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Properties

Label 12.96.1.b.3
Level $12$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $4$

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Invariants

Level: $12$ $\SL_2$-level: $12$ Newform level: $24$
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 $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $1^{4}\cdot2^{6}$
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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 12.96.1.54

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}5&8\\0&11\end{bmatrix}$, $\begin{bmatrix}11&4\\0&11\end{bmatrix}$, $\begin{bmatrix}11&6\\0&7\end{bmatrix}$, $\begin{bmatrix}11&8\\0&7\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2^2\times D_6$
Contains $-I$: yes
Quadratic refinements: 12.192.1-12.b.3.1, 12.192.1-12.b.3.2, 12.192.1-12.b.3.3, 12.192.1-12.b.3.4, 12.192.1-12.b.3.5, 12.192.1-12.b.3.6, 24.192.1-12.b.3.1, 24.192.1-12.b.3.2, 24.192.1-12.b.3.3, 24.192.1-12.b.3.4, 24.192.1-12.b.3.5, 24.192.1-12.b.3.6, 24.192.1-12.b.3.7, 24.192.1-12.b.3.8, 24.192.1-12.b.3.9, 24.192.1-12.b.3.10, 24.192.1-12.b.3.11, 24.192.1-12.b.3.12, 24.192.1-12.b.3.13, 24.192.1-12.b.3.14, 24.192.1-12.b.3.15, 24.192.1-12.b.3.16, 24.192.1-12.b.3.17, 24.192.1-12.b.3.18, 24.192.1-12.b.3.19, 24.192.1-12.b.3.20, 24.192.1-12.b.3.21, 24.192.1-12.b.3.22, 60.192.1-12.b.3.1, 60.192.1-12.b.3.2, 60.192.1-12.b.3.3, 60.192.1-12.b.3.4, 60.192.1-12.b.3.5, 60.192.1-12.b.3.6, 84.192.1-12.b.3.1, 84.192.1-12.b.3.2, 84.192.1-12.b.3.3, 84.192.1-12.b.3.4, 84.192.1-12.b.3.5, 84.192.1-12.b.3.6, 120.192.1-12.b.3.1, 120.192.1-12.b.3.2, 120.192.1-12.b.3.3, 120.192.1-12.b.3.4, 120.192.1-12.b.3.5, 120.192.1-12.b.3.6, 120.192.1-12.b.3.7, 120.192.1-12.b.3.8, 120.192.1-12.b.3.9, 120.192.1-12.b.3.10, 120.192.1-12.b.3.11, 120.192.1-12.b.3.12, 120.192.1-12.b.3.13, 120.192.1-12.b.3.14, 120.192.1-12.b.3.15, 120.192.1-12.b.3.16, 120.192.1-12.b.3.17, 120.192.1-12.b.3.18, 120.192.1-12.b.3.19, 120.192.1-12.b.3.20, 120.192.1-12.b.3.21, 120.192.1-12.b.3.22, 132.192.1-12.b.3.1, 132.192.1-12.b.3.2, 132.192.1-12.b.3.3, 132.192.1-12.b.3.4, 132.192.1-12.b.3.5, 132.192.1-12.b.3.6, 156.192.1-12.b.3.1, 156.192.1-12.b.3.2, 156.192.1-12.b.3.3, 156.192.1-12.b.3.4, 156.192.1-12.b.3.5, 156.192.1-12.b.3.6, 168.192.1-12.b.3.1, 168.192.1-12.b.3.2, 168.192.1-12.b.3.3, 168.192.1-12.b.3.4, 168.192.1-12.b.3.5, 168.192.1-12.b.3.6, 168.192.1-12.b.3.7, 168.192.1-12.b.3.8, 168.192.1-12.b.3.9, 168.192.1-12.b.3.10, 168.192.1-12.b.3.11, 168.192.1-12.b.3.12, 168.192.1-12.b.3.13, 168.192.1-12.b.3.14, 168.192.1-12.b.3.15, 168.192.1-12.b.3.16, 168.192.1-12.b.3.17, 168.192.1-12.b.3.18, 168.192.1-12.b.3.19, 168.192.1-12.b.3.20, 168.192.1-12.b.3.21, 168.192.1-12.b.3.22, 204.192.1-12.b.3.1, 204.192.1-12.b.3.2, 204.192.1-12.b.3.3, 204.192.1-12.b.3.4, 204.192.1-12.b.3.5, 204.192.1-12.b.3.6, 228.192.1-12.b.3.1, 228.192.1-12.b.3.2, 228.192.1-12.b.3.3, 228.192.1-12.b.3.4, 228.192.1-12.b.3.5, 228.192.1-12.b.3.6, 264.192.1-12.b.3.1, 264.192.1-12.b.3.2, 264.192.1-12.b.3.3, 264.192.1-12.b.3.4, 264.192.1-12.b.3.5, 264.192.1-12.b.3.6, 264.192.1-12.b.3.7, 264.192.1-12.b.3.8, 264.192.1-12.b.3.9, 264.192.1-12.b.3.10, 264.192.1-12.b.3.11, 264.192.1-12.b.3.12, 264.192.1-12.b.3.13, 264.192.1-12.b.3.14, 264.192.1-12.b.3.15, 264.192.1-12.b.3.16, 264.192.1-12.b.3.17, 264.192.1-12.b.3.18, 264.192.1-12.b.3.19, 264.192.1-12.b.3.20, 264.192.1-12.b.3.21, 264.192.1-12.b.3.22, 276.192.1-12.b.3.1, 276.192.1-12.b.3.2, 276.192.1-12.b.3.3, 276.192.1-12.b.3.4, 276.192.1-12.b.3.5, 276.192.1-12.b.3.6, 312.192.1-12.b.3.1, 312.192.1-12.b.3.2, 312.192.1-12.b.3.3, 312.192.1-12.b.3.4, 312.192.1-12.b.3.5, 312.192.1-12.b.3.6, 312.192.1-12.b.3.7, 312.192.1-12.b.3.8, 312.192.1-12.b.3.9, 312.192.1-12.b.3.10, 312.192.1-12.b.3.11, 312.192.1-12.b.3.12, 312.192.1-12.b.3.13, 312.192.1-12.b.3.14, 312.192.1-12.b.3.15, 312.192.1-12.b.3.16, 312.192.1-12.b.3.17, 312.192.1-12.b.3.18, 312.192.1-12.b.3.19, 312.192.1-12.b.3.20, 312.192.1-12.b.3.21, 312.192.1-12.b.3.22
Cyclic 12-isogeny field degree: $1$
Cyclic 12-torsion field degree: $4$
Full 12-torsion field degree: $48$

Jacobian

Conductor: $2^{3}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 24.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x^{2} + x $
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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
$(0:0:1)$, $(0:1:0)$, $(1:-1:1)$, $(1:1:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{8x^{2}y^{30}+4392x^{2}y^{28}z^{2}+673800x^{2}y^{26}z^{4}+14310376x^{2}y^{24}z^{6}-1443145944x^{2}y^{22}z^{8}-14397985720x^{2}y^{20}z^{10}+16522252968x^{2}y^{18}z^{12}+117561695112x^{2}y^{16}z^{14}-27930135528x^{2}y^{14}z^{16}-72555943816x^{2}y^{12}z^{18}+12479755032x^{2}y^{10}z^{20}+4477615032x^{2}y^{8}z^{22}-362213704x^{2}y^{6}z^{24}-15522792x^{2}y^{4}z^{26}-10184x^{2}y^{2}z^{28}+728x^{2}z^{30}-8xy^{30}z+2808xy^{28}z^{3}+2122008xy^{26}z^{5}+253278680xy^{24}z^{7}+4341629592xy^{22}z^{9}-2345120488xy^{20}z^{11}-100182389832xy^{18}z^{13}-42515561736xy^{16}z^{15}+155294579304xy^{14}z^{17}+14334927976xy^{12}z^{19}-30388920120xy^{10}z^{21}+668111880xy^{8}z^{23}+535235848xy^{6}z^{25}+2274696xy^{4}z^{27}-169880xy^{2}z^{29}-728xz^{31}-y^{32}-816y^{30}z^{2}-232152y^{28}z^{4}-24195824y^{26}z^{6}-501182044y^{24}z^{8}+3310876816y^{22}z^{10}+39275932824y^{20}z^{12}+6652822992y^{18}z^{14}-102301473222y^{16}z^{16}-4922919568y^{14}z^{18}+25478378520y^{12}z^{20}-812419536y^{10}z^{22}-521790556y^{8}z^{24}-2094800y^{6}z^{26}+169640y^{4}z^{28}+752y^{2}z^{30}-z^{32}}{z^{2}y^{2}(y-z)^{6}(y+z)^{6}(6x^{2}y^{14}-116x^{2}y^{12}z^{2}+1146x^{2}y^{10}z^{4}-9816x^{2}y^{8}z^{6}+9946x^{2}y^{6}z^{8}-3700x^{2}y^{4}z^{10}+486x^{2}y^{2}z^{12}+3xy^{14}z+235xy^{12}z^{3}-2097xy^{10}z^{5}+807xy^{8}z^{7}+3913xy^{6}z^{9}-4319xy^{4}z^{11}+1701xy^{2}z^{13}-243xz^{15}-y^{16}-55y^{14}z^{2}+831y^{12}z^{4}-5391y^{10}z^{6}+1397y^{8}z^{8}+2387y^{6}z^{10}-1459y^{4}z^{12}+243y^{2}z^{14})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.0.a.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
12.48.0.c.3 $12$ $2$ $2$ $0$ $0$ full Jacobian
12.48.1.b.1 $12$ $2$ $2$ $1$ $0$ 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
12.192.5.b.1 $12$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
12.192.5.e.3 $12$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
12.288.9.b.1 $12$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
24.192.5.f.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.bv.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.cb.4 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.cc.3 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.cd.1 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.ce.2 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.cj.2 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.ck.1 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.cl.3 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.cm.4 $24$ $2$ $2$ $5$ $0$ $2^{2}$
24.192.5.cw.4 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.cx.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.de.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.df.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.9.bt.2 $24$ $2$ $2$ $9$ $0$ $1^{4}\cdot4$
24.192.9.cd.1 $24$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
24.192.9.dx.1 $24$ $2$ $2$ $9$ $1$ $1^{4}\cdot4$
24.192.9.eb.1 $24$ $2$ $2$ $9$ $2$ $1^{4}\cdot4$
36.288.9.b.2 $36$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
36.288.17.b.3 $36$ $3$ $3$ $17$ $0$ $1^{8}\cdot4^{2}$
36.288.17.f.4 $36$ $3$ $3$ $17$ $1$ $1^{8}\cdot4^{2}$
60.192.5.h.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.192.5.k.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.480.33.b.4 $60$ $5$ $5$ $33$ $1$ $1^{16}\cdot8^{2}$
60.576.33.b.3 $60$ $6$ $6$ $33$ $0$ $1^{16}\cdot8^{2}$
60.960.65.b.2 $60$ $10$ $10$ $65$ $2$ $1^{32}\cdot8^{4}$
84.192.5.h.1 $84$ $2$ $2$ $5$ $?$ not computed
84.192.5.k.1 $84$ $2$ $2$ $5$ $?$ not computed
120.192.5.ix.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.jo.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kx.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ky.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.kz.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.la.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.lf.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.lg.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.lh.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.li.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.lw.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ly.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mm.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.mo.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.9.ho.4 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.hr.4 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.iq.4 $120$ $2$ $2$ $9$ $?$ not computed
120.192.9.it.4 $120$ $2$ $2$ $9$ $?$ not computed
132.192.5.h.1 $132$ $2$ $2$ $5$ $?$ not computed
132.192.5.k.1 $132$ $2$ $2$ $5$ $?$ not computed
156.192.5.h.2 $156$ $2$ $2$ $5$ $?$ not computed
156.192.5.k.1 $156$ $2$ $2$ $5$ $?$ not computed
168.192.5.ix.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.jo.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kx.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ky.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.kz.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.la.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.lf.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.lg.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.lh.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.li.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.lw.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ly.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.mm.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.mo.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.9.ho.3 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.hr.2 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.iq.2 $168$ $2$ $2$ $9$ $?$ not computed
168.192.9.it.2 $168$ $2$ $2$ $9$ $?$ not computed
204.192.5.h.2 $204$ $2$ $2$ $5$ $?$ not computed
204.192.5.k.1 $204$ $2$ $2$ $5$ $?$ not computed
228.192.5.h.2 $228$ $2$ $2$ $5$ $?$ not computed
228.192.5.k.1 $228$ $2$ $2$ $5$ $?$ not computed
264.192.5.ix.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.jo.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kx.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ky.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.kz.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.la.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.lf.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.lg.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.lh.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.li.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.lw.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ly.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.mm.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.mo.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.9.ho.3 $264$ $2$ $2$ $9$ $?$ not computed
264.192.9.hr.2 $264$ $2$ $2$ $9$ $?$ not computed
264.192.9.iq.2 $264$ $2$ $2$ $9$ $?$ not computed
264.192.9.it.2 $264$ $2$ $2$ $9$ $?$ not computed
276.192.5.h.2 $276$ $2$ $2$ $5$ $?$ not computed
276.192.5.k.1 $276$ $2$ $2$ $5$ $?$ not computed
312.192.5.ix.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.jo.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kx.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ky.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.kz.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.la.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.lf.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.lg.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.lh.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.li.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.lw.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ly.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.mm.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.mo.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.9.ho.2 $312$ $2$ $2$ $9$ $?$ not computed
312.192.9.hr.2 $312$ $2$ $2$ $9$ $?$ not computed
312.192.9.iq.4 $312$ $2$ $2$ $9$ $?$ not computed
312.192.9.it.2 $312$ $2$ $2$ $9$ $?$ not computed