Invariants
| Level: | $248$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
Level structure
| $\GL_2(\Z/248\Z)$-generators: | $\begin{bmatrix}57&104\\28&175\end{bmatrix}$, $\begin{bmatrix}103&80\\104&29\end{bmatrix}$, $\begin{bmatrix}105&160\\68&237\end{bmatrix}$, $\begin{bmatrix}143&176\\64&107\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.96.1.g.1 for the level structure with $-I$) |
| Cyclic 248-isogeny field degree: | $32$ |
| Cyclic 248-torsion field degree: | $3840$ |
| Full 248-torsion field degree: | $7142400$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 96 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{174228x^{2}y^{28}z^{2}+3016602930x^{2}y^{24}z^{6}+161742749031x^{2}y^{20}z^{10}+574925030955x^{2}y^{16}z^{14}+272387551224x^{2}y^{12}z^{18}+34999202565x^{2}y^{8}z^{22}+1509948741x^{2}y^{4}z^{26}+16777215x^{2}z^{30}+712xy^{30}z+324760626xy^{26}z^{5}+67075202952xy^{22}z^{9}+588642243017xy^{18}z^{13}+597507073344xy^{14}z^{17}+126615719865xy^{10}z^{21}+8808039100xy^{6}z^{25}+184549377xy^{2}z^{29}+y^{32}+15867528y^{28}z^{4}+15769425308y^{24}z^{8}+271032482654y^{20}z^{12}+399504719004y^{16}z^{16}+97855028624y^{12}z^{20}+7449250366y^{8}z^{24}+167772858y^{4}z^{28}+z^{32}}{zy^{4}(31x^{2}y^{24}z-998x^{2}y^{20}z^{5}+194106x^{2}y^{16}z^{9}+1831018x^{2}y^{12}z^{13}-13958543x^{2}y^{8}z^{17}-27787305x^{2}y^{4}z^{21}-1048575x^{2}z^{25}-xy^{26}+1760xy^{22}z^{4}+54206xy^{18}z^{8}-1438316xy^{14}z^{12}+5176757xy^{10}z^{16}-47710168xy^{6}z^{20}-9437185xy^{2}z^{24}-380y^{24}z^{3}-26676y^{20}z^{7}-598511y^{16}z^{11}+6033852y^{12}z^{15}-27263640y^{8}z^{19}-8388566y^{4}z^{23}-z^{27})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 248.96.0-8.b.1.5 | $248$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 248.96.0-8.b.1.6 | $248$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 248.96.0-8.c.1.6 | $248$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 248.96.0-8.c.1.9 | $248$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 248.96.0-8.k.1.3 | $248$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 248.96.0-8.k.1.5 | $248$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 248.96.0-8.l.1.2 | $248$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 248.96.0-8.l.1.3 | $248$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 248.96.1-8.h.1.9 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-8.h.1.10 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-8.i.2.3 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-8.i.2.6 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-8.k.2.4 | $248$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 248.96.1-8.k.2.5 | $248$ | $2$ | $2$ | $1$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 248.384.5-8.d.1.3 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.384.5-8.d.2.2 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.384.5-248.bb.2.6 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 248.384.5-248.bb.4.5 | $248$ | $2$ | $2$ | $5$ | $?$ | not computed |