Properties

Label 28.168.8.d.1
Level 2828
Index 168168
Genus 88
Analytic rank 88
Cusps 66
Q\Q-cusps 00

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Invariants

Level: 2828 SL2\SL_2-level: 2828 Newform level: 784784
Index: 168168 PSL2\PSL_2-index:168168
Genus: 8=1+1681216403628 = 1 + \frac{ 168 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}
Cusps: 66 (none of which are rational) Cusp widths 28628^{6} Cusp orbits 66
Elliptic points: 1616 of order 22 and 00 of order 33
Analytic rank: 88
Q\Q-gonality: 4γ84 \le \gamma \le 8
Q\overline{\Q}-gonality: 4γ84 \le \gamma \le 8
Rational cusps: 00
Rational CM points: yes (D=\quad(D = 11,43,67,163-11,-43,-67,-163)

Other labels

Cummins and Pauli (CP) label: 28B8
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 28.168.8.2

Level structure

GL2(Z/28Z)\GL_2(\Z/28\Z)-generators: [7241211]\begin{bmatrix}7&24\\12&11\end{bmatrix}, [9102119]\begin{bmatrix}9&10\\21&19\end{bmatrix}, [1913209]\begin{bmatrix}19&13\\20&9\end{bmatrix}
Contains I-I: yes
Quadratic refinements: none in database
Cyclic 28-isogeny field degree: 4848
Cyclic 28-torsion field degree: 576576
Full 28-torsion field degree: 11521152

Jacobian

Conductor: 2307162^{30}\cdot7^{16}
Simple: no
Squarefree: yes
Decomposition: 1621^{6}\cdot2
Newforms: 196.2.a.a, 784.2.a.a, 784.2.a.b, 784.2.a.f, 784.2.a.g, 784.2.a.h, 784.2.a.k

Models

Canonical model in P7\mathbb{P}^{ 7 } defined by 15 equations

0 0 == x2xy2xzxw+xu+ywyu+2zwzuzvwu+wv x^{2} - x y - 2 x z - x w + x u + y w - y u + 2 z w - z u - z v - w u + w v
== xy+2xzxwxtxr3yz+yw+yt2z2zuzv+zr2tvuvv2x y + 2 x z - x w - x t - x r - 3 y z + y w + y t - 2 z^{2} - z u - z v + z r - 2 t v - u v - v^{2}
== xyxw2xry22yz+yw+yu+yv+2zw+2wv2tu2tvurvrx y - x w - 2 x r - y^{2} - 2 y z + y w + y u + y v + 2 z w + 2 w v - 2 t u - 2 t v - u r - v r
== x2+xw+2xu+2xvxry2yz+2yu+yv+zwwuwv2tu+u2++v2x^{2} + x w + 2 x u + 2 x v - x r - y^{2} - y z + 2 y u + y v + z w - w u - w v - 2 t u + u^{2} + \cdots + v^{2}
==\cdots
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Singular plane model

0 0 == 162957x142592048x13y2024800x13z+16575774x12y2+30661734x12yz++192y2z12 162957 x^{14} - 2592048 x^{13} y - 2024800 x^{13} z + 16575774 x^{12} y^{2} + 30661734 x^{12} y z + \cdots + 192 y^{2} z^{12}
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Rational points

This modular curve has 4 rational CM points but no rational cusps or other known rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

X\displaystyle X == x\displaystyle x
Y\displaystyle Y == y\displaystyle y
Z\displaystyle Z == z\displaystyle z

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 28.84.4.a.1 :

X\displaystyle X == x\displaystyle x
Y\displaystyle Y == y\displaystyle -y
Z\displaystyle Z == w\displaystyle w
W\displaystyle W == uv\displaystyle -u-v

Equation of the image curve:

00 == 2X2+12XY+5Y2+12XZ+2YZZ23XW5YWW2 2 X^{2}+12 X Y+5 Y^{2}+12 X Z+2 Y Z-Z^{2}-3 X W-5 Y W-W^{2}
== X2Y+5XY2+2Y33X2ZY2Z3XZ2YZ2+X2W+XYWY2W+5XZW+3YZWZ2WXW2YW2 X^{2} Y+5 X Y^{2}+2 Y^{3}-3 X^{2} Z-Y^{2} Z-3 X Z^{2}-Y Z^{2}+X^{2} W+X Y W-Y^{2} W+5 X Z W+3 Y Z W-Z^{2} W-X W^{2}-Y W^{2}

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
Xns+(14)X_{\mathrm{ns}}^+(14) 1414 44 44 11 11 1521^{5}\cdot2
28.84.4.a.1 2828 22 22 44 44 141^{4}

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
Xns(28)X_{\mathrm{ns}}(28) 2828 22 22 2323 88 17241^{7}\cdot2^{4}
28.336.23.d.1 2828 22 22 2323 1111 17241^{7}\cdot2^{4}
28.336.23.n.1 2828 22 22 2323 1313 17241^{7}\cdot2^{4}
28.336.23.p.1 2828 22 22 2323 1212 17241^{7}\cdot2^{4}
28.504.30.o.1 2828 33 33 3030 1515 112251^{12}\cdot2^{5}
56.336.23.c.1 5656 22 22 2323 1414 17241^{7}\cdot2^{4}
56.336.23.l.1 5656 22 22 2323 1717 17241^{7}\cdot2^{4}
56.336.23.bp.1 5656 22 22 2323 1717 17241^{7}\cdot2^{4}
56.336.23.bv.1 5656 22 22 2323 1414 17241^{7}\cdot2^{4}
Xns+(56)X_{\mathrm{ns}}^+(56) 5656 44 44 4343 4343 1152101^{15}\cdot2^{10}
84.336.23.bz.1 8484 22 22 2323 ?? not computed
84.336.23.cb.1 8484 22 22 2323 ?? not computed
84.336.23.dz.1 8484 22 22 2323 ?? not computed
84.336.23.eb.1 8484 22 22 2323 ?? not computed
140.336.23.v.1 140140 22 22 2323 ?? not computed
140.336.23.x.1 140140 22 22 2323 ?? not computed
140.336.23.bt.1 140140 22 22 2323 ?? not computed
140.336.23.bv.1 140140 22 22 2323 ?? not computed
168.336.23.gr.1 168168 22 22 2323 ?? not computed
168.336.23.gx.1 168168 22 22 2323 ?? not computed
168.336.23.nl.1 168168 22 22 2323 ?? not computed
168.336.23.nr.1 168168 22 22 2323 ?? not computed
280.336.23.cn.1 280280 22 22 2323 ?? not computed
280.336.23.ct.1 280280 22 22 2323 ?? not computed
280.336.23.fh.1 280280 22 22 2323 ?? not computed
280.336.23.fn.1 280280 22 22 2323 ?? not computed
308.336.23.br.1 308308 22 22 2323 ?? not computed
308.336.23.bt.1 308308 22 22 2323 ?? not computed
308.336.23.bv.1 308308 22 22 2323 ?? not computed
308.336.23.bx.1 308308 22 22 2323 ?? not computed