↑

Properties

Label 24.48.1.dt.1
Level $24$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $24$ $\SL_2$-level: $8$ Newform level: $64$
Index: $48$ $\PSL_2$-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 $4^{4}\cdot8^{4}$ Cusp orbits $4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2 \le \gamma \le 4$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 24.48.1.421

Level structure

$\GL_2(\Z/24\Z)$-generators: $\begin{bmatrix}5&8\\8&21\end{bmatrix}$, $\begin{bmatrix}11&19\\2&17\end{bmatrix}$, $\begin{bmatrix}23&4\\20&17\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-24.dt.1.1, 48.96.1-24.dt.1.2, 48.96.1-24.dt.1.3, 48.96.1-24.dt.1.4, 48.96.1-24.dt.1.5, 48.96.1-24.dt.1.6, 48.96.1-24.dt.1.7, 48.96.1-24.dt.1.8, 48.96.1-24.dt.1.9, 48.96.1-24.dt.1.10, 48.96.1-24.dt.1.11, 48.96.1-24.dt.1.12, 48.96.1-24.dt.1.13, 48.96.1-24.dt.1.14, 48.96.1-24.dt.1.15, 48.96.1-24.dt.1.16, 240.96.1-24.dt.1.1, 240.96.1-24.dt.1.2, 240.96.1-24.dt.1.3, 240.96.1-24.dt.1.4, 240.96.1-24.dt.1.5, 240.96.1-24.dt.1.6, 240.96.1-24.dt.1.7, 240.96.1-24.dt.1.8, 240.96.1-24.dt.1.9, 240.96.1-24.dt.1.10, 240.96.1-24.dt.1.11, 240.96.1-24.dt.1.12, 240.96.1-24.dt.1.13, 240.96.1-24.dt.1.14, 240.96.1-24.dt.1.15, 240.96.1-24.dt.1.16
Cyclic 24-isogeny field degree: $16$
Cyclic 24-torsion field degree: $128$
Full 24-torsion field degree: $1536$

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 $ $=$ $ 6 x^{2} - 9 x y + y^{2} - y z + z^{2} - w^{2} $
$=$ $12 x^{2} + 6 x y + y^{2} + 2 y z - 2 z^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 4 x^{3} y - 6 x^{2} y^{2} - 4 x^{2} z^{2} + 20 x y^{3} + 8 x y z^{2} + 61 y^{4} - 28 y^{2} z^{2} + 4 z^{4} $
 Copy content Toggle raw display

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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{3^2}\cdot\frac{2071517651715305088xz^{11}+162352483625984482368xz^{9}w^{2}-365941491495701806656xz^{7}w^{4}-94410834654628527552xz^{5}w^{6}+108678660412265239680xz^{3}w^{8}+4794299741247104448xzw^{10}-10331484645986972028y^{2}z^{10}+125667451587346815303y^{2}z^{8}w^{2}-85877046251119745496y^{2}z^{6}w^{4}-272539066719525205896y^{2}z^{4}w^{6}-31033385495363093148y^{2}z^{2}w^{8}-87776447612042304y^{2}w^{10}+11482568136527601888yz^{11}-32472566997846621732yz^{9}w^{2}-120150307779543006336yz^{7}w^{4}+206669785823165264256yz^{5}w^{6}+76464504590304260304yz^{3}w^{8}+1643427832089838176yzw^{10}-3711765075093504360z^{12}-36993525878856675924z^{10}w^{2}+126803909401025065938z^{8}w^{4}-25554834768095911440z^{6}w^{6}-20748301272977958576z^{4}w^{8}+3097623402261804600z^{2}w^{10}+32249774616256456w^{12}}{34099055995313664xz^{11}-206775311128084224xz^{9}w^{2}+222360524373869568xz^{7}w^{4}-85064451616845696xz^{5}w^{6}+9652957800251712xz^{3}w^{8}+502429903003584xzw^{10}-170065590880443984y^{2}z^{10}+105173744572666656y^{2}z^{8}w^{2}+4064411469422928y^{2}z^{6}w^{4}-14389217657561302y^{2}z^{4}w^{6}+1890019436634408y^{2}z^{2}w^{8}+219222571291259y^{2}w^{10}+189013467267944064yz^{11}-228251507616250848yz^{9}w^{2}+116319444978093216yz^{7}w^{4}-23210741638551896yz^{5}w^{6}-930727168596976yz^{3}w^{8}+659167204608236yzw^{10}-61099013581786080z^{12}+142304928970137408z^{10}w^{2}-111326581834561320z^{8}w^{4}+33817970267381048z^{6}w^{6}-2368466342869420z^{4}w^{8}-231545605160556z^{2}w^{10}-53802473566302w^{12}}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.1.k.1 $8$ $2$ $2$ $1$ $0$ dimension zero
24.24.0.em.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.em.2 $24$ $2$ $2$ $0$ $0$ full Jacobian

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
24.144.9.xh.1 $24$ $3$ $3$ $9$ $2$ $1^{4}\cdot2^{2}$
24.192.9.iw.1 $24$ $4$ $4$ $9$ $0$ $1^{4}\cdot2^{2}$
48.96.5.du.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.du.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.eo.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.eo.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.96.5.fk.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.fk.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.fv.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.96.5.fv.2 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
120.240.17.od.1 $120$ $5$ $5$ $17$ $?$ not computed
120.288.17.qsb.1 $120$ $6$ $6$ $17$ $?$ not computed
240.96.5.mj.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.mj.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.ng.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.ng.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.om.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.om.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.pl.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.pl.2 $240$ $2$ $2$ $5$ $?$ not computed