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Properties

Label 56.192.1-56.bo.2.5
Level $56$
Index $192$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $8$ Newform level: $64$
Index: $192$ $\PSL_2$-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 $4^{8}\cdot8^{8}$ 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: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.192.1.1139

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}17&4\\18&27\end{bmatrix}$, $\begin{bmatrix}27&12\\38&45\end{bmatrix}$, $\begin{bmatrix}31&30\\4&13\end{bmatrix}$
Contains $-I$: no $\quad$ (see 56.96.1.bo.2 for the level structure with $-I$)
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $192$
Full 56-torsion field degree: $16128$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

$ 0 $ $=$ $ 4 y^{2} + y z + 2 y w + 2 z w + 2 w^{2} $
$=$ $14 x^{2} - 6 y^{2} + 2 y z + 4 y w + z^{2} + 4 z w + 4 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 20 x^{3} z + 7 x^{2} y^{2} - 18 x^{2} z^{2} + 28 x y^{2} z + 4 x z^{3} + 28 y^{2} z^{2} + 2 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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^2\cdot7^4}\cdot\frac{281350500975999yz^{23}+3927623872926978yz^{22}w-8965985762452068yz^{21}w^{2}-612096725684054520yz^{20}w^{3}-6583010327445993840yz^{19}w^{4}-37756819346982105888yz^{18}w^{5}-136882737550492645824yz^{17}w^{6}-309182839400300148864yz^{16}w^{7}-185580681605842586112yz^{15}w^{8}+1936132664062391424000yz^{14}w^{9}+10415963415614828746752yz^{13}w^{10}+32026789848749642698752yz^{12}w^{11}+72265593689317736128512yz^{11}w^{12}+130407077938137434800128yz^{10}w^{13}+197473018198311562936320yz^{9}w^{14}+256671081086445832568832yz^{8}w^{15}+285416828088137924935680yz^{7}w^{16}+265223281844414044569600yz^{6}w^{17}+199110157287377704058880yz^{5}w^{18}+116179047569908912619520yz^{4}w^{19}+50311320791111156367360yz^{3}w^{20}+15146457074217548513280yz^{2}w^{21}+2824064376316184494080yzw^{22}+245570815331842129920yw^{23}+70368744177664z^{24}+562701001951998z^{23}w-11849743284177162z^{22}w^{2}-238815111724362096z^{21}w^{3}-1904400413787546024z^{20}w^{4}-8100196924848266784z^{19}w^{5}-10681390895198834272z^{18}w^{6}+90168444301424767488z^{17}w^{7}+707249685421677084288z^{16}w^{8}+2887830452433469914112z^{15}w^{9}+8412484247535400295424z^{14}w^{10}+18879266075014107291648z^{13}w^{11}+32647000134289202049024z^{12}w^{12}+40516290683848668659712z^{11}w^{13}+26544302657484371214336z^{10}w^{14}-19198180992410376470528z^{9}w^{15}-84015615650936286019584z^{8}w^{16}-134431572879065273204736z^{7}w^{17}-142540734408506387595264z^{6}w^{18}-110018658505245858988032z^{5}w^{19}-62906941233139397689344z^{4}w^{20}-26192507692358127583232z^{3}w^{21}-7545903496790821306368z^{2}w^{22}-1347330374047610437632zw^{23}-112277531170634203136w^{24}}{z^{4}(6561yz^{19}+1124118yz^{18}w+54887868yz^{17}w^{2}+467911080yz^{16}w^{3}-9866308032yz^{15}w^{4}-273873979008yz^{14}w^{5}-3281877220608yz^{13}w^{6}-25245722402304yz^{12}w^{7}-138034564674048yz^{11}w^{8}-558854637014016yz^{10}w^{9}-1708870629193728yz^{9}w^{10}-3986255602667520yz^{8}w^{11}-7119370209509376yz^{7}w^{12}-9713366558343168yz^{6}w^{13}-10033332357758976yz^{5}w^{14}-7705824848117760yz^{4}w^{15}-4261801500278784yz^{3}w^{16}-1603614176575488yz^{2}w^{17}-367408588259328yzw^{18}-38674588237824yw^{19}+13122z^{19}w+2340090z^{18}w^{2}+126799344z^{17}w^{3}+1947726648z^{16}w^{4}+2010137472z^{15}w^{5}-310168317312z^{14}w^{6}-4794497777664z^{13}w^{7}-39202795824640z^{12}w^{8}-211079111914496z^{11}w^{9}-809731435312128z^{10}w^{10}-2302201177808896z^{9}w^{11}-4956010893783040z^{8}w^{12}-8160358967312384z^{7}w^{13}-10290790640091136z^{6}w^{14}-9867599070887936z^{5}w^{15}-7070500671324160z^{4}w^{16}-3666888919744512z^{3}w^{17}-1300072850915328z^{2}w^{18}-281886552227840zw^{19}-28188655222784w^{20})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 56.96.1.bo.2 :

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle x$
$\displaystyle Z$ $=$ $\displaystyle w$

Equation of the image curve:

$0$ $=$ $ X^{4}+7X^{2}Y^{2}-20X^{3}Z+28XY^{2}Z-18X^{2}Z^{2}+28Y^{2}Z^{2}+4XZ^{3}+2Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.1-8.n.1.3 $8$ $2$ $2$ $1$ $0$ dimension zero
56.96.0-56.m.2.3 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.m.2.12 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.n.1.3 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.n.1.15 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.u.2.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.u.2.14 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.v.1.7 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.v.1.12 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.1-8.n.1.8 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1-56.x.1.6 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1-56.x.1.7 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1-56.ba.1.8 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1-56.ba.1.11 $56$ $2$ $2$ $1$ $0$ 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
56.1536.49-56.hr.2.8 $56$ $8$ $8$ $49$ $8$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.4032.145-56.ti.1.3 $56$ $21$ $21$ $145$ $23$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.5376.193-56.uc.2.6 $56$ $28$ $28$ $193$ $31$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$