Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $24$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-8$) |
Other labels
| Cummins and Pauli (CP) label: | 12T1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.72.1.44 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}3&4\\4&3\end{bmatrix}$, $\begin{bmatrix}3&11\\2&3\end{bmatrix}$, $\begin{bmatrix}9&10\\8&3\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $C_2^3.D_4$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $4$ |
| Cyclic 12-torsion field degree: | $16$ |
| Full 12-torsion field degree: | $64$ |
Jacobian
| Conductor: | $2^{3}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 24.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} + x $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| 256.a1 | $-8$ | $8000$ | $= 2^{6} \cdot 5^{3}$ | $8.987$ | $(0:0:1)$, $(1:-1:1)$, $(1:1:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{17960x^{2}y^{30}+1240304x^{2}y^{29}z+18454560x^{2}y^{28}z^{2}+134052288x^{2}y^{27}z^{3}+588305112x^{2}y^{26}z^{4}+1602983088x^{2}y^{25}z^{5}+2371713040x^{2}y^{24}z^{6}-7534144344x^{2}y^{22}z^{8}-13715702864x^{2}y^{21}z^{9}-6810703360x^{2}y^{20}z^{10}-4296639552x^{2}y^{19}z^{11}-97226035560x^{2}y^{18}z^{12}-443432812432x^{2}y^{17}z^{13}-1156669570896x^{2}y^{16}z^{14}-2047505560064x^{2}y^{15}z^{15}-2359736471880x^{2}y^{14}z^{16}-1210991126832x^{2}y^{13}z^{17}+1099436432288x^{2}y^{12}z^{18}+2730294807872x^{2}y^{11}z^{19}+2261655650760x^{2}y^{10}z^{20}+504000495120x^{2}y^{9}z^{21}-674174824080x^{2}y^{8}z^{22}-662566473216x^{2}y^{7}z^{23}-208568536456x^{2}y^{6}z^{24}+33199332496x^{2}y^{5}z^{25}+47323053120x^{2}y^{4}z^{26}+14135786816x^{2}y^{3}z^{27}+930037960x^{2}y^{2}z^{28}-372010800x^{2}yz^{29}-66430000x^{2}z^{30}+2100xy^{31}+384832xy^{30}z+7258292xy^{29}z^{2}+58958640xy^{28}z^{3}+286067508xy^{27}z^{4}+928953504xy^{26}z^{5}+2218566900xy^{25}z^{6}+4674114320xy^{24}z^{7}+11389219716xy^{23}z^{8}+32698274304xy^{22}z^{9}+89615387780xy^{21}z^{10}+213372149936xy^{20}z^{11}+450476043588xy^{19}z^{12}+879125169888xy^{18}z^{13}+1538769766532xy^{17}z^{14}+2139245097744xy^{16}z^{15}+1861133388956xy^{15}z^{16}-6778081728xy^{14}z^{17}-2773174089444xy^{13}z^{18}-4245058056176xy^{12}z^{19}-2951917849828xy^{11}z^{20}-161763159456xy^{10}z^{21}+1543913716572xy^{9}z^{22}+1317796869552xy^{8}z^{23}+372216578220xy^{7}z^{24}-133597428608xy^{6}z^{25}-144933655892xy^{5}z^{26}-43663356912xy^{4}z^{27}-744420500xy^{3}z^{28}+2893680160xy^{2}z^{29}+744019500xyz^{30}+66430000xz^{31}+125y^{32}+98692y^{31}z+3181608y^{30}z^{2}+35677908y^{29}z^{3}+222553224y^{28}z^{4}+881373380y^{27}z^{5}+2176431544y^{26}z^{6}+2366759508y^{25}z^{7}-5191991284y^{24}z^{8}-32016921068y^{23}z^{9}-88805302808y^{22}z^{10}-184064594396y^{21}z^{11}-331582570440y^{20}z^{12}-529101719276y^{19}z^{13}-658335689160y^{18}z^{14}-404759174300y^{17}z^{15}+532971840366y^{16}z^{16}+1881795403404y^{15}z^{17}+2578626428984y^{14}z^{18}+1733818657660y^{13}z^{19}-62847224520y^{12}z^{20}-1213714987572y^{11}z^{21}-1029010798104y^{10}z^{22}-286899050116y^{9}z^{23}+125387475980y^{8}z^{24}+131541860060y^{7}z^{25}+39905813944y^{6}z^{26}+372332684y^{5}z^{27}-2893531768y^{4}z^{28}-744086692y^{3}z^{29}-66414040y^{2}z^{30}-2100yz^{31}+125z^{32}}{32x^{2}y^{30}-15888x^{2}y^{28}z^{2}+100672x^{2}y^{26}z^{4}-330928x^{2}y^{24}z^{6}+2856672x^{2}y^{22}z^{8}-25576848x^{2}y^{20}z^{10}+131956352x^{2}y^{18}z^{12}-372225456x^{2}y^{16}z^{14}+394470368x^{2}y^{14}z^{16}-33650608x^{2}y^{12}z^{18}-294737088x^{2}y^{10}z^{20}+305395312x^{2}y^{8}z^{22}-155248352x^{2}y^{6}z^{24}+45525840x^{2}y^{4}z^{26}-7440128x^{2}y^{2}z^{28}+531440x^{2}z^{30}-464xy^{30}z+53712xy^{28}z^{3}-384176xy^{26}z^{5}+330928xy^{24}z^{7}+7414704xy^{22}z^{9}-49511088xy^{20}z^{11}+199305040xy^{18}z^{13}-607028304xy^{16}z^{15}+1188948752xy^{14}z^{17}-1463279888xy^{12}z^{19}+1174512240xy^{10}z^{21}-633244784xy^{8}z^{23}+230712080xy^{6}z^{25}-55268880xy^{4}z^{27}+7971568xy^{2}z^{29}-531440xz^{31}-y^{32}+3584y^{30}z^{2}-90232y^{28}z^{4}+884000y^{26}z^{6}-4583580y^{24}z^{8}+13969728y^{22}z^{10}-29478856y^{20}z^{12}+94368736y^{18}z^{14}-422093126y^{16}z^{16}+828648320y^{14}z^{18}-862769224y^{12}z^{20}+541736672y^{10}z^{22}-215762588y^{8}z^{24}+54219072y^{6}z^{26}-7972600y^{4}z^{28}+531488y^{2}z^{30}-z^{32}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.36.0.c.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.36.0.d.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.36.1.bo.1 | $12$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 12.144.5.d.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 12.144.5.i.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 12.144.5.y.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 12.144.5.ba.1 | $12$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ca.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ck.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.5.hy.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.5.il.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.si.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.sj.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bdj.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bdl.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.9.cll.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.cln.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dxs.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.dxt.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 36.216.9.i.1 | $36$ | $3$ | $3$ | $9$ | $1$ | $1^{6}\cdot2$ |
| 36.216.9.m.1 | $36$ | $3$ | $3$ | $9$ | $5$ | $1^{6}\cdot2$ |
| 36.216.9.u.1 | $36$ | $3$ | $3$ | $9$ | $6$ | $1^{4}\cdot2^{2}$ |
| 60.144.5.re.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.144.5.rf.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.144.5.rm.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.144.5.rn.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.360.25.cfw.1 | $60$ | $5$ | $5$ | $25$ | $12$ | $1^{24}$ |
| 60.432.25.bkq.1 | $60$ | $6$ | $6$ | $25$ | $2$ | $1^{24}$ |
| 60.720.49.ekg.1 | $60$ | $10$ | $10$ | $49$ | $23$ | $1^{48}$ |
| 84.144.5.ia.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.144.5.ib.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.144.5.ii.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 84.144.5.ij.1 | $84$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.evy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ewf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eyc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eyj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.hsg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.hsh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ine.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.inf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bciy.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bciz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bdqu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bdqv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 132.144.5.ia.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.144.5.ib.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.144.5.ii.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.144.5.ij.1 | $132$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.144.5.ia.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.144.5.ib.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.144.5.ii.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.144.5.ij.1 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.chy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cif.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ckc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ckj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.esg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.esh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.fnd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.fne.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.yfs.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.yft.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.zno.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.znp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 204.144.5.ia.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.144.5.ib.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.144.5.ii.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.144.5.ij.1 | $204$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.144.5.ia.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.144.5.ib.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.144.5.ii.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 228.144.5.ij.1 | $228$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.chy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cif.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ckc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ckj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.esg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.esh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.fne.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.fnf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.yls.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.ylt.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.zto.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.ztp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 276.144.5.ia.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.144.5.ib.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.144.5.ii.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.144.5.ij.1 | $276$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.chy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cif.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ckc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ckj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.esg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.esh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.fne.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.fnf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.yga.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.ygb.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.znw.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.znx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |