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Properties

Label 60.96.1-60.bo.1.6
Level $60$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $12$ Newform level: $24$
Index: $96$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ Cusp orbits $2^{4}$
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: 12P1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.96.1.621

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}7&2\\9&25\end{bmatrix}$, $\begin{bmatrix}23&0\\24&7\end{bmatrix}$, $\begin{bmatrix}23&12\\24&19\end{bmatrix}$, $\begin{bmatrix}37&22\\15&41\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.48.1.bo.1 for the level structure with $-I$)
Cyclic 60-isogeny field degree: $12$
Cyclic 60-torsion field degree: $192$
Full 60-torsion field degree: $23040$

Jacobian

Conductor: $2^{3}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 24.2.a.a

Models

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

$ 0 $ $=$ $ 5 x y + 3 y^{2} - 2 y z + 2 z^{2} $
$=$ $11 x^{2} + 3 x y - x w - 3 y^{2} + 6 y z + y w - 6 z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 21 x^{4} - 8 x^{3} y - 48 x^{3} z + x^{2} y^{2} + 2 x^{2} y z + 4 x^{2} z^{2} - 2 x y z^{2} + \cdots - 44 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 to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{11^2}\cdot\frac{7473600624187097731531895904000xz^{11}-1001611539966672757734068160000xz^{10}w+1219265454827229865012329863040xz^{9}w^{2}+145007396113879170640524238560xz^{8}w^{3}+37162651467661753365195494400xz^{7}w^{4}+24875906179590719836466637600xz^{6}w^{5}+24479662709578882378169520xz^{5}w^{6}+894567985385641315326838920xz^{4}w^{7}+1484763844635347065854240xz^{3}w^{8}+13989191084457871929440200xz^{2}w^{9}+94707978555377061992880xw^{11}+1796598668534340400565621670720y^{2}z^{10}-728794893875343904105713612000y^{2}z^{9}w+328482426991153127267849612976y^{2}z^{8}w^{2}-7563442146038240567949600768y^{2}z^{7}w^{3}-11747482208489642215524960960y^{2}z^{6}w^{4}+8384786217829880392335245136y^{2}z^{5}w^{5}-1705206935916549280923841980y^{2}z^{4}w^{6}+292365131464594790918814168y^{2}z^{3}w^{7}-25333347735622897438355712y^{2}z^{2}w^{8}+2001022918641813840176370y^{2}zw^{9}-56342707056019416184685y^{2}w^{10}-1883292604493746203100199767680yz^{11}-829477350887571419563616407200yz^{10}w+752158412657164865046578121216yz^{9}w^{2}-461798905807853095394691380448yz^{8}w^{3}+106145489386588157939769240960yz^{7}w^{4}-7064996035223502421786241424yz^{6}w^{5}-5255518106296919848295739600yz^{5}w^{6}+2021515919693733786777773088yz^{4}w^{7}-394170741996931997870458752yz^{3}w^{8}+47722323178755950894224550yz^{2}w^{9}-3523972418566471364170720yzw^{10}+141529923990479826728120yw^{11}+5366707968590748164780513743680z^{12}-762208908781234632371843904000z^{11}w+1110248828153080818631698322944z^{10}w^{2}+80903093266200079707437708928z^{9}w^{3}+69927031597763409714066668400z^{8}w^{4}+18286720336157442339208477824z^{7}w^{5}+4055335889052431695905380880z^{6}w^{6}+630127383910708201349372112z^{5}w^{7}+191467392554438546006749332z^{4}w^{8}+5460697902452662674880240z^{3}w^{9}+3861418746892686606410320z^{2}w^{10}+26188141295979673364655w^{12}}{64324876642921877400xz^{11}+1277459134521678068400xz^{10}w+703997796165224647680xz^{9}w^{2}+5639717443806550035855xz^{8}w^{3}+5030999016765893498250xz^{7}w^{4}+1898083995416099845200xz^{6}w^{5}+2359289759449537183500xz^{5}w^{6}+1245434329184866335735xz^{4}w^{7}+161536997773869198360xz^{3}w^{8}+139828900371111100175xz^{2}w^{9}+39287015461100956032y^{2}z^{10}+125462675961826775550y^{2}z^{9}w+412649358873032378592y^{2}z^{8}w^{2}+1814979297799396636776y^{2}z^{7}w^{3}-1462857222029481691875y^{2}z^{6}w^{4}+2232730200923075467152y^{2}z^{5}w^{5}+382240425192800172075y^{2}z^{4}w^{6}-123959435968729037556y^{2}z^{3}w^{7}+60581355082166811492y^{2}z^{2}w^{8}+563033767830024480y^{2}zw^{9}-197584283358089584y^{2}w^{10}-89464625493230389248yz^{11}+328406650556687579250yz^{10}w-2521078568304899513328yz^{9}w^{2}+1573810950281631891291yz^{8}w^{3}-3031446375983075184090yz^{7}w^{4}-930672792136932640488yz^{6}w^{5}-852532398090605647800yz^{5}w^{6}-405347128108540300221yz^{4}w^{7}+50651243076981513672yz^{3}w^{8}+15279325940482657225yz^{2}w^{9}-3480663696402806144yzw^{10}+319315162673372800yw^{11}+79624444979751868548z^{12}+610975054906045612200z^{11}w+1538242457523050946348z^{10}w^{2}+2554202471076900014184z^{9}w^{3}+4385423644368907375620z^{8}w^{4}+1466490430293236316738z^{7}w^{5}+2170231036759256434425z^{6}w^{6}+725666037037751442336z^{5}w^{7}+352401997358433571503z^{4}w^{8}+41246378532638149310z^{3}w^{9}+40193617735918533044z^{2}w^{10}}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.48.1.bo.1 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 5w$
$\displaystyle Z$ $=$ $\displaystyle z$

Equation of the image curve:

$0$ $=$ $ 21X^{4}-8X^{3}Y+X^{2}Y^{2}-48X^{3}Z+2X^{2}YZ+4X^{2}Z^{2}-2XYZ^{2}+88XZ^{3}-44Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.48.1-12.k.1.2 $12$ $2$ $2$ $1$ $0$ dimension zero
60.48.0-30.a.1.8 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.48.0-30.a.1.10 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.48.0-60.q.1.7 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.48.0-60.q.1.13 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.48.1-12.k.1.1 $60$ $2$ $2$ $1$ $0$ 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
60.192.1-60.p.1.4 $60$ $2$ $2$ $1$ $0$ dimension zero
60.192.1-60.p.2.4 $60$ $2$ $2$ $1$ $0$ dimension zero
60.192.1-60.p.3.3 $60$ $2$ $2$ $1$ $0$ dimension zero
60.192.1-60.p.4.2 $60$ $2$ $2$ $1$ $0$ dimension zero
60.288.5-60.jw.1.7 $60$ $3$ $3$ $5$ $0$ $1^{4}$
60.480.17-60.mk.1.6 $60$ $5$ $5$ $17$ $1$ $1^{16}$
60.576.17-60.ia.1.20 $60$ $6$ $6$ $17$ $0$ $1^{16}$
60.960.33-60.on.1.20 $60$ $10$ $10$ $33$ $2$ $1^{32}$
120.192.1-120.tn.1.5 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-120.tn.2.5 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-120.tn.3.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.1-120.tn.4.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.192.5-120.bq.1.23 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.bs.1.11 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.hg.1.23 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.hi.1.7 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.oy.1.10 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.oy.2.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.oz.1.6 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.oz.2.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.pa.1.6 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.pa.2.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.pb.1.10 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.pb.2.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.ua.1.6 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.uc.1.22 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.vg.1.10 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5-120.vi.1.22 $120$ $2$ $2$ $5$ $?$ not computed
180.288.5-180.bo.1.9 $180$ $3$ $3$ $5$ $?$ not computed
180.288.9-180.dw.1.6 $180$ $3$ $3$ $9$ $?$ not computed
180.288.9-180.es.1.8 $180$ $3$ $3$ $9$ $?$ not computed