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Properties

Label 24.96.1.cy.3
Level $24$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $2$

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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: $16$ (of which $2$ are rational) Cusp widths $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $1^{2}\cdot2^{3}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.96.1.1636

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}1&21\\12&1\end{bmatrix}$, $\begin{bmatrix}13&9\\0&1\end{bmatrix}$, $\begin{bmatrix}17&7\\12&23\end{bmatrix}$, $\begin{bmatrix}17&9\\12&1\end{bmatrix}$, $\begin{bmatrix}23&19\\12&1\end{bmatrix}$
$\GL_2(\Z/24\Z)$-subgroup: Group 768.1035865
Contains $-I$: yes
Quadratic refinements: 24.192.1-24.cy.3.1, 24.192.1-24.cy.3.2, 24.192.1-24.cy.3.3, 24.192.1-24.cy.3.4, 24.192.1-24.cy.3.5, 24.192.1-24.cy.3.6, 24.192.1-24.cy.3.7, 24.192.1-24.cy.3.8, 24.192.1-24.cy.3.9, 24.192.1-24.cy.3.10, 24.192.1-24.cy.3.11, 24.192.1-24.cy.3.12, 24.192.1-24.cy.3.13, 24.192.1-24.cy.3.14, 24.192.1-24.cy.3.15, 24.192.1-24.cy.3.16, 120.192.1-24.cy.3.1, 120.192.1-24.cy.3.2, 120.192.1-24.cy.3.3, 120.192.1-24.cy.3.4, 120.192.1-24.cy.3.5, 120.192.1-24.cy.3.6, 120.192.1-24.cy.3.7, 120.192.1-24.cy.3.8, 120.192.1-24.cy.3.9, 120.192.1-24.cy.3.10, 120.192.1-24.cy.3.11, 120.192.1-24.cy.3.12, 120.192.1-24.cy.3.13, 120.192.1-24.cy.3.14, 120.192.1-24.cy.3.15, 120.192.1-24.cy.3.16, 168.192.1-24.cy.3.1, 168.192.1-24.cy.3.2, 168.192.1-24.cy.3.3, 168.192.1-24.cy.3.4, 168.192.1-24.cy.3.5, 168.192.1-24.cy.3.6, 168.192.1-24.cy.3.7, 168.192.1-24.cy.3.8, 168.192.1-24.cy.3.9, 168.192.1-24.cy.3.10, 168.192.1-24.cy.3.11, 168.192.1-24.cy.3.12, 168.192.1-24.cy.3.13, 168.192.1-24.cy.3.14, 168.192.1-24.cy.3.15, 168.192.1-24.cy.3.16, 264.192.1-24.cy.3.1, 264.192.1-24.cy.3.2, 264.192.1-24.cy.3.3, 264.192.1-24.cy.3.4, 264.192.1-24.cy.3.5, 264.192.1-24.cy.3.6, 264.192.1-24.cy.3.7, 264.192.1-24.cy.3.8, 264.192.1-24.cy.3.9, 264.192.1-24.cy.3.10, 264.192.1-24.cy.3.11, 264.192.1-24.cy.3.12, 264.192.1-24.cy.3.13, 264.192.1-24.cy.3.14, 264.192.1-24.cy.3.15, 264.192.1-24.cy.3.16, 312.192.1-24.cy.3.1, 312.192.1-24.cy.3.2, 312.192.1-24.cy.3.3, 312.192.1-24.cy.3.4, 312.192.1-24.cy.3.5, 312.192.1-24.cy.3.6, 312.192.1-24.cy.3.7, 312.192.1-24.cy.3.8, 312.192.1-24.cy.3.9, 312.192.1-24.cy.3.10, 312.192.1-24.cy.3.11, 312.192.1-24.cy.3.12, 312.192.1-24.cy.3.13, 312.192.1-24.cy.3.14, 312.192.1-24.cy.3.15, 312.192.1-24.cy.3.16
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

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} + 3x + 3 $
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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
$(0:1:0)$, $(-1:0:1)$

Maps to other modular curves

$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2}\cdot\frac{1424x^{2}y^{30}+4713533568x^{2}y^{28}z^{2}+3114827842560x^{2}y^{26}z^{4}-175678795669504x^{2}y^{24}z^{6}+4382066161287168x^{2}y^{22}z^{8}-61197925909790720x^{2}y^{20}z^{10}+556586302665916416x^{2}y^{18}z^{12}-4387841166129758208x^{2}y^{16}z^{14}+26539718714490617856x^{2}y^{14}z^{16}-136466211611453947904x^{2}y^{12}z^{18}+682640218614361227264x^{2}y^{10}z^{20}-2571063788570387939328x^{2}y^{8}z^{22}+9089333106487882416128x^{2}y^{6}z^{24}-28540031775227026341888x^{2}y^{4}z^{26}+47709019379019286577152x^{2}y^{2}z^{28}-27262293209257745580032x^{2}z^{30}+696960xy^{30}z+80329098240xy^{28}z^{3}+7035275231232xy^{26}z^{5}-511465450242048xy^{24}z^{7}+12865298775932928xy^{22}z^{9}-181360668383903744xy^{20}z^{11}+1919706570103455744xy^{18}z^{13}-15872319312246079488xy^{16}z^{15}+99288102075887517696xy^{14}z^{17}-553491471043563356160xy^{12}z^{19}+2463534633442426748928xy^{10}z^{21}-8715702302110925193216xy^{8}z^{23}+27475325513483717443584xy^{6}z^{25}-57080425017100707299328xy^{4}z^{27}+47709013256938543120384xy^{2}z^{29}+y^{32}+126304624y^{30}z^{2}+696824764800y^{28}z^{4}-19415084180480y^{26}z^{6}+64679912775680y^{24}z^{8}+4585348482138112y^{22}z^{10}-125485464999690240y^{20}z^{12}+1739076433109581824y^{18}z^{14}-14662208592656990208y^{16}z^{16}+100083949683850870784y^{14}z^{18}-534480564287319310336y^{12}z^{20}+2144045432344890310656y^{10}z^{22}-7156050200114235441152y^{8}z^{24}+17115589712238331559936y^{6}z^{26}-16612416994345735946240y^{4}z^{28}-13631128238386642419712y^{2}z^{30}+27262293490732722290688z^{32}}{y^{2}(y^{2}-8z^{2})^{2}(x^{2}y^{24}-22352x^{2}y^{22}z^{2}-7456896x^{2}y^{20}z^{4}-1186939904x^{2}y^{18}z^{6}-22356955136x^{2}y^{16}z^{8}+2941123952640x^{2}y^{14}z^{10}-27150550827008x^{2}y^{12}z^{12}-234300238725120x^{2}y^{10}z^{14}+3932702893932544x^{2}y^{8}z^{16}-15408019349176320x^{2}y^{6}z^{18}+15407832249663488x^{2}y^{4}z^{20}-360777252864x^{2}y^{2}z^{22}+68719476736x^{2}z^{24}-62xy^{24}z+126064xy^{22}z^{3}+29352320xy^{20}z^{5}-614542336xy^{18}z^{7}-309903802368xy^{16}z^{9}+1983838027776xy^{14}z^{11}+129795701604352xy^{12}z^{13}-2106281112371200xy^{10}z^{15}+11234061045465088xy^{8}z^{17}-19258585305317376xy^{6}z^{19}-88046829568xy^{4}z^{21}+17179869184xy^{2}z^{23}+1625y^{24}z^{2}+114208y^{22}z^{4}+104882944y^{20}z^{6}+20011343872y^{18}z^{8}+84244926464y^{16}z^{10}-21866665476096y^{14}z^{12}+366100751581184y^{12}z^{14}-2234439320993792y^{10}z^{16}+4415848261353472y^{8}z^{18}+3848548932124672y^{6}z^{20}-15404497207558144y^{4}z^{22}-1099511627776y^{2}z^{24}+206158430208z^{26})}$

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