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Properties

Label 12.96.1.b.4
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: 12.96.1.56

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}1&10\\0&7\end{bmatrix}$, $\begin{bmatrix}5&10\\0&7\end{bmatrix}$, $\begin{bmatrix}7&0\\0&5\end{bmatrix}$, $\begin{bmatrix}11&6\\0&5\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2^2\times D_6$
Contains $-I$: yes
Quadratic refinements: 12.192.1-12.b.4.1, 12.192.1-12.b.4.2, 12.192.1-12.b.4.3, 12.192.1-12.b.4.4, 12.192.1-12.b.4.5, 12.192.1-12.b.4.6, 24.192.1-12.b.4.1, 24.192.1-12.b.4.2, 24.192.1-12.b.4.3, 24.192.1-12.b.4.4, 24.192.1-12.b.4.5, 24.192.1-12.b.4.6, 24.192.1-12.b.4.7, 24.192.1-12.b.4.8, 24.192.1-12.b.4.9, 24.192.1-12.b.4.10, 24.192.1-12.b.4.11, 24.192.1-12.b.4.12, 24.192.1-12.b.4.13, 24.192.1-12.b.4.14, 24.192.1-12.b.4.15, 24.192.1-12.b.4.16, 24.192.1-12.b.4.17, 24.192.1-12.b.4.18, 24.192.1-12.b.4.19, 24.192.1-12.b.4.20, 24.192.1-12.b.4.21, 24.192.1-12.b.4.22, 60.192.1-12.b.4.1, 60.192.1-12.b.4.2, 60.192.1-12.b.4.3, 60.192.1-12.b.4.4, 60.192.1-12.b.4.5, 60.192.1-12.b.4.6, 84.192.1-12.b.4.1, 84.192.1-12.b.4.2, 84.192.1-12.b.4.3, 84.192.1-12.b.4.4, 84.192.1-12.b.4.5, 84.192.1-12.b.4.6, 120.192.1-12.b.4.1, 120.192.1-12.b.4.2, 120.192.1-12.b.4.3, 120.192.1-12.b.4.4, 120.192.1-12.b.4.5, 120.192.1-12.b.4.6, 120.192.1-12.b.4.7, 120.192.1-12.b.4.8, 120.192.1-12.b.4.9, 120.192.1-12.b.4.10, 120.192.1-12.b.4.11, 120.192.1-12.b.4.12, 120.192.1-12.b.4.13, 120.192.1-12.b.4.14, 120.192.1-12.b.4.15, 120.192.1-12.b.4.16, 120.192.1-12.b.4.17, 120.192.1-12.b.4.18, 120.192.1-12.b.4.19, 120.192.1-12.b.4.20, 120.192.1-12.b.4.21, 120.192.1-12.b.4.22, 132.192.1-12.b.4.1, 132.192.1-12.b.4.2, 132.192.1-12.b.4.3, 132.192.1-12.b.4.4, 132.192.1-12.b.4.5, 132.192.1-12.b.4.6, 156.192.1-12.b.4.1, 156.192.1-12.b.4.2, 156.192.1-12.b.4.3, 156.192.1-12.b.4.4, 156.192.1-12.b.4.5, 156.192.1-12.b.4.6, 168.192.1-12.b.4.1, 168.192.1-12.b.4.2, 168.192.1-12.b.4.3, 168.192.1-12.b.4.4, 168.192.1-12.b.4.5, 168.192.1-12.b.4.6, 168.192.1-12.b.4.7, 168.192.1-12.b.4.8, 168.192.1-12.b.4.9, 168.192.1-12.b.4.10, 168.192.1-12.b.4.11, 168.192.1-12.b.4.12, 168.192.1-12.b.4.13, 168.192.1-12.b.4.14, 168.192.1-12.b.4.15, 168.192.1-12.b.4.16, 168.192.1-12.b.4.17, 168.192.1-12.b.4.18, 168.192.1-12.b.4.19, 168.192.1-12.b.4.20, 168.192.1-12.b.4.21, 168.192.1-12.b.4.22, 204.192.1-12.b.4.1, 204.192.1-12.b.4.2, 204.192.1-12.b.4.3, 204.192.1-12.b.4.4, 204.192.1-12.b.4.5, 204.192.1-12.b.4.6, 228.192.1-12.b.4.1, 228.192.1-12.b.4.2, 228.192.1-12.b.4.3, 228.192.1-12.b.4.4, 228.192.1-12.b.4.5, 228.192.1-12.b.4.6, 264.192.1-12.b.4.1, 264.192.1-12.b.4.2, 264.192.1-12.b.4.3, 264.192.1-12.b.4.4, 264.192.1-12.b.4.5, 264.192.1-12.b.4.6, 264.192.1-12.b.4.7, 264.192.1-12.b.4.8, 264.192.1-12.b.4.9, 264.192.1-12.b.4.10, 264.192.1-12.b.4.11, 264.192.1-12.b.4.12, 264.192.1-12.b.4.13, 264.192.1-12.b.4.14, 264.192.1-12.b.4.15, 264.192.1-12.b.4.16, 264.192.1-12.b.4.17, 264.192.1-12.b.4.18, 264.192.1-12.b.4.19, 264.192.1-12.b.4.20, 264.192.1-12.b.4.21, 264.192.1-12.b.4.22, 276.192.1-12.b.4.1, 276.192.1-12.b.4.2, 276.192.1-12.b.4.3, 276.192.1-12.b.4.4, 276.192.1-12.b.4.5, 276.192.1-12.b.4.6, 312.192.1-12.b.4.1, 312.192.1-12.b.4.2, 312.192.1-12.b.4.3, 312.192.1-12.b.4.4, 312.192.1-12.b.4.5, 312.192.1-12.b.4.6, 312.192.1-12.b.4.7, 312.192.1-12.b.4.8, 312.192.1-12.b.4.9, 312.192.1-12.b.4.10, 312.192.1-12.b.4.11, 312.192.1-12.b.4.12, 312.192.1-12.b.4.13, 312.192.1-12.b.4.14, 312.192.1-12.b.4.15, 312.192.1-12.b.4.16, 312.192.1-12.b.4.17, 312.192.1-12.b.4.18, 312.192.1-12.b.4.19, 312.192.1-12.b.4.20, 312.192.1-12.b.4.21, 312.192.1-12.b.4.22
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
$(1:1:1)$, $(1:-1:1)$, $(0:0:1)$, $(0:1:0)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{8x^{2}y^{30}-176328x^{2}y^{28}z^{2}+57062760x^{2}y^{26}z^{4}-751512584x^{2}y^{24}z^{6}+2155992936x^{2}y^{22}z^{8}-2690410600x^{2}y^{20}z^{10}+8578240968x^{2}y^{18}z^{12}-39436660008x^{2}y^{16}z^{14}+88915974552x^{2}y^{14}z^{16}-95486976856x^{2}y^{12}z^{18}+27036105912x^{2}y^{10}z^{20}+46490190312x^{2}y^{8}z^{22}-55788536584x^{2}y^{6}z^{24}+25957171848x^{2}y^{4}z^{26}-5423886824x^{2}y^{2}z^{28}+387420488x^{2}z^{30}-728xy^{30}z+1013688xy^{28}z^{3}-234900552xy^{26}z^{5}+1623139880xy^{24}z^{7}-4389622488xy^{22}z^{9}+4808593592xy^{20}z^{11}-1613615112xy^{18}z^{13}+22675872744xy^{16}z^{15}-112244580936xy^{14}z^{17}+231775406056xy^{12}z^{19}-256087963800xy^{10}z^{21}+155528084280xy^{8}z^{23}-44165950472xy^{6}z^{25}+936xy^{4}z^{27}+2711943400xy^{2}z^{29}-387420488xz^{31}-y^{32}+4944y^{30}z^{2}-15695592y^{28}z^{4}+532853056y^{26}z^{6}-2153021884y^{24}z^{8}+3849513136y^{22}z^{10}-2729488536y^{20}z^{12}-13319673888y^{18}z^{14}+64498712058y^{16}z^{16}-133215120208y^{14}z^{18}+150890568360y^{12}z^{20}-93537492096y^{10}z^{22}+25182049604y^{8}z^{24}+2324537680y^{6}z^{26}-2711944360y^{4}z^{28}+387420512y^{2}z^{30}-z^{32}}{zy^{4}(y-z)^{2}(y+z)^{2}(28x^{2}y^{20}z+3056x^{2}y^{18}z^{3}-62817x^{2}y^{16}z^{5}-25856x^{2}y^{14}z^{7}+1414956x^{2}y^{12}z^{9}-2123520x^{2}y^{10}z^{11}+532186x^{2}y^{8}z^{13}-128x^{2}y^{6}z^{15}-64x^{2}y^{4}z^{17}+16x^{2}y^{2}z^{19}-x^{2}z^{21}-xy^{22}-233xy^{20}z^{2}+4615xy^{18}z^{4}+136361xy^{16}z^{6}-1213342xy^{14}z^{8}+1599354xy^{12}z^{10}+529754xy^{10}z^{12}-532362xy^{8}z^{14}+79xy^{6}z^{16}+79xy^{4}z^{18}-17xy^{2}z^{20}+xz^{22}+7y^{22}z-400y^{20}z^{3}-24871y^{18}z^{5}+320563y^{16}z^{7}-554982y^{14}z^{9}-536136y^{12}z^{11}+532918y^{10}z^{13}+886y^{8}z^{15}-65y^{6}z^{17}-80y^{4}z^{19}+17y^{2}z^{21}-z^{23})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

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