Normalized defining polynomial
\( x^{26} - x^{25} + 11 x^{24} - 20 x^{23} + 107 x^{22} - 382 x^{21} + 1248 x^{20} - 3197 x^{19} + \cdots + 361 \)
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: |
\(-11543464978247129822340588525652022819543\)
\(\medspace = -\,23^{13}\cdot 73^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.74\) | 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: | $23^{1/2}73^{1/2}\approx 40.97560249709576$ | ||
| Ramified primes: |
\(23\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23}a^{14}+\frac{8}{23}a^{13}-\frac{9}{23}a^{12}+\frac{1}{23}a^{11}+\frac{7}{23}a^{10}+\frac{7}{23}a^{9}-\frac{9}{23}a^{8}+\frac{10}{23}a^{7}+\frac{9}{23}a^{6}-\frac{11}{23}a^{5}+\frac{7}{23}a^{4}-\frac{11}{23}a^{3}+\frac{10}{23}a^{2}-\frac{5}{23}a+\frac{8}{23}$, $\frac{1}{23}a^{15}-\frac{4}{23}a^{13}+\frac{4}{23}a^{12}-\frac{1}{23}a^{11}-\frac{3}{23}a^{10}+\frac{4}{23}a^{9}-\frac{10}{23}a^{8}-\frac{2}{23}a^{7}+\frac{9}{23}a^{6}+\frac{3}{23}a^{5}+\frac{2}{23}a^{4}+\frac{6}{23}a^{3}+\frac{7}{23}a^{2}+\frac{2}{23}a+\frac{5}{23}$, $\frac{1}{23}a^{16}-\frac{10}{23}a^{13}+\frac{9}{23}a^{12}+\frac{1}{23}a^{11}+\frac{9}{23}a^{10}-\frac{5}{23}a^{9}+\frac{8}{23}a^{8}+\frac{3}{23}a^{7}-\frac{7}{23}a^{6}+\frac{4}{23}a^{5}+\frac{11}{23}a^{4}+\frac{9}{23}a^{3}-\frac{4}{23}a^{2}+\frac{8}{23}a+\frac{9}{23}$, $\frac{1}{23}a^{17}-\frac{3}{23}a^{13}+\frac{3}{23}a^{12}-\frac{4}{23}a^{11}-\frac{4}{23}a^{10}+\frac{9}{23}a^{9}+\frac{5}{23}a^{8}+\frac{1}{23}a^{7}+\frac{2}{23}a^{6}-\frac{7}{23}a^{5}+\frac{10}{23}a^{4}+\frac{1}{23}a^{3}-\frac{7}{23}a^{2}+\frac{5}{23}a+\frac{11}{23}$, $\frac{1}{23}a^{18}+\frac{4}{23}a^{13}-\frac{8}{23}a^{12}-\frac{1}{23}a^{11}+\frac{7}{23}a^{10}+\frac{3}{23}a^{9}-\frac{3}{23}a^{8}+\frac{9}{23}a^{7}-\frac{3}{23}a^{6}-\frac{1}{23}a^{4}+\frac{6}{23}a^{3}-\frac{11}{23}a^{2}-\frac{4}{23}a+\frac{1}{23}$, $\frac{1}{23}a^{19}+\frac{6}{23}a^{13}-\frac{11}{23}a^{12}+\frac{3}{23}a^{11}-\frac{2}{23}a^{10}-\frac{8}{23}a^{9}-\frac{1}{23}a^{8}+\frac{3}{23}a^{7}+\frac{10}{23}a^{6}-\frac{3}{23}a^{5}+\frac{1}{23}a^{4}+\frac{10}{23}a^{3}+\frac{2}{23}a^{2}-\frac{2}{23}a-\frac{9}{23}$, $\frac{1}{529}a^{20}+\frac{7}{529}a^{19}-\frac{5}{529}a^{18}-\frac{6}{529}a^{17}-\frac{3}{529}a^{16}+\frac{5}{529}a^{15}+\frac{9}{529}a^{14}+\frac{132}{529}a^{13}+\frac{75}{529}a^{12}+\frac{89}{529}a^{11}+\frac{199}{529}a^{10}-\frac{93}{529}a^{9}+\frac{156}{529}a^{8}+\frac{129}{529}a^{7}-\frac{228}{529}a^{6}-\frac{54}{529}a^{5}-\frac{109}{529}a^{4}-\frac{155}{529}a^{3}+\frac{209}{529}a^{2}+\frac{214}{529}a+\frac{256}{529}$, $\frac{1}{529}a^{21}-\frac{8}{529}a^{19}+\frac{6}{529}a^{18}-\frac{7}{529}a^{17}+\frac{3}{529}a^{16}-\frac{3}{529}a^{15}+\frac{117}{529}a^{13}+\frac{139}{529}a^{12}-\frac{194}{529}a^{11}-\frac{198}{529}a^{10}+\frac{209}{529}a^{9}+\frac{95}{529}a^{8}+\frac{65}{529}a^{7}+\frac{139}{529}a^{6}+\frac{131}{529}a^{5}+\frac{56}{529}a^{4}+\frac{144}{529}a^{3}+\frac{39}{529}a^{2}-\frac{9}{23}a-\frac{205}{529}$, $\frac{1}{529}a^{22}-\frac{7}{529}a^{19}-\frac{1}{529}a^{18}+\frac{1}{529}a^{17}-\frac{4}{529}a^{16}-\frac{6}{529}a^{15}+\frac{5}{529}a^{14}-\frac{162}{529}a^{13}-\frac{31}{529}a^{12}-\frac{38}{529}a^{11}+\frac{76}{529}a^{10}-\frac{74}{529}a^{9}+\frac{71}{529}a^{8}-\frac{255}{529}a^{7}+\frac{101}{529}a^{6}-\frac{100}{529}a^{5}+\frac{77}{529}a^{4}-\frac{143}{529}a^{3}-\frac{168}{529}a^{2}+\frac{58}{529}a+\frac{139}{529}$, $\frac{1}{529}a^{23}+\frac{2}{529}a^{19}-\frac{11}{529}a^{18}-\frac{4}{529}a^{16}-\frac{6}{529}a^{15}-\frac{7}{529}a^{14}+\frac{203}{529}a^{13}+\frac{142}{529}a^{12}-\frac{14}{529}a^{11}+\frac{261}{529}a^{10}+\frac{87}{529}a^{9}-\frac{198}{529}a^{8}+\frac{84}{529}a^{7}+\frac{236}{529}a^{6}+\frac{44}{529}a^{5}-\frac{239}{529}a^{4}+\frac{35}{529}a^{3}-\frac{227}{529}a^{2}-\frac{88}{529}a-\frac{255}{529}$, $\frac{1}{371887}a^{24}+\frac{216}{371887}a^{23}-\frac{351}{371887}a^{22}-\frac{168}{371887}a^{21}-\frac{211}{371887}a^{20}+\frac{4295}{371887}a^{19}-\frac{1784}{371887}a^{18}+\frac{144}{371887}a^{17}-\frac{7565}{371887}a^{16}-\frac{1713}{371887}a^{15}-\frac{4751}{371887}a^{14}+\frac{14532}{371887}a^{13}+\frac{184625}{371887}a^{12}+\frac{183370}{371887}a^{11}+\frac{39892}{371887}a^{10}-\frac{131436}{371887}a^{9}-\frac{5227}{19573}a^{8}-\frac{131348}{371887}a^{7}-\frac{27069}{371887}a^{6}+\frac{255}{529}a^{5}+\frac{3554}{10051}a^{4}+\frac{127138}{371887}a^{3}-\frac{31860}{371887}a^{2}-\frac{100613}{371887}a+\frac{4752}{19573}$, $\frac{1}{26\!\cdots\!69}a^{25}+\frac{20\!\cdots\!62}{26\!\cdots\!69}a^{24}-\frac{23\!\cdots\!16}{26\!\cdots\!69}a^{23}-\frac{10\!\cdots\!50}{26\!\cdots\!69}a^{22}-\frac{76\!\cdots\!24}{26\!\cdots\!69}a^{21}+\frac{20\!\cdots\!78}{26\!\cdots\!69}a^{20}-\frac{61\!\cdots\!80}{26\!\cdots\!69}a^{19}-\frac{15\!\cdots\!21}{26\!\cdots\!69}a^{18}+\frac{31\!\cdots\!15}{26\!\cdots\!69}a^{17}-\frac{19\!\cdots\!30}{26\!\cdots\!69}a^{16}-\frac{42\!\cdots\!08}{26\!\cdots\!69}a^{15}-\frac{11\!\cdots\!28}{18\!\cdots\!87}a^{14}+\frac{12\!\cdots\!11}{26\!\cdots\!69}a^{13}+\frac{15\!\cdots\!54}{26\!\cdots\!69}a^{12}+\frac{20\!\cdots\!05}{26\!\cdots\!69}a^{11}+\frac{81\!\cdots\!87}{26\!\cdots\!69}a^{10}-\frac{10\!\cdots\!52}{26\!\cdots\!69}a^{9}-\frac{99\!\cdots\!42}{26\!\cdots\!69}a^{8}-\frac{73\!\cdots\!93}{26\!\cdots\!69}a^{7}-\frac{11\!\cdots\!19}{26\!\cdots\!69}a^{6}-\frac{20\!\cdots\!34}{70\!\cdots\!37}a^{5}+\frac{12\!\cdots\!85}{26\!\cdots\!69}a^{4}-\frac{46\!\cdots\!06}{26\!\cdots\!69}a^{3}+\frac{30\!\cdots\!63}{70\!\cdots\!37}a^{2}-\frac{10\!\cdots\!20}{26\!\cdots\!69}a-\frac{55\!\cdots\!85}{13\!\cdots\!51}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{19\!\cdots\!55}{26\!\cdots\!69}a^{25}-\frac{12\!\cdots\!99}{26\!\cdots\!69}a^{24}+\frac{21\!\cdots\!45}{26\!\cdots\!69}a^{23}-\frac{31\!\cdots\!18}{26\!\cdots\!69}a^{22}+\frac{20\!\cdots\!45}{26\!\cdots\!69}a^{21}-\frac{68\!\cdots\!27}{26\!\cdots\!69}a^{20}+\frac{22\!\cdots\!79}{26\!\cdots\!69}a^{19}-\frac{55\!\cdots\!74}{26\!\cdots\!69}a^{18}+\frac{11\!\cdots\!98}{26\!\cdots\!69}a^{17}-\frac{17\!\cdots\!15}{26\!\cdots\!69}a^{16}+\frac{21\!\cdots\!42}{26\!\cdots\!69}a^{15}-\frac{30\!\cdots\!68}{35\!\cdots\!53}a^{14}+\frac{19\!\cdots\!49}{26\!\cdots\!69}a^{13}-\frac{20\!\cdots\!98}{26\!\cdots\!69}a^{12}+\frac{20\!\cdots\!24}{26\!\cdots\!69}a^{11}-\frac{16\!\cdots\!93}{26\!\cdots\!69}a^{10}+\frac{97\!\cdots\!86}{26\!\cdots\!69}a^{9}-\frac{23\!\cdots\!14}{26\!\cdots\!69}a^{8}-\frac{27\!\cdots\!61}{26\!\cdots\!69}a^{7}+\frac{91\!\cdots\!25}{26\!\cdots\!69}a^{6}-\frac{33\!\cdots\!12}{70\!\cdots\!37}a^{5}+\frac{87\!\cdots\!58}{26\!\cdots\!69}a^{4}+\frac{11\!\cdots\!23}{26\!\cdots\!69}a^{3}-\frac{63\!\cdots\!10}{26\!\cdots\!69}a^{2}+\frac{26\!\cdots\!29}{26\!\cdots\!69}a+\frac{24\!\cdots\!99}{13\!\cdots\!51}$, $\frac{87\!\cdots\!03}{26\!\cdots\!69}a^{25}+\frac{14\!\cdots\!16}{26\!\cdots\!69}a^{24}+\frac{92\!\cdots\!43}{26\!\cdots\!69}a^{23}-\frac{67\!\cdots\!08}{26\!\cdots\!69}a^{22}+\frac{79\!\cdots\!22}{26\!\cdots\!69}a^{21}-\frac{23\!\cdots\!35}{26\!\cdots\!69}a^{20}+\frac{76\!\cdots\!20}{26\!\cdots\!69}a^{19}-\frac{17\!\cdots\!78}{26\!\cdots\!69}a^{18}+\frac{16\!\cdots\!39}{13\!\cdots\!51}a^{17}-\frac{44\!\cdots\!78}{26\!\cdots\!69}a^{16}+\frac{51\!\cdots\!36}{26\!\cdots\!69}a^{15}-\frac{71\!\cdots\!60}{35\!\cdots\!53}a^{14}+\frac{47\!\cdots\!91}{26\!\cdots\!69}a^{13}-\frac{60\!\cdots\!25}{26\!\cdots\!69}a^{12}+\frac{54\!\cdots\!13}{26\!\cdots\!69}a^{11}-\frac{32\!\cdots\!62}{26\!\cdots\!69}a^{10}+\frac{30\!\cdots\!45}{26\!\cdots\!69}a^{9}-\frac{65\!\cdots\!12}{26\!\cdots\!69}a^{8}-\frac{46\!\cdots\!79}{26\!\cdots\!69}a^{7}+\frac{16\!\cdots\!09}{26\!\cdots\!69}a^{6}-\frac{88\!\cdots\!02}{70\!\cdots\!37}a^{5}+\frac{25\!\cdots\!87}{26\!\cdots\!69}a^{4}+\frac{10\!\cdots\!04}{26\!\cdots\!69}a^{3}+\frac{26\!\cdots\!45}{26\!\cdots\!69}a^{2}+\frac{54\!\cdots\!47}{26\!\cdots\!69}a-\frac{13\!\cdots\!41}{13\!\cdots\!51}$, $\frac{17\!\cdots\!31}{26\!\cdots\!69}a^{25}+\frac{87\!\cdots\!51}{26\!\cdots\!69}a^{24}+\frac{19\!\cdots\!28}{26\!\cdots\!69}a^{23}-\frac{15\!\cdots\!66}{26\!\cdots\!69}a^{22}+\frac{17\!\cdots\!08}{26\!\cdots\!69}a^{21}-\frac{49\!\cdots\!24}{26\!\cdots\!69}a^{20}+\frac{16\!\cdots\!90}{26\!\cdots\!69}a^{19}-\frac{38\!\cdots\!67}{26\!\cdots\!69}a^{18}+\frac{38\!\cdots\!30}{13\!\cdots\!51}a^{17}-\frac{10\!\cdots\!94}{26\!\cdots\!69}a^{16}+\frac{12\!\cdots\!46}{26\!\cdots\!69}a^{15}-\frac{18\!\cdots\!78}{35\!\cdots\!53}a^{14}+\frac{11\!\cdots\!92}{26\!\cdots\!69}a^{13}-\frac{13\!\cdots\!37}{26\!\cdots\!69}a^{12}+\frac{12\!\cdots\!43}{26\!\cdots\!69}a^{11}-\frac{91\!\cdots\!46}{26\!\cdots\!69}a^{10}+\frac{64\!\cdots\!65}{26\!\cdots\!69}a^{9}-\frac{10\!\cdots\!56}{26\!\cdots\!69}a^{8}-\frac{16\!\cdots\!19}{26\!\cdots\!69}a^{7}+\frac{56\!\cdots\!66}{26\!\cdots\!69}a^{6}-\frac{21\!\cdots\!10}{70\!\cdots\!37}a^{5}+\frac{32\!\cdots\!18}{26\!\cdots\!69}a^{4}+\frac{12\!\cdots\!26}{26\!\cdots\!69}a^{3}-\frac{18\!\cdots\!46}{26\!\cdots\!69}a^{2}+\frac{14\!\cdots\!07}{26\!\cdots\!69}a-\frac{30\!\cdots\!19}{13\!\cdots\!51}$, $\frac{35\!\cdots\!55}{26\!\cdots\!69}a^{25}-\frac{19\!\cdots\!19}{26\!\cdots\!69}a^{24}+\frac{38\!\cdots\!90}{26\!\cdots\!69}a^{23}-\frac{54\!\cdots\!90}{26\!\cdots\!69}a^{22}+\frac{96\!\cdots\!67}{70\!\cdots\!37}a^{21}-\frac{12\!\cdots\!58}{26\!\cdots\!69}a^{20}+\frac{39\!\cdots\!69}{26\!\cdots\!69}a^{19}-\frac{96\!\cdots\!28}{26\!\cdots\!69}a^{18}+\frac{18\!\cdots\!83}{26\!\cdots\!69}a^{17}-\frac{79\!\cdots\!57}{70\!\cdots\!37}a^{16}+\frac{36\!\cdots\!28}{26\!\cdots\!69}a^{15}-\frac{53\!\cdots\!32}{35\!\cdots\!53}a^{14}+\frac{36\!\cdots\!03}{26\!\cdots\!69}a^{13}-\frac{38\!\cdots\!37}{26\!\cdots\!69}a^{12}+\frac{20\!\cdots\!58}{13\!\cdots\!51}a^{11}-\frac{29\!\cdots\!53}{26\!\cdots\!69}a^{10}+\frac{21\!\cdots\!06}{26\!\cdots\!69}a^{9}-\frac{76\!\cdots\!00}{26\!\cdots\!69}a^{8}-\frac{32\!\cdots\!16}{26\!\cdots\!69}a^{7}+\frac{12\!\cdots\!12}{26\!\cdots\!69}a^{6}-\frac{59\!\cdots\!73}{70\!\cdots\!37}a^{5}+\frac{13\!\cdots\!72}{26\!\cdots\!69}a^{4}+\frac{11\!\cdots\!35}{26\!\cdots\!69}a^{3}-\frac{55\!\cdots\!53}{26\!\cdots\!69}a^{2}+\frac{45\!\cdots\!36}{26\!\cdots\!69}a-\frac{11\!\cdots\!14}{13\!\cdots\!51}$, $\frac{11\!\cdots\!75}{11\!\cdots\!03}a^{25}-\frac{28\!\cdots\!08}{11\!\cdots\!03}a^{24}+\frac{12\!\cdots\!71}{11\!\cdots\!03}a^{23}-\frac{13\!\cdots\!45}{11\!\cdots\!03}a^{22}+\frac{57\!\cdots\!93}{60\!\cdots\!37}a^{21}-\frac{93\!\cdots\!86}{30\!\cdots\!19}a^{20}+\frac{11\!\cdots\!74}{11\!\cdots\!03}a^{19}-\frac{11\!\cdots\!13}{49\!\cdots\!61}a^{18}+\frac{52\!\cdots\!75}{11\!\cdots\!03}a^{17}-\frac{77\!\cdots\!73}{11\!\cdots\!03}a^{16}+\frac{94\!\cdots\!91}{11\!\cdots\!03}a^{15}-\frac{13\!\cdots\!60}{15\!\cdots\!11}a^{14}+\frac{45\!\cdots\!81}{60\!\cdots\!37}a^{13}-\frac{95\!\cdots\!17}{11\!\cdots\!03}a^{12}+\frac{91\!\cdots\!11}{11\!\cdots\!03}a^{11}-\frac{65\!\cdots\!51}{11\!\cdots\!03}a^{10}+\frac{46\!\cdots\!24}{11\!\cdots\!03}a^{9}-\frac{23\!\cdots\!20}{30\!\cdots\!19}a^{8}-\frac{15\!\cdots\!98}{11\!\cdots\!03}a^{7}+\frac{17\!\cdots\!81}{49\!\cdots\!61}a^{6}-\frac{16\!\cdots\!61}{30\!\cdots\!19}a^{5}+\frac{29\!\cdots\!35}{11\!\cdots\!03}a^{4}+\frac{86\!\cdots\!17}{11\!\cdots\!03}a^{3}-\frac{14\!\cdots\!10}{11\!\cdots\!03}a^{2}+\frac{99\!\cdots\!27}{11\!\cdots\!03}a-\frac{23\!\cdots\!51}{60\!\cdots\!37}$, $\frac{37\!\cdots\!00}{26\!\cdots\!69}a^{25}-\frac{29\!\cdots\!75}{26\!\cdots\!69}a^{24}+\frac{40\!\cdots\!98}{26\!\cdots\!69}a^{23}-\frac{37\!\cdots\!00}{26\!\cdots\!69}a^{22}+\frac{36\!\cdots\!28}{26\!\cdots\!69}a^{21}-\frac{10\!\cdots\!47}{26\!\cdots\!69}a^{20}+\frac{36\!\cdots\!96}{26\!\cdots\!69}a^{19}-\frac{85\!\cdots\!23}{26\!\cdots\!69}a^{18}+\frac{16\!\cdots\!76}{26\!\cdots\!69}a^{17}-\frac{24\!\cdots\!94}{26\!\cdots\!69}a^{16}+\frac{28\!\cdots\!35}{26\!\cdots\!69}a^{15}-\frac{40\!\cdots\!29}{35\!\cdots\!53}a^{14}+\frac{26\!\cdots\!57}{26\!\cdots\!69}a^{13}-\frac{30\!\cdots\!33}{26\!\cdots\!69}a^{12}+\frac{28\!\cdots\!70}{26\!\cdots\!69}a^{11}-\frac{20\!\cdots\!11}{26\!\cdots\!69}a^{10}+\frac{15\!\cdots\!60}{26\!\cdots\!69}a^{9}-\frac{31\!\cdots\!34}{26\!\cdots\!69}a^{8}-\frac{32\!\cdots\!25}{26\!\cdots\!69}a^{7}+\frac{11\!\cdots\!34}{26\!\cdots\!69}a^{6}-\frac{46\!\cdots\!64}{70\!\cdots\!37}a^{5}+\frac{69\!\cdots\!15}{26\!\cdots\!69}a^{4}+\frac{31\!\cdots\!48}{26\!\cdots\!69}a^{3}-\frac{42\!\cdots\!25}{26\!\cdots\!69}a^{2}+\frac{33\!\cdots\!10}{26\!\cdots\!69}a-\frac{74\!\cdots\!64}{13\!\cdots\!51}$, $\frac{17\!\cdots\!42}{26\!\cdots\!69}a^{25}+\frac{93\!\cdots\!35}{26\!\cdots\!69}a^{24}+\frac{19\!\cdots\!18}{26\!\cdots\!69}a^{23}-\frac{51\!\cdots\!88}{26\!\cdots\!69}a^{22}+\frac{17\!\cdots\!98}{26\!\cdots\!69}a^{21}-\frac{40\!\cdots\!61}{26\!\cdots\!69}a^{20}+\frac{14\!\cdots\!29}{26\!\cdots\!69}a^{19}-\frac{31\!\cdots\!42}{26\!\cdots\!69}a^{18}+\frac{59\!\cdots\!02}{26\!\cdots\!69}a^{17}-\frac{81\!\cdots\!12}{26\!\cdots\!69}a^{16}+\frac{49\!\cdots\!19}{13\!\cdots\!51}a^{15}-\frac{12\!\cdots\!31}{35\!\cdots\!53}a^{14}+\frac{79\!\cdots\!68}{26\!\cdots\!69}a^{13}-\frac{10\!\cdots\!77}{26\!\cdots\!69}a^{12}+\frac{76\!\cdots\!58}{26\!\cdots\!69}a^{11}-\frac{60\!\cdots\!14}{26\!\cdots\!69}a^{10}+\frac{37\!\cdots\!17}{26\!\cdots\!69}a^{9}+\frac{75\!\cdots\!46}{26\!\cdots\!69}a^{8}-\frac{56\!\cdots\!26}{13\!\cdots\!51}a^{7}+\frac{53\!\cdots\!62}{26\!\cdots\!69}a^{6}-\frac{14\!\cdots\!24}{70\!\cdots\!37}a^{5}+\frac{11\!\cdots\!56}{26\!\cdots\!69}a^{4}+\frac{72\!\cdots\!41}{13\!\cdots\!51}a^{3}-\frac{16\!\cdots\!65}{26\!\cdots\!69}a^{2}+\frac{81\!\cdots\!87}{26\!\cdots\!69}a-\frac{21\!\cdots\!96}{13\!\cdots\!51}$, $\frac{24\!\cdots\!80}{26\!\cdots\!69}a^{25}-\frac{94\!\cdots\!99}{26\!\cdots\!69}a^{24}+\frac{26\!\cdots\!03}{26\!\cdots\!69}a^{23}-\frac{30\!\cdots\!37}{26\!\cdots\!69}a^{22}+\frac{24\!\cdots\!06}{26\!\cdots\!69}a^{21}-\frac{40\!\cdots\!64}{13\!\cdots\!51}a^{20}+\frac{25\!\cdots\!22}{26\!\cdots\!69}a^{19}-\frac{60\!\cdots\!71}{26\!\cdots\!69}a^{18}+\frac{11\!\cdots\!10}{26\!\cdots\!69}a^{17}-\frac{16\!\cdots\!71}{26\!\cdots\!69}a^{16}+\frac{19\!\cdots\!45}{26\!\cdots\!69}a^{15}-\frac{23\!\cdots\!03}{35\!\cdots\!53}a^{14}+\frac{13\!\cdots\!83}{26\!\cdots\!69}a^{13}-\frac{14\!\cdots\!60}{26\!\cdots\!69}a^{12}+\frac{13\!\cdots\!66}{26\!\cdots\!69}a^{11}-\frac{82\!\cdots\!10}{26\!\cdots\!69}a^{10}-\frac{12\!\cdots\!66}{26\!\cdots\!69}a^{9}+\frac{40\!\cdots\!80}{26\!\cdots\!69}a^{8}-\frac{78\!\cdots\!36}{26\!\cdots\!69}a^{7}+\frac{15\!\cdots\!55}{26\!\cdots\!69}a^{6}-\frac{39\!\cdots\!34}{70\!\cdots\!37}a^{5}+\frac{72\!\cdots\!54}{26\!\cdots\!69}a^{4}+\frac{22\!\cdots\!18}{26\!\cdots\!69}a^{3}-\frac{55\!\cdots\!38}{26\!\cdots\!69}a^{2}+\frac{14\!\cdots\!66}{26\!\cdots\!69}a+\frac{30\!\cdots\!89}{13\!\cdots\!51}$, $\frac{13\!\cdots\!98}{11\!\cdots\!03}a^{25}-\frac{30\!\cdots\!93}{11\!\cdots\!03}a^{24}+\frac{14\!\cdots\!58}{11\!\cdots\!03}a^{23}-\frac{15\!\cdots\!22}{11\!\cdots\!03}a^{22}+\frac{68\!\cdots\!86}{60\!\cdots\!37}a^{21}-\frac{40\!\cdots\!26}{11\!\cdots\!03}a^{20}+\frac{58\!\cdots\!50}{49\!\cdots\!61}a^{19}-\frac{31\!\cdots\!21}{11\!\cdots\!03}a^{18}+\frac{61\!\cdots\!61}{11\!\cdots\!03}a^{17}-\frac{92\!\cdots\!62}{11\!\cdots\!03}a^{16}+\frac{11\!\cdots\!35}{11\!\cdots\!03}a^{15}-\frac{15\!\cdots\!80}{15\!\cdots\!11}a^{14}+\frac{55\!\cdots\!55}{60\!\cdots\!37}a^{13}-\frac{11\!\cdots\!47}{11\!\cdots\!03}a^{12}+\frac{10\!\cdots\!77}{11\!\cdots\!03}a^{11}-\frac{81\!\cdots\!88}{11\!\cdots\!03}a^{10}+\frac{55\!\cdots\!42}{11\!\cdots\!03}a^{9}-\frac{12\!\cdots\!54}{11\!\cdots\!03}a^{8}-\frac{16\!\cdots\!93}{11\!\cdots\!03}a^{7}+\frac{47\!\cdots\!70}{11\!\cdots\!03}a^{6}-\frac{18\!\cdots\!39}{30\!\cdots\!19}a^{5}+\frac{35\!\cdots\!17}{11\!\cdots\!03}a^{4}+\frac{85\!\cdots\!87}{11\!\cdots\!03}a^{3}-\frac{79\!\cdots\!32}{49\!\cdots\!61}a^{2}+\frac{12\!\cdots\!86}{11\!\cdots\!03}a-\frac{28\!\cdots\!91}{60\!\cdots\!37}$, $\frac{10\!\cdots\!10}{26\!\cdots\!69}a^{25}+\frac{45\!\cdots\!82}{70\!\cdots\!37}a^{24}+\frac{10\!\cdots\!52}{26\!\cdots\!69}a^{23}-\frac{91\!\cdots\!74}{26\!\cdots\!69}a^{22}+\frac{96\!\cdots\!01}{26\!\cdots\!69}a^{21}-\frac{28\!\cdots\!29}{26\!\cdots\!69}a^{20}+\frac{95\!\cdots\!70}{26\!\cdots\!69}a^{19}-\frac{22\!\cdots\!62}{26\!\cdots\!69}a^{18}+\frac{41\!\cdots\!50}{26\!\cdots\!69}a^{17}-\frac{58\!\cdots\!18}{26\!\cdots\!69}a^{16}+\frac{66\!\cdots\!40}{26\!\cdots\!69}a^{15}-\frac{86\!\cdots\!40}{35\!\cdots\!53}a^{14}+\frac{50\!\cdots\!30}{26\!\cdots\!69}a^{13}-\frac{56\!\cdots\!95}{26\!\cdots\!69}a^{12}+\frac{46\!\cdots\!71}{26\!\cdots\!69}a^{11}-\frac{22\!\cdots\!33}{26\!\cdots\!69}a^{10}+\frac{46\!\cdots\!81}{13\!\cdots\!51}a^{9}+\frac{21\!\cdots\!69}{26\!\cdots\!69}a^{8}-\frac{33\!\cdots\!80}{26\!\cdots\!69}a^{7}+\frac{25\!\cdots\!29}{13\!\cdots\!51}a^{6}-\frac{15\!\cdots\!37}{70\!\cdots\!37}a^{5}+\frac{22\!\cdots\!41}{26\!\cdots\!69}a^{4}+\frac{10\!\cdots\!20}{26\!\cdots\!69}a^{3}-\frac{14\!\cdots\!60}{26\!\cdots\!69}a^{2}+\frac{71\!\cdots\!91}{13\!\cdots\!51}a-\frac{26\!\cdots\!40}{13\!\cdots\!51}$, $\frac{19\!\cdots\!33}{26\!\cdots\!69}a^{25}-\frac{13\!\cdots\!53}{26\!\cdots\!69}a^{24}+\frac{19\!\cdots\!66}{26\!\cdots\!69}a^{23}-\frac{34\!\cdots\!89}{26\!\cdots\!69}a^{22}+\frac{18\!\cdots\!45}{26\!\cdots\!69}a^{21}-\frac{69\!\cdots\!64}{26\!\cdots\!69}a^{20}+\frac{21\!\cdots\!19}{26\!\cdots\!69}a^{19}-\frac{53\!\cdots\!48}{26\!\cdots\!69}a^{18}+\frac{10\!\cdots\!33}{26\!\cdots\!69}a^{17}-\frac{15\!\cdots\!06}{26\!\cdots\!69}a^{16}+\frac{18\!\cdots\!80}{26\!\cdots\!69}a^{15}-\frac{26\!\cdots\!55}{35\!\cdots\!53}a^{14}+\frac{17\!\cdots\!18}{26\!\cdots\!69}a^{13}-\frac{19\!\cdots\!75}{26\!\cdots\!69}a^{12}+\frac{19\!\cdots\!48}{26\!\cdots\!69}a^{11}-\frac{12\!\cdots\!95}{26\!\cdots\!69}a^{10}+\frac{10\!\cdots\!44}{26\!\cdots\!69}a^{9}-\frac{28\!\cdots\!69}{26\!\cdots\!69}a^{8}-\frac{30\!\cdots\!76}{26\!\cdots\!69}a^{7}+\frac{60\!\cdots\!64}{26\!\cdots\!69}a^{6}-\frac{35\!\cdots\!97}{70\!\cdots\!37}a^{5}+\frac{56\!\cdots\!43}{26\!\cdots\!69}a^{4}+\frac{20\!\cdots\!27}{26\!\cdots\!69}a^{3}-\frac{24\!\cdots\!40}{26\!\cdots\!69}a^{2}+\frac{25\!\cdots\!67}{26\!\cdots\!69}a-\frac{49\!\cdots\!90}{13\!\cdots\!51}$, $\frac{40\!\cdots\!48}{26\!\cdots\!69}a^{25}-\frac{33\!\cdots\!40}{70\!\cdots\!37}a^{24}+\frac{44\!\cdots\!18}{26\!\cdots\!69}a^{23}-\frac{50\!\cdots\!24}{26\!\cdots\!69}a^{22}+\frac{40\!\cdots\!14}{26\!\cdots\!69}a^{21}-\frac{12\!\cdots\!76}{26\!\cdots\!69}a^{20}+\frac{42\!\cdots\!00}{26\!\cdots\!69}a^{19}-\frac{10\!\cdots\!18}{26\!\cdots\!69}a^{18}+\frac{19\!\cdots\!15}{26\!\cdots\!69}a^{17}-\frac{15\!\cdots\!75}{13\!\cdots\!51}a^{16}+\frac{37\!\cdots\!29}{26\!\cdots\!69}a^{15}-\frac{53\!\cdots\!13}{35\!\cdots\!53}a^{14}+\frac{35\!\cdots\!62}{26\!\cdots\!69}a^{13}-\frac{38\!\cdots\!17}{26\!\cdots\!69}a^{12}+\frac{37\!\cdots\!70}{26\!\cdots\!69}a^{11}-\frac{28\!\cdots\!45}{26\!\cdots\!69}a^{10}+\frac{19\!\cdots\!20}{26\!\cdots\!69}a^{9}-\frac{57\!\cdots\!79}{26\!\cdots\!69}a^{8}-\frac{40\!\cdots\!51}{26\!\cdots\!69}a^{7}+\frac{14\!\cdots\!11}{26\!\cdots\!69}a^{6}-\frac{60\!\cdots\!25}{70\!\cdots\!37}a^{5}+\frac{12\!\cdots\!00}{26\!\cdots\!69}a^{4}+\frac{12\!\cdots\!55}{26\!\cdots\!69}a^{3}-\frac{62\!\cdots\!80}{26\!\cdots\!69}a^{2}+\frac{49\!\cdots\!55}{26\!\cdots\!69}a-\frac{99\!\cdots\!59}{13\!\cdots\!51}$
| 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: | \( 1140159040.1752596 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 1140159040.1752596 \cdot 3}{2\cdot\sqrt{11543464978247129822340588525652022819543}}\cr\approx \mathstrut & 0.378640345672729 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 52 |
| The 16 conjugacy class representatives for $D_{26}$ |
| Character table for $D_{26}$ |
Intermediate fields
| \(\Q(\sqrt{-23}) \), 13.1.22402896724819285921.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 26 sibling: | deg 26 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.13.0.1}{13} }^{2}$ | ${\href{/padicField/3.13.0.1}{13} }^{2}$ | $26$ | $26$ | $26$ | ${\href{/padicField/13.2.0.1}{2} }^{12}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $26$ | ${\href{/padicField/19.2.0.1}{2} }^{13}$ | R | ${\href{/padicField/29.2.0.1}{2} }^{12}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{12}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{13}$ | ${\href{/padicField/41.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/47.2.0.1}{2} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $26$ | ${\href{/padicField/59.2.0.1}{2} }^{12}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(73\)
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |