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Properties

Label 48.96.1.q.1
Level $48$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $288$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $2^{8}\cdot4^{4}\cdot16^{4}$ Cusp orbits $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16M1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.96.1.512

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}7&32\\0&13\end{bmatrix}$, $\begin{bmatrix}13&22\\24&7\end{bmatrix}$, $\begin{bmatrix}17&4\\32&27\end{bmatrix}$, $\begin{bmatrix}25&46\\0&41\end{bmatrix}$, $\begin{bmatrix}47&32\\32&3\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.q.1.1, 48.192.1-48.q.1.2, 48.192.1-48.q.1.3, 48.192.1-48.q.1.4, 48.192.1-48.q.1.5, 48.192.1-48.q.1.6, 48.192.1-48.q.1.7, 48.192.1-48.q.1.8, 48.192.1-48.q.1.9, 48.192.1-48.q.1.10, 48.192.1-48.q.1.11, 48.192.1-48.q.1.12, 48.192.1-48.q.1.13, 48.192.1-48.q.1.14, 48.192.1-48.q.1.15, 48.192.1-48.q.1.16, 240.192.1-48.q.1.1, 240.192.1-48.q.1.2, 240.192.1-48.q.1.3, 240.192.1-48.q.1.4, 240.192.1-48.q.1.5, 240.192.1-48.q.1.6, 240.192.1-48.q.1.7, 240.192.1-48.q.1.8, 240.192.1-48.q.1.9, 240.192.1-48.q.1.10, 240.192.1-48.q.1.11, 240.192.1-48.q.1.12, 240.192.1-48.q.1.13, 240.192.1-48.q.1.14, 240.192.1-48.q.1.15, 240.192.1-48.q.1.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $64$
Full 48-torsion field degree: $12288$

Jacobian

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

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ x^{2} - x y - y z + z^{2} $
$=$ $12 x^{2} - 24 x z - 18 y^{2} + 24 y z + 12 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 4 x^{3} z + 6 x^{2} y^{2} + 6 x^{2} z^{2} + 12 x y^{2} z - 4 x z^{3} + 6 y^{2} z^{2} - 3 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{6}w$
$\displaystyle Z$ $=$ $\displaystyle z$

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{2}{3^2}\cdot\frac{858183596538964198073303040xz^{23}-230409809596269027544530944xz^{21}w^{2}+28555222366796168415936512xz^{19}w^{4}-2157706363160147924090880xz^{17}w^{6}+110165439667928929665024xz^{15}w^{8}-3972001214269456646144xz^{13}w^{10}+102479348617065136128xz^{11}w^{12}-1871713352464793600xz^{9}w^{14}+23302353607661568xz^{7}w^{16}-182441457543168xz^{5}w^{18}+759591149824xz^{3}w^{20}-1118942080xzw^{22}+188517116942399912678522880y^{2}z^{22}-48313462034853885955276800y^{2}z^{20}w^{2}+5714095552854090785488896y^{2}z^{18}w^{4}-411362471472909288210432y^{2}z^{16}w^{6}+19934513190772550664192y^{2}z^{14}w^{8}-677820752069913673728y^{2}z^{12}w^{10}+16323748091293433856y^{2}z^{10}w^{12}-273659240767832064y^{2}z^{8}w^{14}+3037578982313472y^{2}z^{6}w^{16}-20029794197376y^{2}z^{4}w^{18}+61473817152y^{2}z^{2}w^{20}-40398240y^{2}w^{22}-1538631550731643641617448960yz^{23}+405431381675647508236206080yz^{21}w^{2}-49328557350236182701670400yz^{19}w^{4}+3658524084375679647350784yz^{17}w^{6}-183166696361079877926912yz^{15}w^{8}+6464362818234340868096yz^{13}w^{10}-162782399073294483456yz^{11}w^{12}+2888348276041318400yz^{9}w^{14}-34668674543788032yz^{7}w^{16}+258164933488128yz^{5}w^{18}-995937344512yz^{3}w^{20}+1280535040yzw^{22}+251356155923199952240508928z^{24}-25592778124974477042253824z^{22}w^{2}-2256510285503310747140096z^{20}w^{4}+610928822411993886490624z^{18}w^{6}-56271958659868717547520z^{16}w^{8}+3080052035506145460224z^{14}w^{10}-112544634563491987456z^{12}w^{12}+2838266383359492096z^{10}w^{14}-49020831160893696z^{8}w^{16}+553487998927104z^{6}w^{18}-3661761083520z^{4}w^{20}+11041392352z^{2}w^{22}-6910187w^{24}}{w^{4}(609147530481037344768xz^{19}-133811385701113790464xz^{17}w^{2}+12145242339235143680xz^{15}w^{4}-588628231171940352xz^{13}w^{6}+16438377130858496xz^{11}w^{8}-266309848239616xz^{9}w^{10}+2396025681408xz^{7}w^{12}-10786681344xz^{5}w^{14}+19666176xz^{3}w^{16}-8880xzw^{18}+133811385701462999040y^{2}z^{18}-27761316959621406720y^{2}z^{16}w^{2}+2351141349962950656y^{2}z^{14}w^{4}-104586467975827968y^{2}z^{12}w^{6}+2618819101082112y^{2}z^{10}w^{8}-36741114893376y^{2}z^{8}w^{10}+270911134080y^{2}z^{6}w^{12}-907963488y^{2}z^{4}w^{14}+1012608y^{2}z^{2}w^{16}-150y^{2}w^{18}-1092136476657013751808yz^{19}+234466130542322483200yz^{17}w^{2}-20714985192742486016yz^{15}w^{4}+972250885036836864yz^{13}w^{6}-26118021236928512yz^{11}w^{8}+403403431737088yz^{9}w^{10}-3419016855552yz^{7}w^{12}+14267932032yz^{5}w^{14}-23624448yz^{3}w^{16}+9480yzw^{18}+178415180934775996416z^{20}-9456583983118565376z^{18}w^{2}-2529454098844292608z^{16}w^{4}+335007710607608832z^{14}w^{6}-17336655523069696z^{12}w^{8}+464232161992448z^{10}w^{10}-6697158143328z^{8}w^{12}+49546501440z^{6}w^{14}-163438320z^{4}w^{16}+176448z^{2}w^{18}-25w^{20})}$

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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
16.48.0.d.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bb.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.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
48.192.5.z.3 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.bh.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.cg.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.ci.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.dz.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.ef.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.el.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.eo.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.288.17.ex.2 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.io.2 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
240.192.5.bfy.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bfz.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bgc.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bgd.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bgw.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bgx.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bhe.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bhf.2 $240$ $2$ $2$ $5$ $?$ not computed