Normalized defining polynomial
\( x^{26} - x^{25} + 37 x^{24} - 118 x^{23} + 1008 x^{22} - 3235 x^{21} + 16642 x^{20} - 51633 x^{19} + 196206 x^{18} - 526339 x^{17} + 1528709 x^{16} - 3470974 x^{15} + \cdots + 85849 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-5567067695110660347277981181082226208360544116097363\) \(\medspace = -\,3^{13}\cdot 79^{24}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(97.77\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $3^{1/2}79^{12/13}\approx 97.77242883415123$ | ||
Ramified primes: | \(3\), \(79\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Gal(K/\Q) }$: | $26$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(237=3\cdot 79\) | ||
Dirichlet character group: | $\lbrace$$\chi_{237}(64,·)$, $\chi_{237}(1,·)$, $\chi_{237}(131,·)$, $\chi_{237}(196,·)$, $\chi_{237}(65,·)$, $\chi_{237}(8,·)$, $\chi_{237}(10,·)$, $\chi_{237}(143,·)$, $\chi_{237}(80,·)$, $\chi_{237}(146,·)$, $\chi_{237}(67,·)$, $\chi_{237}(22,·)$, $\chi_{237}(89,·)$, $\chi_{237}(220,·)$, $\chi_{237}(223,·)$, $\chi_{237}(97,·)$, $\chi_{237}(100,·)$, $\chi_{237}(101,·)$, $\chi_{237}(38,·)$, $\chi_{237}(46,·)$, $\chi_{237}(176,·)$, $\chi_{237}(179,·)$, $\chi_{237}(52,·)$, $\chi_{237}(125,·)$, $\chi_{237}(62,·)$, $\chi_{237}(166,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{4096}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23}a^{20}-\frac{11}{23}a^{19}-\frac{7}{23}a^{18}+\frac{2}{23}a^{17}-\frac{11}{23}a^{16}+\frac{11}{23}a^{15}+\frac{6}{23}a^{14}-\frac{9}{23}a^{13}-\frac{10}{23}a^{12}-\frac{5}{23}a^{11}+\frac{10}{23}a^{10}+\frac{6}{23}a^{9}-\frac{9}{23}a^{8}-\frac{7}{23}a^{7}+\frac{11}{23}a^{6}+\frac{2}{23}a^{5}+\frac{6}{23}a^{4}-\frac{11}{23}a^{3}-\frac{5}{23}a^{2}-\frac{8}{23}a+\frac{6}{23}$, $\frac{1}{23}a^{21}+\frac{10}{23}a^{19}-\frac{6}{23}a^{18}+\frac{11}{23}a^{17}+\frac{5}{23}a^{16}-\frac{11}{23}a^{15}+\frac{11}{23}a^{14}+\frac{6}{23}a^{13}+\frac{1}{23}a^{11}+\frac{1}{23}a^{10}+\frac{11}{23}a^{9}+\frac{9}{23}a^{8}+\frac{3}{23}a^{7}+\frac{8}{23}a^{6}+\frac{5}{23}a^{5}+\frac{9}{23}a^{4}-\frac{11}{23}a^{3}+\frac{6}{23}a^{2}+\frac{10}{23}a-\frac{3}{23}$, $\frac{1}{23}a^{22}-\frac{11}{23}a^{19}-\frac{11}{23}a^{18}+\frac{8}{23}a^{17}+\frac{7}{23}a^{16}-\frac{7}{23}a^{15}-\frac{8}{23}a^{14}-\frac{2}{23}a^{13}+\frac{9}{23}a^{12}+\frac{5}{23}a^{11}+\frac{3}{23}a^{10}-\frac{5}{23}a^{9}+\frac{1}{23}a^{8}+\frac{9}{23}a^{7}+\frac{10}{23}a^{6}-\frac{11}{23}a^{5}-\frac{2}{23}a^{4}+\frac{1}{23}a^{3}-\frac{9}{23}a^{2}+\frac{8}{23}a+\frac{9}{23}$, $\frac{1}{428789}a^{23}-\frac{5069}{428789}a^{22}+\frac{8906}{428789}a^{21}-\frac{2061}{428789}a^{20}-\frac{117957}{428789}a^{19}-\frac{124539}{428789}a^{18}-\frac{35045}{428789}a^{17}+\frac{168026}{428789}a^{16}-\frac{102429}{428789}a^{15}-\frac{171887}{428789}a^{14}+\frac{157703}{428789}a^{13}+\frac{8136}{18643}a^{12}+\frac{204425}{428789}a^{11}+\frac{39296}{428789}a^{10}+\frac{91347}{428789}a^{9}+\frac{88070}{428789}a^{8}-\frac{179757}{428789}a^{7}-\frac{126778}{428789}a^{6}+\frac{190188}{428789}a^{5}-\frac{7532}{18643}a^{4}+\frac{26249}{428789}a^{3}+\frac{54161}{428789}a^{2}-\frac{176238}{428789}a+\frac{94462}{428789}$, $\frac{1}{226829381}a^{24}-\frac{172}{226829381}a^{23}+\frac{4213164}{226829381}a^{22}+\frac{4106104}{226829381}a^{21}-\frac{1858610}{226829381}a^{20}-\frac{47869979}{226829381}a^{19}+\frac{33616546}{226829381}a^{18}-\frac{4581589}{9862147}a^{17}+\frac{111770088}{226829381}a^{16}-\frac{1873298}{226829381}a^{15}+\frac{14662768}{226829381}a^{14}+\frac{74613943}{226829381}a^{13}-\frac{82731845}{226829381}a^{12}+\frac{58182867}{226829381}a^{11}+\frac{95505687}{226829381}a^{10}+\frac{90643238}{226829381}a^{9}-\frac{96218222}{226829381}a^{8}+\frac{86317315}{226829381}a^{7}+\frac{105801460}{226829381}a^{6}+\frac{31000145}{226829381}a^{5}+\frac{25263251}{226829381}a^{4}-\frac{90571594}{226829381}a^{3}-\frac{76881084}{226829381}a^{2}-\frac{109374321}{226829381}a+\frac{26296928}{226829381}$, $\frac{1}{49\!\cdots\!27}a^{25}+\frac{50\!\cdots\!64}{49\!\cdots\!27}a^{24}-\frac{54\!\cdots\!08}{49\!\cdots\!27}a^{23}+\frac{31\!\cdots\!48}{27\!\cdots\!67}a^{22}-\frac{89\!\cdots\!94}{49\!\cdots\!27}a^{21}+\frac{94\!\cdots\!00}{49\!\cdots\!27}a^{20}+\frac{18\!\cdots\!26}{49\!\cdots\!27}a^{19}-\frac{39\!\cdots\!38}{49\!\cdots\!27}a^{18}-\frac{14\!\cdots\!89}{49\!\cdots\!27}a^{17}-\frac{46\!\cdots\!76}{49\!\cdots\!27}a^{16}-\frac{77\!\cdots\!52}{21\!\cdots\!49}a^{15}-\frac{94\!\cdots\!55}{49\!\cdots\!27}a^{14}-\frac{17\!\cdots\!74}{49\!\cdots\!27}a^{13}+\frac{20\!\cdots\!43}{49\!\cdots\!27}a^{12}-\frac{46\!\cdots\!31}{21\!\cdots\!49}a^{11}-\frac{19\!\cdots\!17}{49\!\cdots\!27}a^{10}-\frac{11\!\cdots\!59}{49\!\cdots\!27}a^{9}+\frac{63\!\cdots\!70}{49\!\cdots\!27}a^{8}-\frac{36\!\cdots\!87}{49\!\cdots\!27}a^{7}+\frac{17\!\cdots\!97}{49\!\cdots\!27}a^{6}-\frac{20\!\cdots\!72}{49\!\cdots\!27}a^{5}+\frac{23\!\cdots\!00}{49\!\cdots\!27}a^{4}+\frac{12\!\cdots\!56}{49\!\cdots\!27}a^{3}-\frac{54\!\cdots\!66}{49\!\cdots\!27}a^{2}+\frac{19\!\cdots\!90}{49\!\cdots\!27}a+\frac{18\!\cdots\!62}{16\!\cdots\!39}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{57473}$, which has order $57473$ (assuming GRH)
Unit group
Rank: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{745437435626525585791097888709341304681484807006785254875549}{408484780404460089039416995905839193109216802389807767712954131881} a^{25} - \frac{800600851644091915550693456619893048232672919112354031509062}{408484780404460089039416995905839193109216802389807767712954131881} a^{24} + \frac{27633491585445404582125825140468867200147918128306953923210841}{408484780404460089039416995905839193109216802389807767712954131881} a^{23} - \frac{89932332821032303795495171336158580850423032047539225217894220}{408484780404460089039416995905839193109216802389807767712954131881} a^{22} + \frac{757736344994103711379521133711857773162434426200308472994013920}{408484780404460089039416995905839193109216802389807767712954131881} a^{21} - \frac{2464288908392365058113282363130840345871255333259680285988693784}{408484780404460089039416995905839193109216802389807767712954131881} a^{20} + \frac{12573204623928256474854921990373761147483149692735488756348734413}{408484780404460089039416995905839193109216802389807767712954131881} a^{19} - \frac{39330911361288242705889494852756800570596591189199619922913689537}{408484780404460089039416995905839193109216802389807767712954131881} a^{18} + \frac{148845080043023163759577860857899614387193792271760207280706212906}{408484780404460089039416995905839193109216802389807767712954131881} a^{17} - \frac{401933652602234683585264079828767046313057346437865059333368878931}{408484780404460089039416995905839193109216802389807767712954131881} a^{16} + \frac{1164644218294350376610152560579646099693911031526036638568843250948}{408484780404460089039416995905839193109216802389807767712954131881} a^{15} - \frac{2657455733490793958085602735923700346711360631808299696136786022297}{408484780404460089039416995905839193109216802389807767712954131881} a^{14} + \frac{6207029985588047906671405250161930715896574910644884226087144376576}{408484780404460089039416995905839193109216802389807767712954131881} a^{13} - \frac{11598064215401636134133795453151717443426255953600115214049227112301}{408484780404460089039416995905839193109216802389807767712954131881} a^{12} + \frac{20736838298104688526927598480458816125218529837050164805240642217353}{408484780404460089039416995905839193109216802389807767712954131881} a^{11} - \frac{28650958448430114553819787935661239455313835538399976808195141005716}{408484780404460089039416995905839193109216802389807767712954131881} a^{10} + \frac{37238537460543477723099146238779884229080542315073926124361974436810}{408484780404460089039416995905839193109216802389807767712954131881} a^{9} - \frac{36549407897392657125357430970714790943501781202717326421796785248527}{408484780404460089039416995905839193109216802389807767712954131881} a^{8} + \frac{37429046557079935182304712656079883099486240822683688346665522654115}{408484780404460089039416995905839193109216802389807767712954131881} a^{7} - \frac{28167755699276684561756024859865494506498191349179582440485315490428}{408484780404460089039416995905839193109216802389807767712954131881} a^{6} + \frac{26158145806174808843074478835580681763373827382739869584454294313980}{408484780404460089039416995905839193109216802389807767712954131881} a^{5} - \frac{13699740916400386158370648641827002546266938032398234891827527703172}{408484780404460089039416995905839193109216802389807767712954131881} a^{4} + \frac{10905024500702360702073152572174989223827794283152320479011396627999}{408484780404460089039416995905839193109216802389807767712954131881} a^{3} - \frac{2604055139933440516041199089854875479500442801564892131076838478333}{408484780404460089039416995905839193109216802389807767712954131881} a^{2} + \frac{3161861817590699445201304561941655417888284129745114459011791636626}{408484780404460089039416995905839193109216802389807767712954131881} a - \frac{78298508111038596112801991594433972609517842594379667909711265}{1394146008206348426755689405821976768290842328975453132126123317} \) (order $6$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{52\!\cdots\!84}{49\!\cdots\!27}a^{25}-\frac{98\!\cdots\!42}{49\!\cdots\!27}a^{24}-\frac{89\!\cdots\!79}{49\!\cdots\!27}a^{23}-\frac{34\!\cdots\!06}{49\!\cdots\!27}a^{22}+\frac{53\!\cdots\!91}{49\!\cdots\!27}a^{21}-\frac{73\!\cdots\!09}{49\!\cdots\!27}a^{20}+\frac{14\!\cdots\!52}{49\!\cdots\!27}a^{19}-\frac{43\!\cdots\!85}{21\!\cdots\!49}a^{18}+\frac{23\!\cdots\!78}{49\!\cdots\!27}a^{17}-\frac{96\!\cdots\!38}{49\!\cdots\!27}a^{16}+\frac{19\!\cdots\!27}{49\!\cdots\!27}a^{15}-\frac{59\!\cdots\!77}{49\!\cdots\!27}a^{14}+\frac{10\!\cdots\!56}{49\!\cdots\!27}a^{13}-\frac{24\!\cdots\!12}{49\!\cdots\!27}a^{12}+\frac{34\!\cdots\!06}{49\!\cdots\!27}a^{11}-\frac{51\!\cdots\!17}{49\!\cdots\!27}a^{10}+\frac{42\!\cdots\!90}{49\!\cdots\!27}a^{9}-\frac{47\!\cdots\!87}{49\!\cdots\!27}a^{8}+\frac{26\!\cdots\!47}{49\!\cdots\!27}a^{7}-\frac{35\!\cdots\!61}{49\!\cdots\!27}a^{6}+\frac{12\!\cdots\!14}{49\!\cdots\!27}a^{5}-\frac{13\!\cdots\!18}{49\!\cdots\!27}a^{4}+\frac{90\!\cdots\!70}{49\!\cdots\!27}a^{3}-\frac{56\!\cdots\!37}{49\!\cdots\!27}a^{2}+\frac{35\!\cdots\!81}{49\!\cdots\!27}a-\frac{17\!\cdots\!11}{73\!\cdots\!93}$, $\frac{37\!\cdots\!49}{73\!\cdots\!93}a^{25}-\frac{11\!\cdots\!77}{16\!\cdots\!39}a^{24}+\frac{29\!\cdots\!71}{16\!\cdots\!39}a^{23}-\frac{11\!\cdots\!15}{16\!\cdots\!39}a^{22}+\frac{81\!\cdots\!00}{16\!\cdots\!39}a^{21}-\frac{29\!\cdots\!64}{16\!\cdots\!39}a^{20}+\frac{13\!\cdots\!83}{16\!\cdots\!39}a^{19}-\frac{44\!\cdots\!69}{16\!\cdots\!39}a^{18}+\frac{67\!\cdots\!41}{73\!\cdots\!93}a^{17}-\frac{43\!\cdots\!81}{16\!\cdots\!39}a^{16}+\frac{11\!\cdots\!01}{16\!\cdots\!39}a^{15}-\frac{26\!\cdots\!39}{16\!\cdots\!39}a^{14}+\frac{57\!\cdots\!56}{16\!\cdots\!39}a^{13}-\frac{10\!\cdots\!90}{16\!\cdots\!39}a^{12}+\frac{17\!\cdots\!51}{16\!\cdots\!39}a^{11}-\frac{22\!\cdots\!38}{16\!\cdots\!39}a^{10}+\frac{23\!\cdots\!54}{16\!\cdots\!39}a^{9}-\frac{21\!\cdots\!73}{16\!\cdots\!39}a^{8}+\frac{15\!\cdots\!06}{16\!\cdots\!39}a^{7}-\frac{13\!\cdots\!57}{16\!\cdots\!39}a^{6}+\frac{68\!\cdots\!59}{16\!\cdots\!39}a^{5}-\frac{43\!\cdots\!11}{16\!\cdots\!39}a^{4}-\frac{44\!\cdots\!41}{16\!\cdots\!39}a^{3}-\frac{76\!\cdots\!20}{16\!\cdots\!39}a^{2}+\frac{10\!\cdots\!10}{16\!\cdots\!39}a+\frac{89\!\cdots\!41}{16\!\cdots\!39}$, $\frac{64\!\cdots\!40}{73\!\cdots\!93}a^{25}-\frac{98\!\cdots\!55}{16\!\cdots\!39}a^{24}+\frac{55\!\cdots\!97}{16\!\cdots\!39}a^{23}-\frac{15\!\cdots\!49}{16\!\cdots\!39}a^{22}+\frac{14\!\cdots\!67}{16\!\cdots\!39}a^{21}-\frac{42\!\cdots\!60}{16\!\cdots\!39}a^{20}+\frac{23\!\cdots\!54}{16\!\cdots\!39}a^{19}-\frac{66\!\cdots\!60}{16\!\cdots\!39}a^{18}+\frac{11\!\cdots\!15}{73\!\cdots\!93}a^{17}-\frac{66\!\cdots\!90}{16\!\cdots\!39}a^{16}+\frac{19\!\cdots\!41}{16\!\cdots\!39}a^{15}-\frac{41\!\cdots\!07}{16\!\cdots\!39}a^{14}+\frac{95\!\cdots\!09}{16\!\cdots\!39}a^{13}-\frac{15\!\cdots\!25}{16\!\cdots\!39}a^{12}+\frac{27\!\cdots\!86}{16\!\cdots\!39}a^{11}-\frac{28\!\cdots\!12}{16\!\cdots\!39}a^{10}+\frac{27\!\cdots\!78}{16\!\cdots\!39}a^{9}-\frac{18\!\cdots\!85}{16\!\cdots\!39}a^{8}-\frac{87\!\cdots\!46}{16\!\cdots\!39}a^{7}+\frac{17\!\cdots\!74}{93\!\cdots\!19}a^{6}-\frac{19\!\cdots\!73}{16\!\cdots\!39}a^{5}+\frac{28\!\cdots\!32}{16\!\cdots\!39}a^{4}-\frac{13\!\cdots\!25}{16\!\cdots\!39}a^{3}+\frac{15\!\cdots\!46}{16\!\cdots\!39}a^{2}-\frac{22\!\cdots\!18}{16\!\cdots\!39}a+\frac{62\!\cdots\!82}{16\!\cdots\!39}$, $\frac{75\!\cdots\!22}{49\!\cdots\!27}a^{25}+\frac{53\!\cdots\!62}{49\!\cdots\!27}a^{24}+\frac{28\!\cdots\!14}{49\!\cdots\!27}a^{23}-\frac{44\!\cdots\!97}{49\!\cdots\!27}a^{22}+\frac{68\!\cdots\!94}{49\!\cdots\!27}a^{21}-\frac{14\!\cdots\!06}{49\!\cdots\!27}a^{20}+\frac{10\!\cdots\!36}{49\!\cdots\!27}a^{19}-\frac{10\!\cdots\!52}{21\!\cdots\!49}a^{18}+\frac{11\!\cdots\!02}{49\!\cdots\!27}a^{17}-\frac{25\!\cdots\!45}{49\!\cdots\!27}a^{16}+\frac{82\!\cdots\!80}{49\!\cdots\!27}a^{15}-\frac{16\!\cdots\!50}{49\!\cdots\!27}a^{14}+\frac{42\!\cdots\!91}{49\!\cdots\!27}a^{13}-\frac{70\!\cdots\!40}{49\!\cdots\!27}a^{12}+\frac{13\!\cdots\!82}{49\!\cdots\!27}a^{11}-\frac{16\!\cdots\!11}{49\!\cdots\!27}a^{10}+\frac{24\!\cdots\!64}{49\!\cdots\!27}a^{9}-\frac{21\!\cdots\!38}{49\!\cdots\!27}a^{8}+\frac{26\!\cdots\!40}{49\!\cdots\!27}a^{7}-\frac{15\!\cdots\!20}{49\!\cdots\!27}a^{6}+\frac{19\!\cdots\!96}{49\!\cdots\!27}a^{5}-\frac{68\!\cdots\!62}{49\!\cdots\!27}a^{4}+\frac{87\!\cdots\!92}{49\!\cdots\!27}a^{3}-\frac{80\!\cdots\!32}{49\!\cdots\!27}a^{2}+\frac{25\!\cdots\!39}{49\!\cdots\!27}a+\frac{12\!\cdots\!90}{73\!\cdots\!93}$, $\frac{12\!\cdots\!44}{49\!\cdots\!27}a^{25}+\frac{11\!\cdots\!96}{49\!\cdots\!27}a^{24}+\frac{44\!\cdots\!98}{49\!\cdots\!27}a^{23}-\frac{71\!\cdots\!87}{49\!\cdots\!27}a^{22}+\frac{97\!\cdots\!98}{49\!\cdots\!27}a^{21}-\frac{20\!\cdots\!07}{49\!\cdots\!27}a^{20}+\frac{13\!\cdots\!78}{49\!\cdots\!27}a^{19}-\frac{32\!\cdots\!52}{49\!\cdots\!27}a^{18}+\frac{13\!\cdots\!67}{49\!\cdots\!27}a^{17}-\frac{29\!\cdots\!17}{49\!\cdots\!27}a^{16}+\frac{86\!\cdots\!69}{49\!\cdots\!27}a^{15}-\frac{72\!\cdots\!75}{21\!\cdots\!49}a^{14}+\frac{20\!\cdots\!11}{27\!\cdots\!67}a^{13}-\frac{56\!\cdots\!46}{49\!\cdots\!27}a^{12}+\frac{88\!\cdots\!54}{49\!\cdots\!27}a^{11}-\frac{87\!\cdots\!04}{49\!\cdots\!27}a^{10}+\frac{99\!\cdots\!27}{49\!\cdots\!27}a^{9}-\frac{68\!\cdots\!92}{49\!\cdots\!27}a^{8}+\frac{68\!\cdots\!94}{49\!\cdots\!27}a^{7}-\frac{28\!\cdots\!90}{49\!\cdots\!27}a^{6}+\frac{17\!\cdots\!90}{49\!\cdots\!27}a^{5}+\frac{17\!\cdots\!43}{49\!\cdots\!27}a^{4}-\frac{27\!\cdots\!36}{49\!\cdots\!27}a^{3}+\frac{80\!\cdots\!33}{49\!\cdots\!27}a^{2}-\frac{70\!\cdots\!08}{49\!\cdots\!27}a+\frac{34\!\cdots\!55}{16\!\cdots\!39}$, $\frac{53\!\cdots\!41}{73\!\cdots\!93}a^{25}-\frac{14\!\cdots\!41}{16\!\cdots\!39}a^{24}+\frac{42\!\cdots\!83}{16\!\cdots\!39}a^{23}-\frac{15\!\cdots\!84}{16\!\cdots\!39}a^{22}+\frac{11\!\cdots\!69}{16\!\cdots\!39}a^{21}-\frac{39\!\cdots\!67}{16\!\cdots\!39}a^{20}+\frac{18\!\cdots\!00}{16\!\cdots\!39}a^{19}-\frac{60\!\cdots\!73}{16\!\cdots\!39}a^{18}+\frac{91\!\cdots\!65}{73\!\cdots\!93}a^{17}-\frac{58\!\cdots\!34}{16\!\cdots\!39}a^{16}+\frac{15\!\cdots\!44}{16\!\cdots\!39}a^{15}-\frac{35\!\cdots\!70}{16\!\cdots\!39}a^{14}+\frac{76\!\cdots\!69}{16\!\cdots\!39}a^{13}-\frac{14\!\cdots\!45}{16\!\cdots\!39}a^{12}+\frac{22\!\cdots\!90}{16\!\cdots\!39}a^{11}-\frac{27\!\cdots\!74}{16\!\cdots\!39}a^{10}+\frac{28\!\cdots\!98}{16\!\cdots\!39}a^{9}-\frac{24\!\cdots\!98}{16\!\cdots\!39}a^{8}+\frac{16\!\cdots\!26}{16\!\cdots\!39}a^{7}-\frac{11\!\cdots\!34}{16\!\cdots\!39}a^{6}+\frac{56\!\cdots\!50}{16\!\cdots\!39}a^{5}-\frac{18\!\cdots\!85}{16\!\cdots\!39}a^{4}-\frac{89\!\cdots\!04}{16\!\cdots\!39}a^{3}+\frac{80\!\cdots\!93}{16\!\cdots\!39}a^{2}-\frac{11\!\cdots\!93}{16\!\cdots\!39}a+\frac{79\!\cdots\!71}{16\!\cdots\!39}$, $\frac{47\!\cdots\!34}{20\!\cdots\!63}a^{25}-\frac{11\!\cdots\!94}{46\!\cdots\!49}a^{24}+\frac{35\!\cdots\!42}{46\!\cdots\!49}a^{23}-\frac{13\!\cdots\!56}{46\!\cdots\!49}a^{22}+\frac{94\!\cdots\!44}{46\!\cdots\!49}a^{21}-\frac{33\!\cdots\!80}{46\!\cdots\!49}a^{20}+\frac{14\!\cdots\!70}{46\!\cdots\!49}a^{19}-\frac{48\!\cdots\!35}{45\!\cdots\!83}a^{18}+\frac{72\!\cdots\!78}{20\!\cdots\!63}a^{17}-\frac{46\!\cdots\!00}{46\!\cdots\!49}a^{16}+\frac{11\!\cdots\!61}{46\!\cdots\!49}a^{15}-\frac{27\!\cdots\!34}{46\!\cdots\!49}a^{14}+\frac{57\!\cdots\!22}{46\!\cdots\!49}a^{13}-\frac{10\!\cdots\!02}{46\!\cdots\!49}a^{12}+\frac{15\!\cdots\!38}{46\!\cdots\!49}a^{11}-\frac{19\!\cdots\!46}{46\!\cdots\!49}a^{10}+\frac{19\!\cdots\!76}{46\!\cdots\!49}a^{9}-\frac{17\!\cdots\!24}{46\!\cdots\!49}a^{8}+\frac{13\!\cdots\!44}{46\!\cdots\!49}a^{7}-\frac{10\!\cdots\!67}{45\!\cdots\!83}a^{6}+\frac{52\!\cdots\!34}{46\!\cdots\!49}a^{5}-\frac{27\!\cdots\!38}{46\!\cdots\!49}a^{4}+\frac{30\!\cdots\!61}{46\!\cdots\!49}a^{3}-\frac{19\!\cdots\!96}{46\!\cdots\!49}a^{2}+\frac{22\!\cdots\!58}{46\!\cdots\!49}a+\frac{47\!\cdots\!04}{46\!\cdots\!49}$, $\frac{16\!\cdots\!94}{49\!\cdots\!27}a^{25}+\frac{22\!\cdots\!43}{49\!\cdots\!27}a^{24}+\frac{83\!\cdots\!66}{49\!\cdots\!27}a^{23}+\frac{64\!\cdots\!34}{49\!\cdots\!27}a^{22}+\frac{32\!\cdots\!92}{49\!\cdots\!27}a^{21}+\frac{13\!\cdots\!06}{49\!\cdots\!27}a^{20}-\frac{10\!\cdots\!52}{49\!\cdots\!27}a^{19}+\frac{73\!\cdots\!88}{21\!\cdots\!49}a^{18}-\frac{28\!\cdots\!50}{49\!\cdots\!27}a^{17}+\frac{16\!\cdots\!23}{49\!\cdots\!27}a^{16}-\frac{28\!\cdots\!83}{49\!\cdots\!27}a^{15}+\frac{10\!\cdots\!82}{49\!\cdots\!27}a^{14}-\frac{17\!\cdots\!47}{49\!\cdots\!27}a^{13}+\frac{45\!\cdots\!01}{49\!\cdots\!27}a^{12}-\frac{58\!\cdots\!57}{49\!\cdots\!27}a^{11}+\frac{10\!\cdots\!12}{49\!\cdots\!27}a^{10}-\frac{78\!\cdots\!73}{49\!\cdots\!27}a^{9}+\frac{10\!\cdots\!83}{49\!\cdots\!27}a^{8}-\frac{48\!\cdots\!26}{49\!\cdots\!27}a^{7}+\frac{86\!\cdots\!03}{49\!\cdots\!27}a^{6}-\frac{18\!\cdots\!72}{49\!\cdots\!27}a^{5}+\frac{44\!\cdots\!59}{49\!\cdots\!27}a^{4}+\frac{47\!\cdots\!77}{49\!\cdots\!27}a^{3}+\frac{16\!\cdots\!27}{49\!\cdots\!27}a^{2}+\frac{44\!\cdots\!51}{49\!\cdots\!27}a+\frac{55\!\cdots\!65}{73\!\cdots\!93}$, $\frac{67\!\cdots\!43}{73\!\cdots\!93}a^{25}-\frac{23\!\cdots\!90}{16\!\cdots\!39}a^{24}+\frac{54\!\cdots\!87}{16\!\cdots\!39}a^{23}-\frac{21\!\cdots\!97}{16\!\cdots\!39}a^{22}+\frac{15\!\cdots\!23}{16\!\cdots\!39}a^{21}-\frac{54\!\cdots\!20}{16\!\cdots\!39}a^{20}+\frac{24\!\cdots\!41}{16\!\cdots\!39}a^{19}-\frac{83\!\cdots\!66}{16\!\cdots\!39}a^{18}+\frac{12\!\cdots\!69}{73\!\cdots\!93}a^{17}-\frac{82\!\cdots\!17}{16\!\cdots\!39}a^{16}+\frac{22\!\cdots\!33}{16\!\cdots\!39}a^{15}-\frac{50\!\cdots\!01}{16\!\cdots\!13}a^{14}+\frac{11\!\cdots\!96}{16\!\cdots\!39}a^{13}-\frac{21\!\cdots\!63}{16\!\cdots\!39}a^{12}+\frac{33\!\cdots\!45}{16\!\cdots\!39}a^{11}-\frac{45\!\cdots\!72}{16\!\cdots\!39}a^{10}+\frac{48\!\cdots\!55}{16\!\cdots\!39}a^{9}-\frac{44\!\cdots\!43}{16\!\cdots\!39}a^{8}+\frac{33\!\cdots\!70}{16\!\cdots\!39}a^{7}-\frac{27\!\cdots\!65}{16\!\cdots\!39}a^{6}+\frac{15\!\cdots\!63}{16\!\cdots\!39}a^{5}-\frac{10\!\cdots\!37}{16\!\cdots\!39}a^{4}+\frac{13\!\cdots\!18}{16\!\cdots\!39}a^{3}-\frac{24\!\cdots\!08}{16\!\cdots\!39}a^{2}+\frac{32\!\cdots\!96}{16\!\cdots\!39}a-\frac{19\!\cdots\!70}{16\!\cdots\!39}$, $\frac{53\!\cdots\!53}{73\!\cdots\!93}a^{25}-\frac{11\!\cdots\!29}{16\!\cdots\!39}a^{24}+\frac{39\!\cdots\!76}{16\!\cdots\!39}a^{23}-\frac{14\!\cdots\!76}{16\!\cdots\!39}a^{22}+\frac{10\!\cdots\!24}{16\!\cdots\!39}a^{21}-\frac{35\!\cdots\!09}{16\!\cdots\!39}a^{20}+\frac{15\!\cdots\!04}{16\!\cdots\!39}a^{19}-\frac{53\!\cdots\!72}{16\!\cdots\!39}a^{18}+\frac{77\!\cdots\!22}{73\!\cdots\!93}a^{17}-\frac{49\!\cdots\!03}{16\!\cdots\!39}a^{16}+\frac{12\!\cdots\!48}{16\!\cdots\!39}a^{15}-\frac{29\!\cdots\!83}{16\!\cdots\!39}a^{14}+\frac{59\!\cdots\!65}{16\!\cdots\!39}a^{13}-\frac{10\!\cdots\!28}{16\!\cdots\!39}a^{12}+\frac{15\!\cdots\!18}{16\!\cdots\!39}a^{11}-\frac{19\!\cdots\!89}{16\!\cdots\!39}a^{10}+\frac{19\!\cdots\!66}{16\!\cdots\!39}a^{9}-\frac{17\!\cdots\!42}{16\!\cdots\!39}a^{8}+\frac{12\!\cdots\!12}{16\!\cdots\!39}a^{7}-\frac{94\!\cdots\!51}{16\!\cdots\!39}a^{6}+\frac{47\!\cdots\!13}{16\!\cdots\!39}a^{5}-\frac{20\!\cdots\!85}{16\!\cdots\!39}a^{4}+\frac{87\!\cdots\!88}{16\!\cdots\!39}a^{3}+\frac{13\!\cdots\!93}{16\!\cdots\!39}a^{2}-\frac{23\!\cdots\!92}{16\!\cdots\!39}a+\frac{86\!\cdots\!94}{16\!\cdots\!39}$, $\frac{18\!\cdots\!06}{49\!\cdots\!27}a^{25}+\frac{11\!\cdots\!38}{49\!\cdots\!27}a^{24}+\frac{63\!\cdots\!72}{49\!\cdots\!27}a^{23}-\frac{11\!\cdots\!50}{49\!\cdots\!27}a^{22}+\frac{14\!\cdots\!34}{49\!\cdots\!27}a^{21}-\frac{32\!\cdots\!94}{49\!\cdots\!27}a^{20}+\frac{20\!\cdots\!58}{49\!\cdots\!27}a^{19}-\frac{51\!\cdots\!60}{49\!\cdots\!27}a^{18}+\frac{20\!\cdots\!06}{49\!\cdots\!27}a^{17}-\frac{46\!\cdots\!84}{49\!\cdots\!27}a^{16}+\frac{13\!\cdots\!94}{49\!\cdots\!27}a^{15}-\frac{11\!\cdots\!57}{21\!\cdots\!49}a^{14}+\frac{61\!\cdots\!10}{49\!\cdots\!27}a^{13}-\frac{94\!\cdots\!58}{49\!\cdots\!27}a^{12}+\frac{15\!\cdots\!72}{49\!\cdots\!27}a^{11}-\frac{15\!\cdots\!62}{49\!\cdots\!27}a^{10}+\frac{17\!\cdots\!68}{49\!\cdots\!27}a^{9}-\frac{12\!\cdots\!00}{49\!\cdots\!27}a^{8}+\frac{14\!\cdots\!04}{49\!\cdots\!27}a^{7}-\frac{81\!\cdots\!32}{49\!\cdots\!27}a^{6}+\frac{75\!\cdots\!00}{49\!\cdots\!27}a^{5}-\frac{24\!\cdots\!34}{49\!\cdots\!27}a^{4}+\frac{21\!\cdots\!80}{49\!\cdots\!27}a^{3}-\frac{12\!\cdots\!72}{49\!\cdots\!27}a^{2}+\frac{10\!\cdots\!52}{49\!\cdots\!27}a-\frac{33\!\cdots\!32}{16\!\cdots\!39}$, $\frac{24\!\cdots\!16}{49\!\cdots\!27}a^{25}+\frac{66\!\cdots\!56}{49\!\cdots\!27}a^{24}+\frac{20\!\cdots\!66}{49\!\cdots\!27}a^{23}+\frac{20\!\cdots\!30}{49\!\cdots\!27}a^{22}+\frac{35\!\cdots\!36}{49\!\cdots\!27}a^{21}+\frac{37\!\cdots\!72}{49\!\cdots\!27}a^{20}-\frac{21\!\cdots\!80}{49\!\cdots\!27}a^{19}+\frac{19\!\cdots\!84}{21\!\cdots\!49}a^{18}-\frac{52\!\cdots\!74}{49\!\cdots\!27}a^{17}+\frac{19\!\cdots\!94}{27\!\cdots\!67}a^{16}-\frac{33\!\cdots\!04}{49\!\cdots\!27}a^{15}+\frac{14\!\cdots\!06}{49\!\cdots\!27}a^{14}-\frac{66\!\cdots\!52}{49\!\cdots\!27}a^{13}+\frac{21\!\cdots\!88}{49\!\cdots\!27}a^{12}+\frac{74\!\cdots\!42}{49\!\cdots\!27}a^{11}-\frac{18\!\cdots\!00}{49\!\cdots\!27}a^{10}+\frac{54\!\cdots\!18}{49\!\cdots\!27}a^{9}-\frac{71\!\cdots\!00}{49\!\cdots\!27}a^{8}+\frac{92\!\cdots\!70}{49\!\cdots\!27}a^{7}-\frac{68\!\cdots\!48}{49\!\cdots\!27}a^{6}+\frac{60\!\cdots\!88}{49\!\cdots\!27}a^{5}-\frac{34\!\cdots\!65}{48\!\cdots\!09}a^{4}+\frac{26\!\cdots\!48}{49\!\cdots\!27}a^{3}-\frac{87\!\cdots\!30}{49\!\cdots\!27}a^{2}+\frac{14\!\cdots\!82}{49\!\cdots\!27}a-\frac{15\!\cdots\!48}{73\!\cdots\!93}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 57529828940.82975 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 57529828940.82975 \cdot 57473}{6\cdot\sqrt{5567067695110660347277981181082226208360544116097363}}\cr\approx \mathstrut & 0.175683403748937 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 26 |
The 26 conjugacy class representatives for $C_{26}$ |
Character table for $C_{26}$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 13.13.59091511031674153381441.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $26$ | R | $26$ | ${\href{/padicField/7.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/13.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/19.13.0.1}{13} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{13}$ | $26$ | ${\href{/padicField/31.13.0.1}{13} }^{2}$ | ${\href{/padicField/37.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/43.13.0.1}{13} }^{2}$ | $26$ | $26$ | $26$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(3\) | Deg $26$ | $2$ | $13$ | $13$ | |||
\(79\) | 79.13.12.1 | $x^{13} + 79$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ |
79.13.12.1 | $x^{13} + 79$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ |