| $\GL_2(\Z/17\Z)$-generators: |
$\begin{bmatrix}13&3\\0&12\end{bmatrix}$, $\begin{bmatrix}13&4\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/17\Z)$-subgroup: |
$C_2\times F_{17}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
17.288.5-17.c.1.1, 17.288.5-17.c.1.2, 34.288.5-17.c.1.1, 34.288.5-17.c.1.2, 51.288.5-17.c.1.1, 51.288.5-17.c.1.2, 68.288.5-17.c.1.1, 68.288.5-17.c.1.2, 68.288.5-17.c.1.3, 68.288.5-17.c.1.4, 85.288.5-17.c.1.1, 85.288.5-17.c.1.2, 102.288.5-17.c.1.1, 102.288.5-17.c.1.2, 119.288.5-17.c.1.1, 119.288.5-17.c.1.2, 136.288.5-17.c.1.1, 136.288.5-17.c.1.2, 136.288.5-17.c.1.3, 136.288.5-17.c.1.4, 136.288.5-17.c.1.5, 136.288.5-17.c.1.6, 136.288.5-17.c.1.7, 136.288.5-17.c.1.8, 170.288.5-17.c.1.1, 170.288.5-17.c.1.2, 187.288.5-17.c.1.1, 187.288.5-17.c.1.2, 204.288.5-17.c.1.1, 204.288.5-17.c.1.2, 204.288.5-17.c.1.3, 204.288.5-17.c.1.4, 221.288.5-17.c.1.1, 221.288.5-17.c.1.2, 238.288.5-17.c.1.1, 238.288.5-17.c.1.2, 255.288.5-17.c.1.1, 255.288.5-17.c.1.2, 323.288.5-17.c.1.1, 323.288.5-17.c.1.2 |
| Cyclic 17-isogeny field degree: |
$1$ |
| Cyclic 17-torsion field degree: |
$4$ |
| Full 17-torsion field degree: |
$544$ |
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x y + x w + y^{2} + y w + y t + z w + z t - t^{2} $ |
| $=$ | $x z + x w + y z - y t + z^{2} + z w - w^{2} - w t$ |
| $=$ | $4 x^{2} + x w + z w + z t - w^{2} - w t - t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 256 x^{8} - 128 x^{7} y + 128 x^{7} z - 164 x^{6} y^{2} - 168 x^{6} y z - 164 x^{6} z^{2} - 68 x^{5} y^{3} + \cdots + z^{8} $ |
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This modular curve has no $\Q_p$ points for $p=3,7$, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^{18}}\cdot\frac{23613348678568254745xw^{17}+87333777061737813496xw^{16}t+135164337115321295632xw^{15}t^{2}+50215391260989028704xw^{14}t^{3}-293358298700631635504xw^{13}t^{4}-1094946298486626669248xw^{12}t^{5}-2687315671724762855680xw^{11}t^{6}-5296139443809732115456xw^{10}t^{7}-8557709715870800673280xw^{9}t^{8}-11333507160695505092608xw^{8}t^{9}-12268388460565050439680xw^{7}t^{10}-10792003701609888237568xw^{6}t^{11}-7628679142859166741760xw^{5}t^{12}-4249663315584005955840xw^{4}t^{13}-1805073885688161045824xw^{3}t^{14}-552072234985446568896xw^{2}t^{15}-108813673559232434880xwt^{16}-10426537159064969280xt^{17}+9477494527456363616y^{2}w^{16}+35365341381405722592y^{2}w^{15}t+69121348590894638832y^{2}w^{14}t^{2}+133126815851961530864y^{2}w^{13}t^{3}+361492058026055176256y^{2}w^{12}t^{4}+1010241537918578548608y^{2}w^{11}t^{5}+2334297517352687926720y^{2}w^{10}t^{6}+4325774029771314361600y^{2}w^{9}t^{7}+6457497213256131624192y^{2}w^{8}t^{8}+7782259364032049548288y^{2}w^{7}t^{9}+7544489848506780241920y^{2}w^{6}t^{10}+5830135429387867541504y^{2}w^{5}t^{11}+3528484155788935081216y^{2}w^{4}t^{12}+1621464508603787361280y^{2}w^{3}t^{13}+535887063745701529920y^{2}w^{2}t^{14}+114255531822903198336y^{2}wt^{15}+11875858975225230400y^{2}t^{16}+969037061539046312yzw^{16}+2316207873503162352yzw^{15}t-17167412073858283760yzw^{14}t^{2}-142574318443461771152yzw^{13}t^{3}-590852720794635171280yzw^{12}t^{4}-1768052767832910188928yzw^{11}t^{5}-4155204705504313621120yzw^{10}t^{6}-7844779497959131180160yzw^{9}t^{7}-11967918879449105604864yzw^{8}t^{8}-14764492282486557864960yzw^{7}t^{9}-14687300771448275268608yzw^{6}t^{10}-11679002294906219961856yzw^{5}t^{11}-7305349383224851211776yzw^{4}t^{12}-3490308183834304101760yzw^{3}t^{13}-1207322062792677973824yzw^{2}t^{14}-272177400196818911632yzwt^{15}-30396842686317959120yzt^{16}+15434138609787025792yw^{17}+64453404976058458664yw^{16}t+135738248886679392768yw^{15}t^{2}+277627476400695389408yw^{14}t^{3}+817971203447810760352yw^{13}t^{4}+2501971600065758910672yw^{12}t^{5}+6323791480895207220224yw^{11}t^{6}+12854040035816894570944yw^{10}t^{7}+21163897172512486691712yw^{9}t^{8}+28343469524287020535040yw^{8}t^{9}+30847898608420695463936yw^{7}t^{10}+27139470383543278922240yw^{6}t^{11}+19084181031066206722304yw^{5}t^{12}+10520811884059359619392yw^{4}t^{13}+4398339419946390737472yw^{3}t^{14}+1314948196526128695184yw^{2}t^{15}+250969760061236149280ywt^{16}+22901761297900292240yt^{17}+3163948202448059840z^{3}w^{15}+12418750233280334928z^{3}w^{14}t+33240964262276968512z^{3}w^{13}t^{2}+106950631809161694144z^{3}w^{12}t^{3}+360147209370756105216z^{3}w^{11}t^{4}+1005656215233174993152z^{3}w^{10}t^{5}+2219511018175202753280z^{3}w^{9}t^{6}+3893214952616989907200z^{3}w^{8}t^{7}+5465710179652902993920z^{3}w^{7}t^{8}+6146314904737336791040z^{3}w^{6}t^{9}+5504884361940399728640z^{3}w^{5}t^{10}+3869026839475195934208z^{3}w^{4}t^{11}+2076640930694464391168z^{3}w^{3}t^{12}+809432311718861296768z^{3}w^{2}t^{13}+206873986613591037888z^{3}wt^{14}+26600122903036196560z^{3}t^{15}+9642372443703245556z^{2}w^{16}+37590881818385166224z^{2}w^{15}t+55790245674909986788z^{2}w^{14}t^{2}-8875026982691497520z^{2}w^{13}t^{3}-262793109722391539984z^{2}w^{12}t^{4}-869895500931352444160z^{2}w^{11}t^{5}-2027103960470855160256z^{2}w^{10}t^{6}-3758091328959915939648z^{2}w^{9}t^{7}-5661786029221492507584z^{2}w^{8}t^{8}-6968061410768689524736z^{2}w^{7}t^{9}-7004046822076814807552z^{2}w^{6}t^{10}-5712436516516135977472z^{2}w^{5}t^{11}-3726500480071933015360z^{2}w^{4}t^{12}-1889511452807708426112z^{2}w^{3}t^{13}-707211688171178903632z^{2}w^{2}t^{14}-176683180047600764432z^{2}wt^{15}-22651401310912818912z^{2}t^{16}+14534595794931359129zw^{17}+59151959434218671425zw^{16}t+85189641876992890576zw^{15}t^{2}-35428603997470953272zw^{14}t^{3}-477677918236472008192zw^{13}t^{4}-1513139368946299595968zw^{12}t^{5}-3528824090130833360192zw^{11}t^{6}-6716393043473511261440zw^{10}t^{7}-10548792638211824353984zw^{9}t^{8}-13667562700919536671296zw^{8}t^{9}-14591312002383590256640zw^{7}t^{10}-12771949035190231134208zw^{6}t^{11}-9076982130028223803136zw^{5}t^{12}-5140710358765895612480zw^{4}t^{13}-2247929431334300621696zw^{3}t^{14}-718358651967986640000zw^{2}t^{15}-150440706880061881216zwt^{16}-15674334848817495088zt^{17}-6899658646206520469w^{18}-31202888704821276377w^{17}t-55314832496628652957w^{16}t^{2}+55028725781960839984w^{15}t^{3}+638502987103349959300w^{14}t^{4}+2271566759330340894688w^{13}t^{5}+5645813205082448909520w^{12}t^{6}+11045971280999728952960w^{11}t^{7}+17505304124040233836992w^{10}t^{8}+22587088617582499073472w^{9}t^{9}+23695211686108199623168w^{8}t^{10}+20111074722609848815104w^{7}t^{11}+13688261201348867861760w^{6}t^{12}+7371711170780505326912w^{5}t^{13}+3081140960815141355520w^{4}t^{14}+977588122475169309840w^{3}t^{15}+231480782016260538080w^{2}t^{16}+40469708627009671376wt^{17}+4627228383069664512t^{18}}{(w+t)^{8}(394152752xw^{9}+2209166320xw^{8}t+5267499504xw^{7}t^{2}+7169315344xw^{6}t^{3}+6286745248xw^{5}t^{4}+3721746624xw^{4}t^{5}+1482528384xw^{3}t^{6}+395159424xw^{2}t^{7}+65345280xwt^{8}+883456xt^{9}-1188943688y^{2}w^{8}-4679830652y^{2}w^{7}t-7703654516y^{2}w^{6}t^{2}-7578322560y^{2}w^{5}t^{3}-4862909568y^{2}w^{4}t^{4}-2031124096y^{2}w^{3}t^{5}-594971712y^{2}w^{2}t^{6}-73251776y^{2}wt^{7}-12049600y^{2}t^{8}+1995763671yzw^{8}+7445372224yzw^{7}t+11907768320yzw^{6}t^{2}+11751163436yzw^{5}t^{3}+7466333904yzw^{4}t^{4}+3250384224yzw^{3}t^{5}+855104080yzw^{2}t^{6}+173420672yzwt^{7}+3406528yzt^{8}-2130999376yw^{9}-11040730431yw^{8}t-23338029232yw^{7}t^{2}-27931606552yw^{6}t^{3}-22007596556yw^{5}t^{4}-11560725072yw^{4}t^{5}-4224093760yw^{3}t^{6}-930384720yw^{2}t^{7}-157821952ywt^{8}-3130048yt^{9}-661535756z^{3}w^{7}-2244604656z^{3}w^{6}t-3104780176z^{3}w^{5}t^{2}-2613548176z^{3}w^{4}t^{3}-1340384448z^{3}w^{3}t^{4}-458968448z^{3}w^{2}t^{5}-73469376z^{3}wt^{6}-10379520z^{3}t^{7}-78026460z^{2}w^{8}-857591947z^{2}w^{7}t-2215953084z^{2}w^{6}t^{2}-2641124468z^{2}w^{5}t^{3}-2023531140z^{2}w^{4}t^{4}-943413552z^{2}w^{3}t^{5}-320672928z^{2}w^{2}t^{6}-25620464z^{2}wt^{7}-13730624z^{2}t^{8}+1563534043zw^{9}+6030833324zw^{8}t+11552849711zw^{7}t^{2}+15441758224zw^{6}t^{3}+14003162276zw^{5}t^{4}+9175263892zw^{4}t^{5}+4011780256zw^{3}t^{6}+1235613216zw^{2}t^{7}+197258992zwt^{8}+21244480zt^{9}-1780016020w^{10}-8548981460w^{9}t-16610726217w^{8}t^{2}-18347748872w^{7}t^{3}-13739040796w^{6}t^{4}-7244613916w^{5}t^{5}-2997648128w^{4}t^{6}-969609760w^{3}t^{7}-269441360w^{2}t^{8}-44404864wt^{9}-3119360t^{10})}$ 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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.
This modular curve is minimally covered by the modular curves in the database listed below.
