Properties

Label 32.96.0-32.c.1.4
Level 3232
Index 9696
Genus 00
Analytic rank 00
Cusps 1010
Q\Q-cusps 00

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Invariants

Level: 3232 SL2\SL_2-level: 88
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 (none of which are rational) Cusp widths 48824^{8}\cdot8^{2} Cusp orbits 282\cdot8
Elliptic points: 00 of order 22 and 00 of order 33
Q\Q-gonality: 11
Q\overline{\Q}-gonality: 11
Rational cusps: 00
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8N0
Rouse and Zureick-Brown (RZB) label: X239c
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 32.96.0.104

Level structure

GL2(Z/32Z)\GL_2(\Z/32\Z)-generators: [1331249]\begin{bmatrix}13&31\\24&9\end{bmatrix}, [1521615]\begin{bmatrix}15&21\\6&15\end{bmatrix}
Contains I-I: no \quad (see 32.48.0.c.1 for the level structure with I-I)
Cyclic 32-isogeny field degree: 1616
Cyclic 32-torsion field degree: 128128
Full 32-torsion field degree: 40964096

Models

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

Rational points

This modular curve has infinitely many rational points, including 2 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 == 25(xy)48(458752x16+524288x15y+917504x14y24587520x13y3+802816x12y4+8945664x11y5+401408x10y65857280x9y7+125440x8y8+1464320x7y9+25088x6y10139776x5y11+3136x4y12+4480x3y13+224x2y1432xy15+7y16)3(xy)48(4x2+y2)8(256x81024x7y1792x6y2+1792x5y3+1120x4y4448x3y5112x2y6+16xy7+y8)4\displaystyle 2^5\,\frac{(x-y)^{48} (458752 x^{16}+524288 x^{15} y+917504 x^{14} y^2-4587520 x^{13} y^3+802816 x^{12} y^4+8945664 x^{11} y^5+401408 x^{10} y^6-5857280 x^9 y^7+125440 x^8 y^8+1464320 x^7 y^9+25088 x^6 y^{10}-139776 x^5 y^{11}+3136 x^4 y^{12}+4480 x^3 y^{13}+224 x^2 y^{14}-32 x y^{15}+7 y^{16})^3}{(x-y)^{48} (4 x^2+y^2)^8 (256 x^8-1024 x^7 y-1792 x^6 y^2+1792 x^5 y^3+1120 x^4 y^4-448 x^3 y^5-112 x^2 y^6+16 x y^7+y^8)^4}

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
16.48.0-16.c.1.4 1616 22 22 00 00
32.48.0-16.c.1.1 3232 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
96.288.8-96.i.1.11 9696 33 33 88
96.384.7-96.dg.1.16 9696 44 44 77
160.480.16-160.i.1.8 160160 55 55 1616