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Properties

Label 48.48.1.cb.1
Level $48$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $288$
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 $2$ are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $1^{2}\cdot2^{3}$
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: 16G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.47

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}19&17\\8&17\end{bmatrix}$, $\begin{bmatrix}19&21\\24&5\end{bmatrix}$, $\begin{bmatrix}25&40\\28&23\end{bmatrix}$, $\begin{bmatrix}29&22\\28&7\end{bmatrix}$, $\begin{bmatrix}35&10\\16&27\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.cb.1.1, 48.96.1-48.cb.1.2, 48.96.1-48.cb.1.3, 48.96.1-48.cb.1.4, 48.96.1-48.cb.1.5, 48.96.1-48.cb.1.6, 48.96.1-48.cb.1.7, 48.96.1-48.cb.1.8, 48.96.1-48.cb.1.9, 48.96.1-48.cb.1.10, 48.96.1-48.cb.1.11, 48.96.1-48.cb.1.12, 48.96.1-48.cb.1.13, 48.96.1-48.cb.1.14, 48.96.1-48.cb.1.15, 48.96.1-48.cb.1.16, 240.96.1-48.cb.1.1, 240.96.1-48.cb.1.2, 240.96.1-48.cb.1.3, 240.96.1-48.cb.1.4, 240.96.1-48.cb.1.5, 240.96.1-48.cb.1.6, 240.96.1-48.cb.1.7, 240.96.1-48.cb.1.8, 240.96.1-48.cb.1.9, 240.96.1-48.cb.1.10, 240.96.1-48.cb.1.11, 240.96.1-48.cb.1.12, 240.96.1-48.cb.1.13, 240.96.1-48.cb.1.14, 240.96.1-48.cb.1.15, 240.96.1-48.cb.1.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $64$
Full 48-torsion field degree: $24576$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 99x + 378 $
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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.

Weierstrass model
$(0:1:0)$, $(6:0:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{3^8}\cdot\frac{72x^{2}y^{14}-1748466x^{2}y^{12}z^{2}+7526831688x^{2}y^{10}z^{4}-11410959965841x^{2}y^{8}z^{6}+7015075177399488x^{2}y^{6}z^{8}-1656889863250248009x^{2}y^{4}z^{10}+165937492907141075388x^{2}y^{2}z^{12}-5991252732528354928665x^{2}z^{14}-2844xy^{14}z+35592696xy^{12}z^{3}-117026713359xy^{10}z^{5}+152659283213934xy^{8}z^{7}-85940430497886024xy^{6}z^{9}+19581156361151976888xy^{4}z^{11}-1926029375706606797097xy^{2}z^{13}+68811223416715148513130xz^{15}-y^{16}+76464y^{14}z^{2}-492853572y^{12}z^{4}+1016332964712y^{10}z^{6}-889134854116572y^{8}z^{8}+356068495914514272y^{6}z^{10}-67262987904343060266y^{4}z^{12}+5925071995289859217392y^{2}z^{14}-197182242129552543183321z^{16}}{z^{5}y^{2}(2763x^{2}y^{6}z-27344304x^{2}y^{4}z^{3}+56815005744x^{2}y^{2}z^{5}-31112228407680x^{2}z^{7}+xy^{8}-77058xy^{6}z^{2}+464495472xy^{4}z^{4}-757384833888xy^{2}z^{6}+357332697441792xz^{8}-72y^{8}z+1500768y^{6}z^{3}-4708943424y^{4}z^{5}+4278306957696y^{2}z^{7}-1023955961974272z^{9})}$

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
$X_{\pm1}(8)$ $8$ $2$ $2$ $0$ $0$ full Jacobian
48.24.0.f.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.24.1.b.1 $48$ $2$ $2$ $1$ $0$ 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
48.96.1.i.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ba.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.bn.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.by.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.dm.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ea.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.ee.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.96.1.eo.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.144.9.jt.2 $48$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
48.192.9.bfu.1 $48$ $4$ $4$ $9$ $1$ $1^{4}\cdot2^{2}$
240.96.1.oq.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.oy.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.pw.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.qe.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.to.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.tw.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.uu.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.vc.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.fn.2 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.ill.1 $240$ $6$ $6$ $17$ $?$ not computed