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Properties

Label 28.96.4.a.1
Level $28$
Index $96$
Genus $4$
Analytic rank $0$
Cusps $10$
$\Q$-cusps $4$

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Invariants

Level: $28$ $\SL_2$-level: $28$ Newform level: $28$
Index: $96$ $\PSL_2$-index:$96$
Genus: $4 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps: $10$ (of which $4$ are rational) Cusp widths $2^{4}\cdot4\cdot14^{4}\cdot28$ Cusp orbits $1^{4}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3 \le \gamma \le 4$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 28D4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 28.96.4.2

Level structure

$\GL_2(\Z/28\Z)$-generators: $\begin{bmatrix}3&18\\20&13\end{bmatrix}$, $\begin{bmatrix}11&26\\0&5\end{bmatrix}$, $\begin{bmatrix}21&20\\24&3\end{bmatrix}$, $\begin{bmatrix}23&16\\24&21\end{bmatrix}$, $\begin{bmatrix}27&0\\6&25\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 28.192.4-28.a.1.1, 28.192.4-28.a.1.2, 28.192.4-28.a.1.3, 28.192.4-28.a.1.4, 28.192.4-28.a.1.5, 28.192.4-28.a.1.6, 28.192.4-28.a.1.7, 28.192.4-28.a.1.8, 28.192.4-28.a.1.9, 28.192.4-28.a.1.10, 28.192.4-28.a.1.11, 28.192.4-28.a.1.12, 28.192.4-28.a.1.13, 28.192.4-28.a.1.14, 28.192.4-28.a.1.15, 28.192.4-28.a.1.16, 56.192.4-28.a.1.1, 56.192.4-28.a.1.2, 56.192.4-28.a.1.3, 56.192.4-28.a.1.4, 56.192.4-28.a.1.5, 56.192.4-28.a.1.6, 56.192.4-28.a.1.7, 56.192.4-28.a.1.8, 56.192.4-28.a.1.9, 56.192.4-28.a.1.10, 56.192.4-28.a.1.11, 56.192.4-28.a.1.12, 56.192.4-28.a.1.13, 56.192.4-28.a.1.14, 56.192.4-28.a.1.15, 56.192.4-28.a.1.16, 84.192.4-28.a.1.1, 84.192.4-28.a.1.2, 84.192.4-28.a.1.3, 84.192.4-28.a.1.4, 84.192.4-28.a.1.5, 84.192.4-28.a.1.6, 84.192.4-28.a.1.7, 84.192.4-28.a.1.8, 84.192.4-28.a.1.9, 84.192.4-28.a.1.10, 84.192.4-28.a.1.11, 84.192.4-28.a.1.12, 84.192.4-28.a.1.13, 84.192.4-28.a.1.14, 84.192.4-28.a.1.15, 84.192.4-28.a.1.16, 140.192.4-28.a.1.1, 140.192.4-28.a.1.2, 140.192.4-28.a.1.3, 140.192.4-28.a.1.4, 140.192.4-28.a.1.5, 140.192.4-28.a.1.6, 140.192.4-28.a.1.7, 140.192.4-28.a.1.8, 140.192.4-28.a.1.9, 140.192.4-28.a.1.10, 140.192.4-28.a.1.11, 140.192.4-28.a.1.12, 140.192.4-28.a.1.13, 140.192.4-28.a.1.14, 140.192.4-28.a.1.15, 140.192.4-28.a.1.16, 168.192.4-28.a.1.1, 168.192.4-28.a.1.2, 168.192.4-28.a.1.3, 168.192.4-28.a.1.4, 168.192.4-28.a.1.5, 168.192.4-28.a.1.6, 168.192.4-28.a.1.7, 168.192.4-28.a.1.8, 168.192.4-28.a.1.9, 168.192.4-28.a.1.10, 168.192.4-28.a.1.11, 168.192.4-28.a.1.12, 168.192.4-28.a.1.13, 168.192.4-28.a.1.14, 168.192.4-28.a.1.15, 168.192.4-28.a.1.16, 280.192.4-28.a.1.1, 280.192.4-28.a.1.2, 280.192.4-28.a.1.3, 280.192.4-28.a.1.4, 280.192.4-28.a.1.5, 280.192.4-28.a.1.6, 280.192.4-28.a.1.7, 280.192.4-28.a.1.8, 280.192.4-28.a.1.9, 280.192.4-28.a.1.10, 280.192.4-28.a.1.11, 280.192.4-28.a.1.12, 280.192.4-28.a.1.13, 280.192.4-28.a.1.14, 280.192.4-28.a.1.15, 280.192.4-28.a.1.16, 308.192.4-28.a.1.1, 308.192.4-28.a.1.2, 308.192.4-28.a.1.3, 308.192.4-28.a.1.4, 308.192.4-28.a.1.5, 308.192.4-28.a.1.6, 308.192.4-28.a.1.7, 308.192.4-28.a.1.8, 308.192.4-28.a.1.9, 308.192.4-28.a.1.10, 308.192.4-28.a.1.11, 308.192.4-28.a.1.12, 308.192.4-28.a.1.13, 308.192.4-28.a.1.14, 308.192.4-28.a.1.15, 308.192.4-28.a.1.16
Cyclic 28-isogeny field degree: $2$
Cyclic 28-torsion field degree: $24$
Full 28-torsion field degree: $2016$

Jacobian

Conductor: $2^{6}\cdot7^{4}$
Simple: no
Squarefree: no
Decomposition: $1^{2}\cdot2$
Newforms: 14.2.a.a$^{2}$, 28.2.d.a

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $ $=$ $ 7 x^{2} + z^{2} - 2 z w $
$=$ $21 x y^{2} + 4 x z w - x w^{2} + 14 y^{3} - 4 y z^{2} + 8 y z w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 7 x^{5} - 6 x^{3} z^{2} - 8 x^{2} y z^{2} - 3 x y^{2} z^{2} - x z^{4} - y^{3} z^{2} $
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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.

Canonical model
$(0:0:0:1)$, $(0:0:2:1)$, $(-1/4:0:1/4:1)$, $(1/4:0:1/4:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle 2y$
$\displaystyle Z$ $=$ $\displaystyle z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{31515596xyz^{14}-494905376xyz^{13}w+2692667278xyz^{12}w^{2}-7218877736xyz^{11}w^{3}+9374505560xyz^{10}w^{4}-183586144xyz^{9}w^{5}-18971299424xyz^{8}w^{6}+28791625856xyz^{7}w^{7}-17251244416xyz^{6}w^{8}-53433856xyz^{5}w^{9}+4900284928xyz^{4}w^{10}-1809446912xyz^{3}w^{11}+90417152xyz^{2}w^{12}+29704192xyzw^{13}-1605632xyw^{14}+52597265y^{2}z^{14}-455339374y^{2}z^{13}w+1508842020y^{2}z^{12}w^{2}-1887318104y^{2}z^{11}w^{3}-2114905856y^{2}z^{10}w^{4}+11579676864y^{2}z^{9}w^{5}-18404792000y^{2}z^{8}w^{6}+12819318400y^{2}z^{7}w^{7}+181375488y^{2}z^{6}w^{8}-6353901568y^{2}z^{5}w^{9}+3524111360y^{2}z^{4}w^{10}-400705536y^{2}z^{3}w^{11}-120020992y^{2}z^{2}w^{12}+18464768y^{2}zw^{13}+4194304z^{16}-12274988z^{15}w-50911885z^{14}w^{2}+388067926z^{13}w^{3}-1167286872z^{12}w^{4}+1940506792z^{11}w^{5}-1297906848z^{10}w^{6}-1390828640z^{9}w^{7}+3976823872z^{8}w^{8}-3733293184z^{7}w^{9}+1384680960z^{6}w^{10}+188682752z^{5}w^{11}-300195840z^{4}w^{12}+75880448z^{3}w^{13}-4612096z^{2}w^{14}-360448zw^{15}+16384w^{16}}{z^{4}(7084xyz^{10}-263536xyz^{9}w+1829198xyz^{8}w^{2}-4918256xyz^{7}w^{3}+4768288xyz^{6}w^{4}+1967840xyz^{5}w^{5}-6722016xyz^{4}w^{6}+3753344xyz^{3}w^{7}-387072xyz^{2}w^{8}-78848xyzw^{9}+3584xyw^{10}+23737y^{2}z^{10}-270074y^{2}z^{9}w+995960y^{2}z^{8}w^{2}-1013712y^{2}z^{7}w^{3}-2049376y^{2}z^{6}w^{4}+5858048y^{2}z^{5}w^{5}-4782400y^{2}z^{4}w^{6}+1000832y^{2}z^{3}w^{7}+274176y^{2}z^{2}w^{8}-60928y^{2}zw^{9}+1012z^{11}w-35269z^{10}w^{2}+259874z^{9}w^{3}-803324z^{8}w^{4}+1126384z^{7}w^{5}-445872z^{6}w^{6}-543840z^{5}w^{7}+621184z^{4}w^{8}-201984z^{3}w^{9}+18944z^{2}w^{10}+512zw^{11})}$

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
14.48.2.a.1 $14$ $2$ $2$ $2$ $0$ $2$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
28.192.9.a.1 $28$ $2$ $2$ $9$ $1$ $1^{3}\cdot2$
28.192.9.b.2 $28$ $2$ $2$ $9$ $0$ $1^{3}\cdot2$
28.192.9.b.3 $28$ $2$ $2$ $9$ $0$ $1^{3}\cdot2$
28.192.9.c.1 $28$ $2$ $2$ $9$ $1$ $1^{3}\cdot2$
28.192.9.c.2 $28$ $2$ $2$ $9$ $1$ $1^{3}\cdot2$
28.192.9.d.1 $28$ $2$ $2$ $9$ $1$ $1^{3}\cdot2$
28.192.9.d.2 $28$ $2$ $2$ $9$ $1$ $1^{3}\cdot2$
28.192.11.f.2 $28$ $2$ $2$ $11$ $1$ $1^{3}\cdot2^{2}$
28.192.11.g.1 $28$ $2$ $2$ $11$ $0$ $1^{3}\cdot2^{2}$
28.192.11.h.1 $28$ $2$ $2$ $11$ $1$ $1^{3}\cdot2^{2}$
28.192.11.i.2 $28$ $2$ $2$ $11$ $1$ $1^{3}\cdot2^{2}$
28.288.10.a.1 $28$ $3$ $3$ $10$ $0$ $2\cdot4$
28.288.10.b.2 $28$ $3$ $3$ $10$ $0$ $2\cdot4$
28.288.10.c.1 $28$ $3$ $3$ $10$ $0$ $1^{2}\cdot4$
28.672.37.a.1 $28$ $7$ $7$ $37$ $1$ $1^{9}\cdot2^{4}\cdot4^{2}\cdot8$
56.192.9.b.3 $56$ $2$ $2$ $9$ $1$ $1^{3}\cdot2$
56.192.9.b.4 $56$ $2$ $2$ $9$ $1$ $1^{3}\cdot2$
56.192.9.d.3 $56$ $2$ $2$ $9$ $1$ $1^{3}\cdot2$
56.192.9.d.4 $56$ $2$ $2$ $9$ $1$ $1^{3}\cdot2$
56.192.9.f.1 $56$ $2$ $2$ $9$ $0$ $1^{3}\cdot2$
56.192.9.f.2 $56$ $2$ $2$ $9$ $0$ $1^{3}\cdot2$
56.192.9.h.1 $56$ $2$ $2$ $9$ $3$ $1^{3}\cdot2$
56.192.9.h.2 $56$ $2$ $2$ $9$ $3$ $1^{3}\cdot2$
56.192.11.v.1 $56$ $2$ $2$ $11$ $3$ $1^{3}\cdot2^{2}$
56.192.11.y.1 $56$ $2$ $2$ $11$ $3$ $1^{3}\cdot2^{2}$
56.192.11.bb.1 $56$ $2$ $2$ $11$ $2$ $1^{3}\cdot2^{2}$
56.192.11.be.1 $56$ $2$ $2$ $11$ $5$ $1^{3}\cdot2^{2}$
84.192.9.c.2 $84$ $2$ $2$ $9$ $?$ not computed
84.192.9.c.3 $84$ $2$ $2$ $9$ $?$ not computed
84.192.9.d.1 $84$ $2$ $2$ $9$ $?$ not computed
84.192.9.d.2 $84$ $2$ $2$ $9$ $?$ not computed
84.192.9.g.1 $84$ $2$ $2$ $9$ $?$ not computed
84.192.9.g.4 $84$ $2$ $2$ $9$ $?$ not computed
84.192.9.h.2 $84$ $2$ $2$ $9$ $?$ not computed
84.192.9.h.3 $84$ $2$ $2$ $9$ $?$ not computed
84.192.11.m.2 $84$ $2$ $2$ $11$ $?$ not computed
84.192.11.n.2 $84$ $2$ $2$ $11$ $?$ not computed
84.192.11.s.2 $84$ $2$ $2$ $11$ $?$ not computed
84.192.11.t.2 $84$ $2$ $2$ $11$ $?$ not computed
84.288.20.a.2 $84$ $3$ $3$ $20$ $?$ not computed
84.384.23.c.2 $84$ $4$ $4$ $23$ $?$ not computed
140.192.9.c.1 $140$ $2$ $2$ $9$ $?$ not computed
140.192.9.c.2 $140$ $2$ $2$ $9$ $?$ not computed
140.192.9.d.2 $140$ $2$ $2$ $9$ $?$ not computed
140.192.9.d.3 $140$ $2$ $2$ $9$ $?$ not computed
140.192.9.g.1 $140$ $2$ $2$ $9$ $?$ not computed
140.192.9.g.4 $140$ $2$ $2$ $9$ $?$ not computed
140.192.9.h.1 $140$ $2$ $2$ $9$ $?$ not computed
140.192.9.h.3 $140$ $2$ $2$ $9$ $?$ not computed
140.192.11.m.1 $140$ $2$ $2$ $11$ $?$ not computed
140.192.11.n.1 $140$ $2$ $2$ $11$ $?$ not computed
140.192.11.s.1 $140$ $2$ $2$ $11$ $?$ not computed
140.192.11.t.1 $140$ $2$ $2$ $11$ $?$ not computed
168.192.9.st.2 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.st.3 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.sw.1 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.sw.2 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.td.1 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.td.3 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.tg.2 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.tg.3 $168$ $2$ $2$ $9$ $?$ not computed
168.192.11.bu.2 $168$ $2$ $2$ $11$ $?$ not computed
168.192.11.bx.1 $168$ $2$ $2$ $11$ $?$ not computed
168.192.11.cm.1 $168$ $2$ $2$ $11$ $?$ not computed
168.192.11.cp.2 $168$ $2$ $2$ $11$ $?$ not computed
252.288.10.a.1 $252$ $3$ $3$ $10$ $?$ not computed
252.288.10.b.2 $252$ $3$ $3$ $10$ $?$ not computed
252.288.10.e.1 $252$ $3$ $3$ $10$ $?$ not computed
252.288.10.f.1 $252$ $3$ $3$ $10$ $?$ not computed
252.288.10.i.1 $252$ $3$ $3$ $10$ $?$ not computed
252.288.10.j.2 $252$ $3$ $3$ $10$ $?$ not computed
280.192.9.f.1 $280$ $2$ $2$ $9$ $?$ not computed
280.192.9.f.2 $280$ $2$ $2$ $9$ $?$ not computed
280.192.9.i.2 $280$ $2$ $2$ $9$ $?$ not computed
280.192.9.i.3 $280$ $2$ $2$ $9$ $?$ not computed
280.192.9.p.1 $280$ $2$ $2$ $9$ $?$ not computed
280.192.9.p.4 $280$ $2$ $2$ $9$ $?$ not computed
280.192.9.s.1 $280$ $2$ $2$ $9$ $?$ not computed
280.192.9.s.3 $280$ $2$ $2$ $9$ $?$ not computed
280.192.11.br.1 $280$ $2$ $2$ $11$ $?$ not computed
280.192.11.bu.1 $280$ $2$ $2$ $11$ $?$ not computed
280.192.11.cj.1 $280$ $2$ $2$ $11$ $?$ not computed
280.192.11.cm.1 $280$ $2$ $2$ $11$ $?$ not computed
308.192.9.e.1 $308$ $2$ $2$ $9$ $?$ not computed
308.192.9.e.3 $308$ $2$ $2$ $9$ $?$ not computed
308.192.9.f.1 $308$ $2$ $2$ $9$ $?$ not computed
308.192.9.f.4 $308$ $2$ $2$ $9$ $?$ not computed
308.192.9.g.2 $308$ $2$ $2$ $9$ $?$ not computed
308.192.9.g.3 $308$ $2$ $2$ $9$ $?$ not computed
308.192.9.h.3 $308$ $2$ $2$ $9$ $?$ not computed
308.192.9.h.4 $308$ $2$ $2$ $9$ $?$ not computed
308.192.11.o.2 $308$ $2$ $2$ $11$ $?$ not computed
308.192.11.p.1 $308$ $2$ $2$ $11$ $?$ not computed
308.192.11.q.1 $308$ $2$ $2$ $11$ $?$ not computed
308.192.11.r.2 $308$ $2$ $2$ $11$ $?$ not computed