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Properties

Label 12.96.3.h.2
Level $12$
Index $96$
Genus $3$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $2$

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Invariants

Level: $12$ $\SL_2$-level: $12$ Newform level: $144$
Index: $96$ $\PSL_2$-index:$96$
Genus: $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $2$ are rational) Cusp widths $4^{6}\cdot12^{6}$ Cusp orbits $1^{2}\cdot2^{5}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12L3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 12.96.3.46

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}5&2\\6&11\end{bmatrix}$, $\begin{bmatrix}5&6\\0&7\end{bmatrix}$, $\begin{bmatrix}11&0\\6&5\end{bmatrix}$, $\begin{bmatrix}11&4\\0&11\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2^2\times D_6$
Contains $-I$: yes
Quadratic refinements: 12.192.3-12.h.2.1, 12.192.3-12.h.2.2, 12.192.3-12.h.2.3, 12.192.3-12.h.2.4, 12.192.3-12.h.2.5, 12.192.3-12.h.2.6, 12.192.3-12.h.2.7, 12.192.3-12.h.2.8, 24.192.3-12.h.2.1, 24.192.3-12.h.2.2, 24.192.3-12.h.2.3, 24.192.3-12.h.2.4, 24.192.3-12.h.2.5, 24.192.3-12.h.2.6, 24.192.3-12.h.2.7, 24.192.3-12.h.2.8, 60.192.3-12.h.2.1, 60.192.3-12.h.2.2, 60.192.3-12.h.2.3, 60.192.3-12.h.2.4, 60.192.3-12.h.2.5, 60.192.3-12.h.2.6, 60.192.3-12.h.2.7, 60.192.3-12.h.2.8, 84.192.3-12.h.2.1, 84.192.3-12.h.2.2, 84.192.3-12.h.2.3, 84.192.3-12.h.2.4, 84.192.3-12.h.2.5, 84.192.3-12.h.2.6, 84.192.3-12.h.2.7, 84.192.3-12.h.2.8, 120.192.3-12.h.2.1, 120.192.3-12.h.2.2, 120.192.3-12.h.2.3, 120.192.3-12.h.2.4, 120.192.3-12.h.2.5, 120.192.3-12.h.2.6, 120.192.3-12.h.2.7, 120.192.3-12.h.2.8, 132.192.3-12.h.2.1, 132.192.3-12.h.2.2, 132.192.3-12.h.2.3, 132.192.3-12.h.2.4, 132.192.3-12.h.2.5, 132.192.3-12.h.2.6, 132.192.3-12.h.2.7, 132.192.3-12.h.2.8, 156.192.3-12.h.2.1, 156.192.3-12.h.2.2, 156.192.3-12.h.2.3, 156.192.3-12.h.2.4, 156.192.3-12.h.2.5, 156.192.3-12.h.2.6, 156.192.3-12.h.2.7, 156.192.3-12.h.2.8, 168.192.3-12.h.2.1, 168.192.3-12.h.2.2, 168.192.3-12.h.2.3, 168.192.3-12.h.2.4, 168.192.3-12.h.2.5, 168.192.3-12.h.2.6, 168.192.3-12.h.2.7, 168.192.3-12.h.2.8, 204.192.3-12.h.2.1, 204.192.3-12.h.2.2, 204.192.3-12.h.2.3, 204.192.3-12.h.2.4, 204.192.3-12.h.2.5, 204.192.3-12.h.2.6, 204.192.3-12.h.2.7, 204.192.3-12.h.2.8, 228.192.3-12.h.2.1, 228.192.3-12.h.2.2, 228.192.3-12.h.2.3, 228.192.3-12.h.2.4, 228.192.3-12.h.2.5, 228.192.3-12.h.2.6, 228.192.3-12.h.2.7, 228.192.3-12.h.2.8, 264.192.3-12.h.2.1, 264.192.3-12.h.2.2, 264.192.3-12.h.2.3, 264.192.3-12.h.2.4, 264.192.3-12.h.2.5, 264.192.3-12.h.2.6, 264.192.3-12.h.2.7, 264.192.3-12.h.2.8, 276.192.3-12.h.2.1, 276.192.3-12.h.2.2, 276.192.3-12.h.2.3, 276.192.3-12.h.2.4, 276.192.3-12.h.2.5, 276.192.3-12.h.2.6, 276.192.3-12.h.2.7, 276.192.3-12.h.2.8, 312.192.3-12.h.2.1, 312.192.3-12.h.2.2, 312.192.3-12.h.2.3, 312.192.3-12.h.2.4, 312.192.3-12.h.2.5, 312.192.3-12.h.2.6, 312.192.3-12.h.2.7, 312.192.3-12.h.2.8
Cyclic 12-isogeny field degree: $2$
Cyclic 12-torsion field degree: $4$
Full 12-torsion field degree: $48$

Jacobian

Conductor: $2^{12}\cdot3^{4}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2$
Newforms: 48.2.c.a, 144.2.a.b

Models

Embedded model Embedded model in $\mathbb{P}^{4}$

$ 0 $ $=$ $ x z t + y^{2} t - y w t + z^{2} t $
$=$ $x y z + y^{3} - y^{2} w + y z^{2}$
$=$ $x z w + y^{2} w - y w^{2} + z^{2} w$
$=$ $x^{2} z + x y^{2} - x y w + x z^{2}$
$=$$\cdots$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 3 x^{5} + 3 x^{4} z - 9 x^{3} y^{2} + 4 x^{3} z^{2} + 9 x^{2} y^{2} z + 4 x^{2} z^{3} + 3 x y^{2} z^{2} + \cdots + z^{5} $
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Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ 3x^{7} + 15x^{6} + 21x^{5} + 30x^{4} + 21x^{3} + 15x^{2} + 3x $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model
$(0:0:0:0:1)$, $(0:1/2:-1/2:1:0)$

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{3}t$
$\displaystyle Z$ $=$ $\displaystyle z$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle \frac{1}{2}y+\frac{1}{2}z$
$\displaystyle Y$ $=$ $\displaystyle -\frac{3}{8}y^{3}t+\frac{3}{8}y^{2}zt+\frac{1}{8}yz^{2}t-\frac{1}{8}z^{3}t$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}y-\frac{1}{2}z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^6\cdot3}\cdot\frac{38999534495662080xzw^{12}+2168946343646352xzw^{10}t^{2}-38994085801116288xzw^{8}t^{4}-10362441095706780xzw^{6}t^{6}-237087690412080xzw^{4}t^{8}+25156869949917xzw^{2}t^{10}+366386257370xzt^{12}+326149079040xw^{13}-6498205122428928xw^{11}t^{2}-3652310971309104xw^{9}t^{4}+4490711298481392xw^{7}t^{6}+1594422159064308xw^{5}t^{8}+83417039701584xw^{3}t^{10}+139100771601xwt^{12}+8666662676594688yzw^{12}-24192880866832320yzw^{10}t^{2}-19204078927117248yzw^{8}t^{4}+7807210722432yzw^{6}t^{6}+653238103516296yzw^{4}t^{8}+32568728204748yzw^{2}t^{10}+203383460946yzt^{12}-12999438241431552yw^{13}-27081508378321872yw^{11}t^{2}+11174092425984816yw^{9}t^{4}+8330299318430316yw^{7}t^{6}-58232624495364yw^{5}t^{8}-141083983773273yw^{3}t^{10}-3853524433159ywt^{12}+17332995580231680z^{2}w^{12}-3971663625132000z^{2}w^{10}t^{2}-21823187925323232z^{2}w^{8}t^{4}-5174287298662680z^{2}w^{6}t^{6}+45353141949264z^{2}w^{4}t^{8}+28848612325290z^{2}w^{2}t^{10}+282495313204z^{2}t^{12}-8666648672993280zw^{13}-25639064764338240zw^{11}t^{2}-1513908758832192zw^{9}t^{4}+8826539453115264zw^{7}t^{6}+1947554501465232zw^{5}t^{8}+48190125843588zw^{3}t^{10}-1848044281032zwt^{12}-188441690112w^{14}+4332287661244416w^{12}t^{2}-995223598831584w^{10}t^{4}-4527987893744112w^{8}t^{6}-903032028901224w^{6}t^{8}+2664625713480w^{4}t^{10}+1759418295754w^{2}t^{12}+2834352t^{14}}{t^{4}(10215360xzw^{8}+27404640xzw^{6}t^{2}+16353864xzw^{4}t^{4}+2201841xzw^{2}t^{6}+22286xzt^{8}+6635520xw^{9}+20560896xw^{7}t^{2}+17839008xw^{5}t^{4}+4150428xw^{3}t^{6}+184317xwt^{8}+4293504yzw^{8}+11144640yzw^{6}t^{2}+6710952yzw^{4}t^{4}+873568yzw^{2}t^{6}+14150yzt^{8}+4744512yw^{9}+14073696yw^{7}t^{2}+9787464yw^{5}t^{4}+2185971yw^{3}t^{6}+51949ywt^{8}+1877760z^{2}w^{8}+5495424z^{2}w^{6}t^{2}+2838264z^{2}w^{4}t^{4}+460154z^{2}w^{2}t^{6}+2300z^{2}t^{8}-4131072zw^{9}-9008832zw^{7}t^{2}-5388816zw^{5}t^{4}-312016zw^{3}t^{6}+16172zwt^{8}-3833856w^{10}-11787840w^{8}t^{2}-9938592w^{6}t^{4}-2313804w^{4}t^{6}-88450w^{2}t^{8})}$

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.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
12.48.1.c.1 $12$ $2$ $2$ $1$ $0$ $2$
12.48.2.a.2 $12$ $2$ $2$ $2$ $0$ $1$

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.c.1 $12$ $2$ $2$ $5$ $0$ $1^{2}$
12.192.5.e.2 $12$ $2$ $2$ $5$ $0$ $1^{2}$
12.192.5.e.3 $12$ $2$ $2$ $5$ $0$ $1^{2}$
12.288.13.n.1 $12$ $3$ $3$ $13$ $0$ $1^{4}\cdot2^{3}$
24.192.5.bo.1 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.bo.3 $24$ $2$ $2$ $5$ $0$ $1^{2}$
24.192.5.bz.1 $24$ $2$ $2$ $5$ $1$ $1^{2}$
24.192.5.bz.3 $24$ $2$ $2$ $5$ $1$ $1^{2}$
36.288.13.h.2 $36$ $3$ $3$ $13$ $0$ $1^{4}\cdot2^{3}$
36.288.19.m.2 $36$ $3$ $3$ $19$ $1$ $1^{8}\cdot2^{2}\cdot4$
36.288.19.q.2 $36$ $3$ $3$ $19$ $2$ $1^{8}\cdot4^{2}$
60.192.5.v.1 $60$ $2$ $2$ $5$ $1$ $1^{2}$
60.192.5.v.2 $60$ $2$ $2$ $5$ $1$ $1^{2}$
60.192.5.w.2 $60$ $2$ $2$ $5$ $1$ $1^{2}$
60.192.5.w.4 $60$ $2$ $2$ $5$ $1$ $1^{2}$
60.480.35.bb.1 $60$ $5$ $5$ $35$ $4$ $1^{16}\cdot2^{4}\cdot8$
60.576.37.cv.1 $60$ $6$ $6$ $37$ $2$ $1^{16}\cdot2\cdot4^{2}\cdot8$
60.960.69.fj.2 $60$ $10$ $10$ $69$ $8$ $1^{32}\cdot2^{5}\cdot4^{2}\cdot8^{2}$
84.192.5.v.2 $84$ $2$ $2$ $5$ $?$ not computed
84.192.5.v.3 $84$ $2$ $2$ $5$ $?$ not computed
84.192.5.w.1 $84$ $2$ $2$ $5$ $?$ not computed
84.192.5.w.3 $84$ $2$ $2$ $5$ $?$ not computed
120.192.5.pl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.pl.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ps.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ps.4 $120$ $2$ $2$ $5$ $?$ not computed
132.192.5.v.1 $132$ $2$ $2$ $5$ $?$ not computed
132.192.5.v.2 $132$ $2$ $2$ $5$ $?$ not computed
132.192.5.w.2 $132$ $2$ $2$ $5$ $?$ not computed
132.192.5.w.3 $132$ $2$ $2$ $5$ $?$ not computed
156.192.5.v.2 $156$ $2$ $2$ $5$ $?$ not computed
156.192.5.v.3 $156$ $2$ $2$ $5$ $?$ not computed
156.192.5.w.2 $156$ $2$ $2$ $5$ $?$ not computed
156.192.5.w.4 $156$ $2$ $2$ $5$ $?$ not computed
168.192.5.pl.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.pl.4 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ps.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ps.4 $168$ $2$ $2$ $5$ $?$ not computed
204.192.5.v.1 $204$ $2$ $2$ $5$ $?$ not computed
204.192.5.v.2 $204$ $2$ $2$ $5$ $?$ not computed
204.192.5.w.2 $204$ $2$ $2$ $5$ $?$ not computed
204.192.5.w.4 $204$ $2$ $2$ $5$ $?$ not computed
228.192.5.v.2 $228$ $2$ $2$ $5$ $?$ not computed
228.192.5.v.3 $228$ $2$ $2$ $5$ $?$ not computed
228.192.5.w.1 $228$ $2$ $2$ $5$ $?$ not computed
228.192.5.w.2 $228$ $2$ $2$ $5$ $?$ not computed
264.192.5.pl.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.pl.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ps.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ps.4 $264$ $2$ $2$ $5$ $?$ not computed
276.192.5.v.1 $276$ $2$ $2$ $5$ $?$ not computed
276.192.5.v.2 $276$ $2$ $2$ $5$ $?$ not computed
276.192.5.w.2 $276$ $2$ $2$ $5$ $?$ not computed
276.192.5.w.3 $276$ $2$ $2$ $5$ $?$ not computed
312.192.5.pl.1 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.pl.4 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ps.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ps.4 $312$ $2$ $2$ $5$ $?$ not computed