 11
240.192.1-240.cm.1.4 240240 22 22 11
240.192.1-240.cm.2.8 240240 22 22 11
240.192.1-240.cu.1.4 240240 22 22 11
240.192.1-240.cu.2.8 240240 22 22 11
272.192.1-272.u.1.2 272272 22 22 11
272.192.1-272.u.2.8 272272 22 22 11
272.192.1-272.y.1.2 272272 22 22 11
272.192.1-272.y.2.2 272272 22 22 11
304.192.1-304.u.1.2 304304 22 22 11
304.192.1-304.u.2.4 304304 22 22 11
304.192.1-304.y.1.2 304304 22 22 11
304.192.1-304.y.2.4 304304 22 22 11