↑

Properties

Label 12.96.1.f.2
Level $12$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $12$ $\SL_2$-level: $12$ Newform level: $72$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $2^{8}$
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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 12.96.1.117

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}1&1\\0&11\end{bmatrix}$, $\begin{bmatrix}1&10\\0&7\end{bmatrix}$, $\begin{bmatrix}11&2\\0&5\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_6:D_4$
Contains $-I$: yes
Quadratic refinements: 12.192.1-12.f.2.1, 12.192.1-12.f.2.2, 12.192.1-12.f.2.3, 12.192.1-12.f.2.4, 24.192.1-12.f.2.1, 24.192.1-12.f.2.2, 24.192.1-12.f.2.3, 24.192.1-12.f.2.4, 24.192.1-12.f.2.5, 24.192.1-12.f.2.6, 24.192.1-12.f.2.7, 24.192.1-12.f.2.8, 24.192.1-12.f.2.9, 24.192.1-12.f.2.10, 24.192.1-12.f.2.11, 24.192.1-12.f.2.12, 60.192.1-12.f.2.1, 60.192.1-12.f.2.2, 60.192.1-12.f.2.3, 60.192.1-12.f.2.4, 84.192.1-12.f.2.1, 84.192.1-12.f.2.2, 84.192.1-12.f.2.3, 84.192.1-12.f.2.4, 120.192.1-12.f.2.1, 120.192.1-12.f.2.2, 120.192.1-12.f.2.3, 120.192.1-12.f.2.4, 120.192.1-12.f.2.5, 120.192.1-12.f.2.6, 120.192.1-12.f.2.7, 120.192.1-12.f.2.8, 120.192.1-12.f.2.9, 120.192.1-12.f.2.10, 120.192.1-12.f.2.11, 120.192.1-12.f.2.12, 132.192.1-12.f.2.1, 132.192.1-12.f.2.2, 132.192.1-12.f.2.3, 132.192.1-12.f.2.4, 156.192.1-12.f.2.1, 156.192.1-12.f.2.2, 156.192.1-12.f.2.3, 156.192.1-12.f.2.4, 168.192.1-12.f.2.1, 168.192.1-12.f.2.2, 168.192.1-12.f.2.3, 168.192.1-12.f.2.4, 168.192.1-12.f.2.5, 168.192.1-12.f.2.6, 168.192.1-12.f.2.7, 168.192.1-12.f.2.8, 168.192.1-12.f.2.9, 168.192.1-12.f.2.10, 168.192.1-12.f.2.11, 168.192.1-12.f.2.12, 204.192.1-12.f.2.1, 204.192.1-12.f.2.2, 204.192.1-12.f.2.3, 204.192.1-12.f.2.4, 228.192.1-12.f.2.1, 228.192.1-12.f.2.2, 228.192.1-12.f.2.3, 228.192.1-12.f.2.4, 264.192.1-12.f.2.1, 264.192.1-12.f.2.2, 264.192.1-12.f.2.3, 264.192.1-12.f.2.4, 264.192.1-12.f.2.5, 264.192.1-12.f.2.6, 264.192.1-12.f.2.7, 264.192.1-12.f.2.8, 264.192.1-12.f.2.9, 264.192.1-12.f.2.10, 264.192.1-12.f.2.11, 264.192.1-12.f.2.12, 276.192.1-12.f.2.1, 276.192.1-12.f.2.2, 276.192.1-12.f.2.3, 276.192.1-12.f.2.4, 312.192.1-12.f.2.1, 312.192.1-12.f.2.2, 312.192.1-12.f.2.3, 312.192.1-12.f.2.4, 312.192.1-12.f.2.5, 312.192.1-12.f.2.6, 312.192.1-12.f.2.7, 312.192.1-12.f.2.8, 312.192.1-12.f.2.9, 312.192.1-12.f.2.10, 312.192.1-12.f.2.11, 312.192.1-12.f.2.12
Cyclic 12-isogeny field degree: $1$
Cyclic 12-torsion field degree: $2$
Full 12-torsion field degree: $48$

Jacobian

Conductor: $2^{3}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 72.2.a.a

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 6 x y + 3 y^{2} - w^{2} $
$=$ $5 x^{2} - 2 x y + 2 x w - 3 y^{2} - z^{2} + w^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 4 x^{3} z + 3 x^{2} y^{2} + 2 x^{2} z^{2} - 12 x z^{3} - 15 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{2}{3}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3\cdot5^6}\cdot\frac{119801030839063200000000000xz^{22}w-108107156420736459120000000000xz^{20}w^{3}+1644435022752296169408000000000xz^{18}w^{5}-11231020911714158792646000000000xz^{16}w^{7}+46835906592999689672882400000000xz^{14}w^{9}-109550397087884427108412752000000xz^{12}w^{11}+345530040061515442525563019200000xz^{10}w^{13}-250186595934089063120954039280000xz^{8}w^{15}+735822657908478161796499602120000xz^{6}w^{17}-171172441081781400166476653412000xz^{4}w^{19}+13706506063459530452633037850080xz^{2}w^{21}-368616128382593393360934781308xw^{23}+14165970885499442400000000000y^{2}z^{22}-455942595926983808880000000000y^{2}z^{20}w^{2}+3880244998031482432440000000000y^{2}z^{18}w^{4}-18477839431943895335943600000000y^{2}z^{16}w^{6}+59576575321409789158509600000000y^{2}z^{14}w^{8}-138763642419735628847652144000000y^{2}z^{12}w^{10}+270288157322185303754484028800000y^{2}z^{10}w^{12}-292735347076476880964176417200000y^{2}z^{8}w^{14}+418821485466071839428194323896000y^{2}z^{6}w^{16}-91825970891356088227156139604000y^{2}z^{4}w^{18}+7199093372095522288106745474960y^{2}z^{2}w^{20}-191538558854395151196622125132y^{2}w^{22}+132449958082804204800000000000yz^{22}w-2060252098802598715200000000000yz^{20}w^{3}+13973503565117113510824000000000yz^{18}w^{5}-59625444775647661446916800000000yz^{16}w^{7}+181069429647495657034684800000000yz^{14}w^{9}-397682044264539008427862368000000yz^{12}w^{11}+781500423016086975396800856000000yz^{10}w^{13}-787007388513061466656627170240000yz^{8}w^{15}+1189879933475431893333685674528000yz^{6}w^{17}-262718478273862284375554968896000yz^{4}w^{19}+20652889368087136745733217594320yz^{2}w^{21}-550277647845268249511718750000yw^{23}-29054883787201800000000000z^{24}+569046356632575360000000000z^{22}w^{2}-47990373782963640120000000000z^{20}w^{4}+693590388954829961245200000000z^{18}w^{6}-5214954453552849384787500000000z^{16}w^{8}+25924613913864014634271008000000z^{14}w^{10}-44421277992516943891045293600000z^{12}w^{12}+222644866657207536532802890800000z^{10}w^{14}-89938439776012871421831714822000z^{8}w^{16}+487118901572318021018926990536000z^{6}w^{18}-116298192908403088419797434242720z^{4}w^{20}+9396561072940564154209622013204z^{2}w^{22}-253856761827701330165497703081w^{24}}{z^{2}(1318839857971200000000xz^{20}w+30072876820070400000000xz^{18}w^{3}+262351899545370624000000xz^{16}w^{5}+1324399075804506931200000xz^{14}w^{7}+4512957435963150451200000xz^{12}w^{9}+11129477563821238919136000xz^{10}w^{11}+20456596550956587283764000xz^{8}w^{13}+28078111036405202125334760xz^{6}w^{15}+28005727590323922749655348xz^{4}w^{17}+18775424626899719238281250xz^{2}w^{19}+6775154505271911621093750xw^{21}+41794220851200000000y^{2}z^{20}+2701880282972160000000y^{2}z^{18}w^{2}+36827752879411200000000y^{2}z^{16}w^{4}+245454736335720652800000y^{2}z^{14}w^{6}+1023977851797736886400000y^{2}z^{12}w^{8}+2969328409382716355376000y^{2}z^{10}w^{10}+6270676703284887839692800y^{2}z^{8}w^{12}+9762506428405522903288200y^{2}z^{6}w^{14}+10991581461032581419683292y^{2}z^{4}w^{16}+8341034749324035644531250y^{2}z^{2}w^{18}+3520473549678039550781250y^{2}w^{20}+715145556787200000000yz^{20}w+18817731843194880000000yz^{18}w^{3}+184903754157637632000000yz^{16}w^{5}+1033317771847916544000000yz^{14}w^{7}+3854700248071734316800000yz^{12}w^{9}+10337634836150086845936000yz^{10}w^{11}+20597418898236363844706400yz^{8}w^{13}+30647649031196600773234080yz^{6}w^{15}+33276915587109375000000000yz^{4}w^{17}+24507861906829833984375000yz^{2}w^{19}+10114088337121582031250000yw^{21}+55725627801600000000z^{22}+3611794648596480000000z^{20}w^{2}+49948274951700480000000z^{18}w^{4}+342056327858697830400000z^{16}w^{6}+1484866293336018777600000z^{14}w^{8}+4543992002339338114368000z^{12}w^{10}+10308928391428546179968400z^{10}w^{12}+17685727580123498289483000z^{8}w^{14}+22860069918882557639967081z^{6}w^{16}+21580919540195560425209736z^{4}w^{18}+13716258119181060791015625z^{2}w^{20}+4665880440818786621093750w^{22})}$

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
$X_{\pm1}(12)$ $12$ $2$ $2$ $0$ $0$ full Jacobian
12.48.0.c.2 $12$ $2$ $2$ $0$ $0$ full Jacobian
12.48.1.k.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
12.288.9.i.2 $12$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
24.192.5.eu.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.eu.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ev.2 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.ev.4 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fj.1 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fj.3 $24$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
24.192.5.fm.1 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
24.192.5.fm.3 $24$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
36.288.9.f.2 $36$ $3$ $3$ $9$ $0$ $1^{4}\cdot2^{2}$
36.288.17.k.1 $36$ $3$ $3$ $17$ $0$ $1^{8}\cdot4^{2}$
36.288.17.l.1 $36$ $3$ $3$ $17$ $1$ $1^{8}\cdot4^{2}$
60.480.33.bu.1 $60$ $5$ $5$ $33$ $2$ $1^{16}\cdot8^{2}$
60.576.33.eh.2 $60$ $6$ $6$ $33$ $2$ $1^{16}\cdot8^{2}$
60.960.65.hg.2 $60$ $10$ $10$ $65$ $6$ $1^{32}\cdot8^{4}$
120.192.5.wa.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wa.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.we.3 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.we.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wq.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wq.4 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wu.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.wu.4 $120$ $2$ $2$ $5$ $?$ not computed
168.192.5.wa.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wa.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.we.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.we.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wq.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wq.3 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wu.2 $168$ $2$ $2$ $5$ $?$ not computed
168.192.5.wu.3 $168$ $2$ $2$ $5$ $?$ not computed
264.192.5.wa.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wa.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.we.2 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.we.4 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wq.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wq.3 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wu.1 $264$ $2$ $2$ $5$ $?$ not computed
264.192.5.wu.3 $264$ $2$ $2$ $5$ $?$ not computed
312.192.5.wa.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wa.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.we.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.we.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wq.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wq.3 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wu.2 $312$ $2$ $2$ $5$ $?$ not computed
312.192.5.wu.3 $312$ $2$ $2$ $5$ $?$ not computed