↑

Properties

Label 12.48.1.d.1
Level $12$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $4$

Related objects

Downloads

Learn more

Invariants

Level: $12$ $\SL_2$-level: $12$ Newform level: $72$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ Cusp orbits $1^{4}\cdot2^{2}$
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: 12P1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 12.48.1.9

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}7&0\\4&7\end{bmatrix}$, $\begin{bmatrix}7&0\\6&5\end{bmatrix}$, $\begin{bmatrix}11&0\\0&5\end{bmatrix}$, $\begin{bmatrix}11&0\\10&1\end{bmatrix}$, $\begin{bmatrix}11&6\\8&11\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2^3\times D_6$
Contains $-I$: yes
Quadratic refinements: 12.96.1-12.d.1.1, 12.96.1-12.d.1.2, 12.96.1-12.d.1.3, 12.96.1-12.d.1.4, 12.96.1-12.d.1.5, 12.96.1-12.d.1.6, 12.96.1-12.d.1.7, 12.96.1-12.d.1.8, 12.96.1-12.d.1.9, 12.96.1-12.d.1.10, 24.96.1-12.d.1.1, 24.96.1-12.d.1.2, 24.96.1-12.d.1.3, 24.96.1-12.d.1.4, 24.96.1-12.d.1.5, 24.96.1-12.d.1.6, 24.96.1-12.d.1.7, 24.96.1-12.d.1.8, 24.96.1-12.d.1.9, 24.96.1-12.d.1.10, 60.96.1-12.d.1.1, 60.96.1-12.d.1.2, 60.96.1-12.d.1.3, 60.96.1-12.d.1.4, 60.96.1-12.d.1.5, 60.96.1-12.d.1.6, 60.96.1-12.d.1.7, 60.96.1-12.d.1.8, 60.96.1-12.d.1.9, 60.96.1-12.d.1.10, 84.96.1-12.d.1.1, 84.96.1-12.d.1.2, 84.96.1-12.d.1.3, 84.96.1-12.d.1.4, 84.96.1-12.d.1.5, 84.96.1-12.d.1.6, 84.96.1-12.d.1.7, 84.96.1-12.d.1.8, 84.96.1-12.d.1.9, 84.96.1-12.d.1.10, 120.96.1-12.d.1.1, 120.96.1-12.d.1.2, 120.96.1-12.d.1.3, 120.96.1-12.d.1.4, 120.96.1-12.d.1.5, 120.96.1-12.d.1.6, 120.96.1-12.d.1.7, 120.96.1-12.d.1.8, 120.96.1-12.d.1.9, 120.96.1-12.d.1.10, 132.96.1-12.d.1.1, 132.96.1-12.d.1.2, 132.96.1-12.d.1.3, 132.96.1-12.d.1.4, 132.96.1-12.d.1.5, 132.96.1-12.d.1.6, 132.96.1-12.d.1.7, 132.96.1-12.d.1.8, 132.96.1-12.d.1.9, 132.96.1-12.d.1.10, 156.96.1-12.d.1.1, 156.96.1-12.d.1.2, 156.96.1-12.d.1.3, 156.96.1-12.d.1.4, 156.96.1-12.d.1.5, 156.96.1-12.d.1.6, 156.96.1-12.d.1.7, 156.96.1-12.d.1.8, 156.96.1-12.d.1.9, 156.96.1-12.d.1.10, 168.96.1-12.d.1.1, 168.96.1-12.d.1.2, 168.96.1-12.d.1.3, 168.96.1-12.d.1.4, 168.96.1-12.d.1.5, 168.96.1-12.d.1.6, 168.96.1-12.d.1.7, 168.96.1-12.d.1.8, 168.96.1-12.d.1.9, 168.96.1-12.d.1.10, 204.96.1-12.d.1.1, 204.96.1-12.d.1.2, 204.96.1-12.d.1.3, 204.96.1-12.d.1.4, 204.96.1-12.d.1.5, 204.96.1-12.d.1.6, 204.96.1-12.d.1.7, 204.96.1-12.d.1.8, 204.96.1-12.d.1.9, 204.96.1-12.d.1.10, 228.96.1-12.d.1.1, 228.96.1-12.d.1.2, 228.96.1-12.d.1.3, 228.96.1-12.d.1.4, 228.96.1-12.d.1.5, 228.96.1-12.d.1.6, 228.96.1-12.d.1.7, 228.96.1-12.d.1.8, 228.96.1-12.d.1.9, 228.96.1-12.d.1.10, 264.96.1-12.d.1.1, 264.96.1-12.d.1.2, 264.96.1-12.d.1.3, 264.96.1-12.d.1.4, 264.96.1-12.d.1.5, 264.96.1-12.d.1.6, 264.96.1-12.d.1.7, 264.96.1-12.d.1.8, 264.96.1-12.d.1.9, 264.96.1-12.d.1.10, 276.96.1-12.d.1.1, 276.96.1-12.d.1.2, 276.96.1-12.d.1.3, 276.96.1-12.d.1.4, 276.96.1-12.d.1.5, 276.96.1-12.d.1.6, 276.96.1-12.d.1.7, 276.96.1-12.d.1.8, 276.96.1-12.d.1.9, 276.96.1-12.d.1.10, 312.96.1-12.d.1.1, 312.96.1-12.d.1.2, 312.96.1-12.d.1.3, 312.96.1-12.d.1.4, 312.96.1-12.d.1.5, 312.96.1-12.d.1.6, 312.96.1-12.d.1.7, 312.96.1-12.d.1.8, 312.96.1-12.d.1.9, 312.96.1-12.d.1.10
Cyclic 12-isogeny field degree: $2$
Cyclic 12-torsion field degree: $8$
Full 12-torsion field degree: $96$

Jacobian

Conductor: $2^{3}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 72.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 39x - 70 $
 Copy content Toggle raw display

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
$(-5:0:1)$, $(-2:0:1)$, $(7:0:1)$, $(0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^2}\cdot\frac{48x^{2}y^{14}+20603970x^{2}y^{12}z^{2}+596598186024x^{2}y^{10}z^{4}+2591415006608565x^{2}y^{8}z^{6}+2999663765443954440x^{2}y^{6}z^{8}+1342582209064924248249x^{2}y^{4}z^{10}+253565669955443818968540x^{2}y^{2}z^{12}+16997644367594655055874037x^{2}z^{14}+7572xy^{14}z+669231720xy^{12}z^{3}+12144823339167xy^{10}z^{5}+33418801143993474xy^{8}z^{7}+30097343519700840576xy^{6}z^{9}+11462543023009458260880xy^{4}z^{11}+1932346127304845969021721xy^{2}z^{13}+118983545466698516454699030xz^{15}+y^{16}+300720y^{14}z^{2}+24011388900y^{12}z^{4}+183330009767520y^{10}z^{6}+297992456081381412y^{8}z^{8}+173882791072168531488y^{6}z^{10}+43785663587859745263402y^{4}z^{12}+4739058019096060967028648y^{2}z^{14}+169976622882007544198261121z^{16}}{zy^{4}(693x^{2}y^{8}z-31104x^{2}y^{6}z^{3}-8398080x^{2}y^{4}z^{5}-362797056x^{2}y^{2}z^{7}+78364164096x^{2}z^{9}+xy^{10}+4662xy^{8}z^{2}+342144xy^{6}z^{4}+57106944xy^{4}z^{6}+3990767616xy^{2}z^{8}-391820820480xz^{10}+44y^{10}z-6975y^{8}z^{3}-311040y^{6}z^{5}+26873856y^{4}z^{7}-16688664576y^{2}z^{9}-1097098297344z^{11})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

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
$X_{\pm1}(2,6)$ $6$ $2$ $2$ $0$ $0$ full Jacobian
12.12.0.b.1 $12$ $4$ $4$ $0$ $0$ full Jacobian
12.24.0.j.1 $12$ $2$ $2$ $0$ $0$ full Jacobian
12.24.1.i.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
12.96.1.d.1 $12$ $2$ $2$ $1$ $0$ dimension zero
12.96.1.d.2 $12$ $2$ $2$ $1$ $0$ dimension zero
12.96.3.d.1 $12$ $2$ $2$ $3$ $0$ $1^{2}$
12.96.3.e.1 $12$ $2$ $2$ $3$ $0$ $1^{2}$
12.96.3.i.1 $12$ $2$ $2$ $3$ $0$ $2$
12.96.3.i.2 $12$ $2$ $2$ $3$ $0$ $2$
12.144.5.e.1 $12$ $3$ $3$ $5$ $0$ $1^{4}$
24.96.1.co.1 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.1.co.2 $24$ $2$ $2$ $1$ $0$ dimension zero
24.96.3.be.1 $24$ $2$ $2$ $3$ $1$ $1^{2}$
24.96.3.bh.1 $24$ $2$ $2$ $3$ $0$ $1^{2}$
24.96.3.cb.1 $24$ $2$ $2$ $3$ $0$ $2$
24.96.3.cb.2 $24$ $2$ $2$ $3$ $0$ $2$
36.144.5.d.1 $36$ $3$ $3$ $5$ $0$ $1^{4}$
36.144.9.d.1 $36$ $3$ $3$ $9$ $1$ $1^{8}$
36.144.9.h.1 $36$ $3$ $3$ $9$ $0$ $1^{8}$
60.96.1.d.1 $60$ $2$ $2$ $1$ $0$ dimension zero
60.96.1.d.2 $60$ $2$ $2$ $1$ $0$ dimension zero
60.96.3.i.1 $60$ $2$ $2$ $3$ $2$ $1^{2}$
60.96.3.j.1 $60$ $2$ $2$ $3$ $0$ $1^{2}$
60.96.3.n.1 $60$ $2$ $2$ $3$ $0$ $2$
60.96.3.n.2 $60$ $2$ $2$ $3$ $0$ $2$
60.240.17.d.1 $60$ $5$ $5$ $17$ $2$ $1^{16}$
60.288.17.d.1 $60$ $6$ $6$ $17$ $2$ $1^{16}$
60.480.33.p.1 $60$ $10$ $10$ $33$ $6$ $1^{32}$
84.96.1.d.1 $84$ $2$ $2$ $1$ $?$ dimension zero
84.96.1.d.2 $84$ $2$ $2$ $1$ $?$ dimension zero
84.96.3.i.1 $84$ $2$ $2$ $3$ $?$ not computed
84.96.3.j.1 $84$ $2$ $2$ $3$ $?$ not computed
84.96.3.n.1 $84$ $2$ $2$ $3$ $?$ not computed
84.96.3.n.2 $84$ $2$ $2$ $3$ $?$ not computed
120.96.1.lk.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.lk.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.3.ei.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.el.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.fh.1 $120$ $2$ $2$ $3$ $?$ not computed
120.96.3.fh.2 $120$ $2$ $2$ $3$ $?$ not computed
132.96.1.d.1 $132$ $2$ $2$ $1$ $?$ dimension zero
132.96.1.d.2 $132$ $2$ $2$ $1$ $?$ dimension zero
132.96.3.i.1 $132$ $2$ $2$ $3$ $?$ not computed
132.96.3.j.1 $132$ $2$ $2$ $3$ $?$ not computed
132.96.3.n.1 $132$ $2$ $2$ $3$ $?$ not computed
132.96.3.n.2 $132$ $2$ $2$ $3$ $?$ not computed
156.96.1.d.1 $156$ $2$ $2$ $1$ $?$ dimension zero
156.96.1.d.2 $156$ $2$ $2$ $1$ $?$ dimension zero
156.96.3.i.1 $156$ $2$ $2$ $3$ $?$ not computed
156.96.3.j.1 $156$ $2$ $2$ $3$ $?$ not computed
156.96.3.n.1 $156$ $2$ $2$ $3$ $?$ not computed
156.96.3.n.2 $156$ $2$ $2$ $3$ $?$ not computed
168.96.1.lk.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.lk.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.3.dk.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.dn.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ej.1 $168$ $2$ $2$ $3$ $?$ not computed
168.96.3.ej.2 $168$ $2$ $2$ $3$ $?$ not computed
204.96.1.d.1 $204$ $2$ $2$ $1$ $?$ dimension zero
204.96.1.d.2 $204$ $2$ $2$ $1$ $?$ dimension zero
204.96.3.i.1 $204$ $2$ $2$ $3$ $?$ not computed
204.96.3.j.1 $204$ $2$ $2$ $3$ $?$ not computed
204.96.3.n.1 $204$ $2$ $2$ $3$ $?$ not computed
204.96.3.n.2 $204$ $2$ $2$ $3$ $?$ not computed
228.96.1.d.1 $228$ $2$ $2$ $1$ $?$ dimension zero
228.96.1.d.2 $228$ $2$ $2$ $1$ $?$ dimension zero
228.96.3.i.1 $228$ $2$ $2$ $3$ $?$ not computed
228.96.3.j.1 $228$ $2$ $2$ $3$ $?$ not computed
228.96.3.n.1 $228$ $2$ $2$ $3$ $?$ not computed
228.96.3.n.2 $228$ $2$ $2$ $3$ $?$ not computed
264.96.1.lk.1 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.1.lk.2 $264$ $2$ $2$ $1$ $?$ dimension zero
264.96.3.dk.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.dn.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ej.1 $264$ $2$ $2$ $3$ $?$ not computed
264.96.3.ej.2 $264$ $2$ $2$ $3$ $?$ not computed
276.96.1.d.1 $276$ $2$ $2$ $1$ $?$ dimension zero
276.96.1.d.2 $276$ $2$ $2$ $1$ $?$ dimension zero
276.96.3.i.1 $276$ $2$ $2$ $3$ $?$ not computed
276.96.3.j.1 $276$ $2$ $2$ $3$ $?$ not computed
276.96.3.n.1 $276$ $2$ $2$ $3$ $?$ not computed
276.96.3.n.2 $276$ $2$ $2$ $3$ $?$ not computed
312.96.1.lk.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.lk.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.ei.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.el.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.fh.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.fh.2 $312$ $2$ $2$ $3$ $?$ not computed