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Properties

Label 24.96.1.cy.4
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $24$ $\SL_2$-level: $12$ Newform level: $192$
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 $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $2^{6}\cdot4$
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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1611

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&2\\0&19\end{bmatrix}$, $\begin{bmatrix}7&19\\12&1\end{bmatrix}$, $\begin{bmatrix}11&12\\0&5\end{bmatrix}$, $\begin{bmatrix}13&5\\0&23\end{bmatrix}$, $\begin{bmatrix}13&8\\12&19\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035865
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cy.4.1, 24.192.1-24.cy.4.2, 24.192.1-24.cy.4.3, 24.192.1-24.cy.4.4, 24.192.1-24.cy.4.5, 24.192.1-24.cy.4.6, 24.192.1-24.cy.4.7, 24.192.1-24.cy.4.8, 24.192.1-24.cy.4.9, 24.192.1-24.cy.4.10, 24.192.1-24.cy.4.11, 24.192.1-24.cy.4.12, 24.192.1-24.cy.4.13, 24.192.1-24.cy.4.14, 24.192.1-24.cy.4.15, 24.192.1-24.cy.4.16, 120.192.1-24.cy.4.1, 120.192.1-24.cy.4.2, 120.192.1-24.cy.4.3, 120.192.1-24.cy.4.4, 120.192.1-24.cy.4.5, 120.192.1-24.cy.4.6, 120.192.1-24.cy.4.7, 120.192.1-24.cy.4.8, 120.192.1-24.cy.4.9, 120.192.1-24.cy.4.10, 120.192.1-24.cy.4.11, 120.192.1-24.cy.4.12, 120.192.1-24.cy.4.13, 120.192.1-24.cy.4.14, 120.192.1-24.cy.4.15, 120.192.1-24.cy.4.16, 168.192.1-24.cy.4.1, 168.192.1-24.cy.4.2, 168.192.1-24.cy.4.3, 168.192.1-24.cy.4.4, 168.192.1-24.cy.4.5, 168.192.1-24.cy.4.6, 168.192.1-24.cy.4.7, 168.192.1-24.cy.4.8, 168.192.1-24.cy.4.9, 168.192.1-24.cy.4.10, 168.192.1-24.cy.4.11, 168.192.1-24.cy.4.12, 168.192.1-24.cy.4.13, 168.192.1-24.cy.4.14, 168.192.1-24.cy.4.15, 168.192.1-24.cy.4.16, 264.192.1-24.cy.4.1, 264.192.1-24.cy.4.2, 264.192.1-24.cy.4.3, 264.192.1-24.cy.4.4, 264.192.1-24.cy.4.5, 264.192.1-24.cy.4.6, 264.192.1-24.cy.4.7, 264.192.1-24.cy.4.8, 264.192.1-24.cy.4.9, 264.192.1-24.cy.4.10, 264.192.1-24.cy.4.11, 264.192.1-24.cy.4.12, 264.192.1-24.cy.4.13, 264.192.1-24.cy.4.14, 264.192.1-24.cy.4.15, 264.192.1-24.cy.4.16, 312.192.1-24.cy.4.1, 312.192.1-24.cy.4.2, 312.192.1-24.cy.4.3, 312.192.1-24.cy.4.4, 312.192.1-24.cy.4.5, 312.192.1-24.cy.4.6, 312.192.1-24.cy.4.7, 312.192.1-24.cy.4.8, 312.192.1-24.cy.4.9, 312.192.1-24.cy.4.10, 312.192.1-24.cy.4.11, 312.192.1-24.cy.4.12, 312.192.1-24.cy.4.13, 312.192.1-24.cy.4.14, 312.192.1-24.cy.4.15, 312.192.1-24.cy.4.16
Cyclic 24-isogeny field degree: $2$
Cyclic 24-torsion field degree: $16$
Full 24-torsion field degree: $768$

Jacobian

Conductor: $2^{6}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 192.2.a.d

Models

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

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

$ 0 $ $=$ $ x^{4} + 4 x^{3} z - 2 x^{2} y^{2} + 5 x^{2} z^{2} - 8 x y^{2} z + 4 x z^{3} - 2 y^{2} z^{2} + 4 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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}y$
$\displaystyle Z$ $=$ $\displaystyle w$

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}{2^3}\cdot\frac{1536411240xy^{23}-299812890672xy^{21}w^{2}+21683912613984xy^{19}w^{4}-728462294685504xy^{17}w^{6}+11688611600058624xy^{15}w^{8}-83068558973185536xy^{13}w^{10}+183198033111149568xy^{11}w^{12}-1501266634734938112xy^{9}w^{14}-118137637404056844288xy^{7}w^{16}-9094578824583091187712xy^{5}w^{18}-692523049000938157449216xy^{3}w^{20}-52255301617417543761936384xyw^{22}+1124731089y^{24}-222330472488y^{22}w^{2}+16431419260872y^{20}w^{4}-573750086257440y^{18}w^{6}+9925963347413232y^{16}w^{8}-83124405356979456y^{14}w^{10}+299585254488383232y^{12}w^{12}-1540978826975032320y^{10}w^{14}-83310742038725427456y^{8}w^{16}-6439727604053349992448y^{6}w^{18}-490159392751184868882432y^{4}w^{20}-36974066661299616766599168y^{2}w^{22}-10900022202368z^{24}+18879018663936z^{23}w+508394732568576z^{22}w^{2}+709219757228032z^{21}w^{3}-10764943932039168z^{20}w^{4}-30183024497491968z^{19}w^{5}+36015146896900096z^{18}w^{6}+429713093451939840z^{17}w^{7}+296857963336568832z^{16}w^{8}+530538456318410752z^{15}w^{9}-8075584789128314880z^{14}w^{10}+16556998558673338368z^{13}w^{11}-107104426073444237312z^{12}w^{12}+451482365449811853312z^{11}w^{13}-1881652363924849852416z^{10}w^{14}+8641836321335725981696z^{9}w^{15}-32906663261282472062976z^{8}w^{16}+151786171570697525624832z^{7}w^{17}-480326584040727178526720z^{6}w^{18}+2351106788726050532917248z^{5}w^{19}-3882910081433901804724224z^{4}w^{20}+27201116774833647098429440z^{3}w^{21}+93628141169228763909439488z^{2}w^{22}+24993589316675847134085120zw^{23}+97061848456802002720915456w^{24}}{w^{4}(1536411240xy^{17}w^{2}-162197735256xy^{15}w^{4}+4863284632944xy^{13}w^{6}-57411850361184xy^{11}w^{8}+98090204955840xy^{9}w^{10}-11095465503745152xy^{7}w^{12}-704632822717526784xy^{5}w^{14}-47027038552925372928xy^{3}w^{16}-3162046733561196463104xyw^{18}+1124731089y^{18}w^{2}-121589186832y^{16}w^{4}+3862395299952y^{14}w^{6}-51180922128960y^{12}w^{8}+182567448427104y^{10}w^{10}-8364578841158400y^{8}w^{12}-494963977627314432y^{6}w^{14}-33126192476876049408y^{4}w^{16}-2227946075958982143744y^{2}w^{18}+2097152z^{20}-58720256z^{19}w-1361544376832z^{18}w^{2}+2348032618496z^{17}w^{3}+19002580762624z^{16}w^{4}+115857510600704z^{15}w^{5}-350758099025920z^{14}w^{6}+519813655293952z^{13}w^{7}-6149294552377344z^{12}w^{8}+26320730040782848z^{11}w^{9}-108131962300503040z^{10}w^{10}+532874781910667264z^{9}w^{11}-1952636119550953472z^{8}w^{12}+9605058572262748160z^{7}w^{13}-28387899765977255936z^{6}w^{14}+148312847582296625152z^{5}w^{15}-213923944262668072960z^{4}w^{16}+1670963727992712228864z^{3}w^{17}+5686927168618402223616z^{2}w^{18}+1531748978626572435456zw^{19}+5874313495444669605888w^{20})}$

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
12.48.0.c.2 $12$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bu.4 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.48.1.es.1 $24$ $2$ $2$ $1$ $0$ 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
24.192.5.dt.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.du.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.dz.2 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.ea.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.el.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.em.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.er.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.es.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.288.9.n.2 $24$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
72.288.9.k.2 $72$ $3$ $3$ $9$ $?$ not computed
72.288.17.bv.1 $72$ $3$ $3$ $17$ $?$ not computed
72.288.17.ch.1 $72$ $3$ $3$ $17$ $?$ not computed
120.192.5.ss.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.st.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.te.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.tf.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ui.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.uj.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.uu.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.uv.3 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.ss.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.st.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.te.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.tf.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.ui.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.uj.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.uu.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.uv.2 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.ss.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.st.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.te.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.tf.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.ui.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.uj.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.uu.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.uv.3 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.ss.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.st.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.te.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.tf.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.ui.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.uj.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.uu.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.uv.3 $312$ $2$ $2$ $5$ $?$ not computed