Invariants
Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (none of which are rational) | Cusp widths | $3^{4}\cdot12^{2}$ | Cusp orbits | $2\cdot4$ | ||
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: | 12K1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.112 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&3\\18&23\end{bmatrix}$, $\begin{bmatrix}1&11\\14&17\end{bmatrix}$, $\begin{bmatrix}17&21\\0&11\end{bmatrix}$, $\begin{bmatrix}21&4\\20&15\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 24-isogeny field degree: | $16$ |
Cyclic 24-torsion field degree: | $128$ |
Full 24-torsion field degree: | $2048$ |
Jacobian
Conductor: | $2^{6}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 576.2.a.f |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - x y - x z + y^{2} + 2 y z + z^{2} + 2 w^{2} $ |
$=$ | $x y + x z + 4 y^{2} - 4 y z + 4 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 7 x^{4} - 8 x^{3} z + 6 x^{2} y^{2} + 18 x^{2} z^{2} + 12 x y^{2} z - 8 x z^{3} + 6 y^{2} z^{2} + 7 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^3\cdot3^3\,\frac{367940880xz^{8}-1128996792xz^{6}w^{2}+1130450580xz^{4}w^{4}+1263869250xz^{2}w^{6}+158120256xw^{8}+1539134784y^{2}z^{7}-1445087952y^{2}z^{5}w^{2}-1942682616y^{2}z^{3}w^{4}+5470435656y^{2}zw^{6}-1232196624yz^{8}+3664034352yz^{6}w^{2}+368443404yz^{4}w^{4}-5040985152yz^{2}w^{6}+421836149yw^{8}+1037477808z^{9}-1497343968z^{7}w^{2}+608329764z^{5}w^{4}+4655010192z^{3}w^{6}+421836149zw^{8}}{45992610xz^{8}-23995440xz^{6}w^{2}-3429216xz^{4}w^{4}+16464xz^{2}w^{6}+192391848y^{2}z^{7}-8639379y^{2}z^{5}w^{2}-9374778y^{2}z^{3}w^{4}+333396y^{2}zw^{6}-154024578yz^{8}+78540111yz^{6}w^{2}+18949140yz^{4}w^{4}+958440yz^{2}w^{6}-2744yw^{8}+129684726z^{9}-35210862z^{7}w^{2}+509166z^{5}w^{4}+822612z^{3}w^{6}-2744zw^{8}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
12.18.0.f.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.0.c.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.18.1.h.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 |
---|---|---|---|---|---|---|
24.72.3.bs.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.dy.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.hc.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.hi.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.72.3.ut.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.uz.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.wc.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.72.3.wi.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
72.108.7.fo.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
72.324.19.fq.1 | $72$ | $9$ | $9$ | $19$ | $?$ | not computed |
120.72.3.dgq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dgs.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dhe.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dhg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dxm.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dxo.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dya.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.72.3.dyc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.180.13.qa.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
120.216.13.oq.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
168.72.3.dcm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dco.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dda.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.ddc.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dqa.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dqc.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dqo.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.72.3.dqq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.288.19.bgq.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
264.72.3.dcm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dco.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dda.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.ddc.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dqa.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dqc.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dqo.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.72.3.dqq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dcm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dco.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dda.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.ddc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dqa.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dqc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dqo.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.72.3.dqq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |