Normalized defining polynomial
\( x^{26} - 11 x^{25} + 8 x^{24} + 220 x^{23} + 18 x^{22} - 3704 x^{21} - 2308 x^{20} + 32061 x^{19} + \cdots + 1492440721 \)
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: |
\(-338317693370822771423393222562696496647525569958371057767\)
\(\medspace = -\,7^{13}\cdot 79^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(149.35\) | 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: | $7^{1/2}79^{12/13}\approx 149.34985200406032$ | ||
| Ramified primes: |
\(7\), \(79\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $26$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(553=7\cdot 79\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{553}(512,·)$, $\chi_{553}(1,·)$, $\chi_{553}(258,·)$, $\chi_{553}(225,·)$, $\chi_{553}(8,·)$, $\chi_{553}(204,·)$, $\chi_{553}(141,·)$, $\chi_{553}(526,·)$, $\chi_{553}(337,·)$, $\chi_{553}(146,·)$, $\chi_{553}(405,·)$, $\chi_{553}(22,·)$, $\chi_{553}(538,·)$, $\chi_{553}(475,·)$, $\chi_{553}(223,·)$, $\chi_{553}(97,·)$, $\chi_{553}(482,·)$, $\chi_{553}(484,·)$, $\chi_{553}(64,·)$, $\chi_{553}(302,·)$, $\chi_{553}(176,·)$, $\chi_{553}(433,·)$, $\chi_{553}(496,·)$, $\chi_{553}(125,·)$, $\chi_{553}(62,·)$, $\chi_{553}(447,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{4096}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{23}a^{15}-\frac{2}{23}a^{14}-\frac{2}{23}a^{13}-\frac{6}{23}a^{12}-\frac{5}{23}a^{11}+\frac{3}{23}a^{10}-\frac{2}{23}a^{9}-\frac{4}{23}a^{8}+\frac{8}{23}a^{7}+\frac{10}{23}a^{6}-\frac{4}{23}a^{5}-\frac{11}{23}a^{4}+\frac{2}{23}a^{3}+\frac{10}{23}a^{2}+\frac{2}{23}a$, $\frac{1}{23}a^{16}-\frac{6}{23}a^{14}-\frac{10}{23}a^{13}+\frac{6}{23}a^{12}-\frac{7}{23}a^{11}+\frac{4}{23}a^{10}-\frac{8}{23}a^{9}+\frac{3}{23}a^{7}-\frac{7}{23}a^{6}+\frac{4}{23}a^{5}+\frac{3}{23}a^{4}-\frac{9}{23}a^{3}-\frac{1}{23}a^{2}+\frac{4}{23}a$, $\frac{1}{23}a^{17}+\frac{1}{23}a^{14}-\frac{6}{23}a^{13}+\frac{3}{23}a^{12}-\frac{3}{23}a^{11}+\frac{10}{23}a^{10}+\frac{11}{23}a^{9}+\frac{2}{23}a^{8}-\frac{5}{23}a^{7}-\frac{5}{23}a^{6}+\frac{2}{23}a^{5}-\frac{6}{23}a^{4}+\frac{11}{23}a^{3}-\frac{5}{23}a^{2}-\frac{11}{23}a$, $\frac{1}{23}a^{18}-\frac{4}{23}a^{14}+\frac{5}{23}a^{13}+\frac{3}{23}a^{12}-\frac{8}{23}a^{11}+\frac{8}{23}a^{10}+\frac{4}{23}a^{9}-\frac{1}{23}a^{8}+\frac{10}{23}a^{7}-\frac{8}{23}a^{6}-\frac{2}{23}a^{5}-\frac{1}{23}a^{4}-\frac{7}{23}a^{3}+\frac{2}{23}a^{2}-\frac{2}{23}a$, $\frac{1}{23}a^{19}-\frac{3}{23}a^{14}-\frac{5}{23}a^{13}-\frac{9}{23}a^{12}+\frac{11}{23}a^{11}-\frac{7}{23}a^{10}-\frac{9}{23}a^{9}-\frac{6}{23}a^{8}+\frac{1}{23}a^{7}-\frac{8}{23}a^{6}+\frac{6}{23}a^{5}-\frac{5}{23}a^{4}+\frac{10}{23}a^{3}-\frac{8}{23}a^{2}+\frac{8}{23}a$, $\frac{1}{23}a^{20}-\frac{11}{23}a^{14}+\frac{8}{23}a^{13}-\frac{7}{23}a^{12}+\frac{1}{23}a^{11}+\frac{11}{23}a^{9}-\frac{11}{23}a^{8}-\frac{7}{23}a^{7}-\frac{10}{23}a^{6}+\frac{6}{23}a^{5}-\frac{2}{23}a^{3}-\frac{8}{23}a^{2}+\frac{6}{23}a$, $\frac{1}{529}a^{21}+\frac{8}{529}a^{20}+\frac{1}{529}a^{19}-\frac{9}{529}a^{18}+\frac{5}{529}a^{17}-\frac{7}{529}a^{16}+\frac{5}{529}a^{15}-\frac{262}{529}a^{14}-\frac{261}{529}a^{13}-\frac{237}{529}a^{12}-\frac{24}{529}a^{11}-\frac{182}{529}a^{10}-\frac{27}{529}a^{9}+\frac{84}{529}a^{8}-\frac{165}{529}a^{7}-\frac{125}{529}a^{6}+\frac{13}{529}a^{5}+\frac{166}{529}a^{4}+\frac{38}{529}a^{3}+\frac{242}{529}a^{2}+\frac{9}{23}a$, $\frac{1}{529}a^{22}+\frac{6}{529}a^{20}+\frac{6}{529}a^{19}+\frac{8}{529}a^{18}-\frac{1}{529}a^{17}-\frac{8}{529}a^{16}-\frac{3}{529}a^{15}+\frac{87}{529}a^{14}+\frac{172}{529}a^{13}-\frac{37}{529}a^{12}+\frac{263}{529}a^{11}+\frac{210}{529}a^{10}-\frac{22}{529}a^{9}-\frac{124}{529}a^{8}-\frac{116}{529}a^{7}+\frac{231}{529}a^{6}-\frac{99}{529}a^{5}+\frac{182}{529}a^{4}+\frac{122}{529}a^{3}+\frac{226}{529}a^{2}-\frac{2}{23}a$, $\frac{1}{529}a^{23}+\frac{4}{529}a^{20}+\frac{2}{529}a^{19}+\frac{7}{529}a^{18}+\frac{8}{529}a^{17}-\frac{7}{529}a^{16}+\frac{11}{529}a^{15}+\frac{249}{529}a^{14}-\frac{173}{529}a^{13}-\frac{224}{529}a^{12}+\frac{124}{529}a^{11}-\frac{218}{529}a^{10}-\frac{261}{529}a^{9}+\frac{254}{529}a^{8}+\frac{232}{529}a^{7}+\frac{191}{529}a^{6}+\frac{35}{529}a^{5}-\frac{9}{23}a^{4}-\frac{2}{529}a^{3}+\frac{43}{529}a^{2}-\frac{3}{23}a$, $\frac{1}{28823623}a^{24}+\frac{22021}{28823623}a^{23}+\frac{18345}{28823623}a^{22}+\frac{16920}{28823623}a^{21}+\frac{441906}{28823623}a^{20}+\frac{556331}{28823623}a^{19}+\frac{210380}{28823623}a^{18}+\frac{137171}{28823623}a^{17}-\frac{96227}{28823623}a^{16}-\frac{575433}{28823623}a^{15}-\frac{2958}{28823623}a^{14}-\frac{12547399}{28823623}a^{13}-\frac{3846494}{28823623}a^{12}+\frac{10541683}{28823623}a^{11}+\frac{403067}{28823623}a^{10}+\frac{8821288}{28823623}a^{9}-\frac{4396675}{28823623}a^{8}+\frac{489294}{28823623}a^{7}-\frac{106738}{279841}a^{6}-\frac{10842518}{28823623}a^{5}-\frac{12333564}{28823623}a^{4}-\frac{2600531}{28823623}a^{3}+\frac{172119}{1253201}a^{2}+\frac{803}{54487}a-\frac{216}{2369}$, $\frac{1}{33\!\cdots\!91}a^{25}+\frac{12\!\cdots\!00}{33\!\cdots\!91}a^{24}+\frac{29\!\cdots\!32}{33\!\cdots\!91}a^{23}-\frac{14\!\cdots\!01}{33\!\cdots\!91}a^{22}+\frac{11\!\cdots\!25}{33\!\cdots\!91}a^{21}+\frac{71\!\cdots\!42}{33\!\cdots\!91}a^{20}+\frac{28\!\cdots\!33}{33\!\cdots\!91}a^{19}-\frac{62\!\cdots\!67}{33\!\cdots\!91}a^{18}+\frac{31\!\cdots\!60}{33\!\cdots\!91}a^{17}+\frac{77\!\cdots\!99}{33\!\cdots\!91}a^{16}-\frac{63\!\cdots\!36}{33\!\cdots\!91}a^{15}-\frac{44\!\cdots\!89}{33\!\cdots\!91}a^{14}+\frac{40\!\cdots\!03}{33\!\cdots\!91}a^{13}-\frac{23\!\cdots\!02}{33\!\cdots\!91}a^{12}+\frac{90\!\cdots\!49}{33\!\cdots\!91}a^{11}-\frac{16\!\cdots\!53}{33\!\cdots\!91}a^{10}-\frac{12\!\cdots\!58}{33\!\cdots\!91}a^{9}-\frac{10\!\cdots\!28}{33\!\cdots\!91}a^{8}-\frac{10\!\cdots\!96}{33\!\cdots\!91}a^{7}+\frac{14\!\cdots\!85}{33\!\cdots\!91}a^{6}+\frac{74\!\cdots\!99}{33\!\cdots\!91}a^{5}-\frac{15\!\cdots\!97}{33\!\cdots\!91}a^{4}+\frac{10\!\cdots\!40}{33\!\cdots\!91}a^{3}-\frac{70\!\cdots\!83}{14\!\cdots\!17}a^{2}-\frac{18\!\cdots\!47}{62\!\cdots\!79}a-\frac{70\!\cdots\!05}{27\!\cdots\!73}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $23$ |
Class group and class number
$C_{8511191}$, which has order $8511191$ (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: |
$\frac{34\!\cdots\!22}{33\!\cdots\!91}a^{25}-\frac{26\!\cdots\!66}{33\!\cdots\!91}a^{24}-\frac{14\!\cdots\!56}{33\!\cdots\!91}a^{23}+\frac{15\!\cdots\!59}{33\!\cdots\!91}a^{22}+\frac{69\!\cdots\!09}{33\!\cdots\!91}a^{21}-\frac{21\!\cdots\!96}{33\!\cdots\!91}a^{20}-\frac{24\!\cdots\!67}{33\!\cdots\!91}a^{19}+\frac{23\!\cdots\!63}{33\!\cdots\!91}a^{18}+\frac{29\!\cdots\!72}{33\!\cdots\!91}a^{17}-\frac{10\!\cdots\!56}{33\!\cdots\!91}a^{16}-\frac{29\!\cdots\!42}{33\!\cdots\!91}a^{15}+\frac{34\!\cdots\!51}{33\!\cdots\!91}a^{14}+\frac{10\!\cdots\!55}{33\!\cdots\!91}a^{13}+\frac{40\!\cdots\!09}{33\!\cdots\!91}a^{12}-\frac{25\!\cdots\!85}{33\!\cdots\!91}a^{11}+\frac{63\!\cdots\!90}{33\!\cdots\!91}a^{10}+\frac{20\!\cdots\!42}{33\!\cdots\!91}a^{9}+\frac{48\!\cdots\!81}{33\!\cdots\!91}a^{8}+\frac{48\!\cdots\!42}{33\!\cdots\!91}a^{7}+\frac{14\!\cdots\!36}{33\!\cdots\!91}a^{6}+\frac{89\!\cdots\!95}{14\!\cdots\!17}a^{5}+\frac{45\!\cdots\!03}{33\!\cdots\!91}a^{4}+\frac{49\!\cdots\!25}{33\!\cdots\!91}a^{3}+\frac{39\!\cdots\!66}{14\!\cdots\!17}a^{2}+\frac{14\!\cdots\!96}{62\!\cdots\!79}a+\frac{61\!\cdots\!20}{27\!\cdots\!73}$, $\frac{31\!\cdots\!56}{33\!\cdots\!91}a^{25}-\frac{20\!\cdots\!15}{33\!\cdots\!91}a^{24}-\frac{18\!\cdots\!70}{33\!\cdots\!91}a^{23}+\frac{15\!\cdots\!72}{33\!\cdots\!91}a^{22}+\frac{12\!\cdots\!24}{33\!\cdots\!91}a^{21}-\frac{21\!\cdots\!70}{33\!\cdots\!91}a^{20}-\frac{33\!\cdots\!46}{33\!\cdots\!91}a^{19}+\frac{23\!\cdots\!24}{33\!\cdots\!91}a^{18}+\frac{36\!\cdots\!09}{33\!\cdots\!91}a^{17}-\frac{10\!\cdots\!31}{33\!\cdots\!91}a^{16}-\frac{34\!\cdots\!96}{33\!\cdots\!91}a^{15}+\frac{30\!\cdots\!57}{33\!\cdots\!91}a^{14}+\frac{11\!\cdots\!05}{33\!\cdots\!91}a^{13}+\frac{70\!\cdots\!95}{33\!\cdots\!91}a^{12}-\frac{30\!\cdots\!13}{33\!\cdots\!91}a^{11}+\frac{60\!\cdots\!99}{33\!\cdots\!91}a^{10}+\frac{21\!\cdots\!00}{33\!\cdots\!91}a^{9}+\frac{55\!\cdots\!78}{33\!\cdots\!91}a^{8}+\frac{52\!\cdots\!89}{33\!\cdots\!91}a^{7}+\frac{15\!\cdots\!45}{33\!\cdots\!91}a^{6}+\frac{96\!\cdots\!27}{14\!\cdots\!17}a^{5}+\frac{50\!\cdots\!46}{33\!\cdots\!91}a^{4}+\frac{54\!\cdots\!61}{33\!\cdots\!91}a^{3}+\frac{44\!\cdots\!65}{14\!\cdots\!17}a^{2}+\frac{16\!\cdots\!61}{62\!\cdots\!79}a+\frac{72\!\cdots\!61}{27\!\cdots\!73}$, $\frac{21\!\cdots\!54}{33\!\cdots\!91}a^{25}-\frac{16\!\cdots\!75}{33\!\cdots\!91}a^{24}-\frac{95\!\cdots\!12}{33\!\cdots\!91}a^{23}+\frac{94\!\cdots\!16}{33\!\cdots\!91}a^{22}+\frac{55\!\cdots\!26}{33\!\cdots\!91}a^{21}-\frac{13\!\cdots\!29}{33\!\cdots\!91}a^{20}-\frac{17\!\cdots\!22}{33\!\cdots\!91}a^{19}+\frac{14\!\cdots\!89}{33\!\cdots\!91}a^{18}+\frac{20\!\cdots\!67}{33\!\cdots\!91}a^{17}-\frac{66\!\cdots\!40}{33\!\cdots\!91}a^{16}-\frac{19\!\cdots\!62}{33\!\cdots\!91}a^{15}+\frac{20\!\cdots\!97}{33\!\cdots\!91}a^{14}+\frac{70\!\cdots\!41}{33\!\cdots\!91}a^{13}+\frac{33\!\cdots\!65}{33\!\cdots\!91}a^{12}-\frac{17\!\cdots\!49}{33\!\cdots\!91}a^{11}+\frac{38\!\cdots\!02}{33\!\cdots\!91}a^{10}+\frac{13\!\cdots\!69}{33\!\cdots\!91}a^{9}+\frac{32\!\cdots\!16}{33\!\cdots\!91}a^{8}+\frac{32\!\cdots\!67}{33\!\cdots\!91}a^{7}+\frac{91\!\cdots\!39}{33\!\cdots\!91}a^{6}+\frac{58\!\cdots\!68}{14\!\cdots\!17}a^{5}+\frac{29\!\cdots\!62}{33\!\cdots\!91}a^{4}+\frac{32\!\cdots\!85}{33\!\cdots\!91}a^{3}+\frac{26\!\cdots\!50}{14\!\cdots\!17}a^{2}+\frac{96\!\cdots\!82}{62\!\cdots\!79}a+\frac{40\!\cdots\!01}{27\!\cdots\!73}$, $\frac{91\!\cdots\!14}{33\!\cdots\!91}a^{25}-\frac{67\!\cdots\!26}{33\!\cdots\!91}a^{24}-\frac{42\!\cdots\!64}{33\!\cdots\!91}a^{23}+\frac{41\!\cdots\!95}{33\!\cdots\!91}a^{22}+\frac{24\!\cdots\!61}{33\!\cdots\!91}a^{21}-\frac{57\!\cdots\!40}{33\!\cdots\!91}a^{20}-\frac{75\!\cdots\!03}{33\!\cdots\!91}a^{19}+\frac{63\!\cdots\!65}{33\!\cdots\!91}a^{18}+\frac{86\!\cdots\!82}{33\!\cdots\!91}a^{17}-\frac{28\!\cdots\!21}{33\!\cdots\!91}a^{16}-\frac{85\!\cdots\!86}{33\!\cdots\!91}a^{15}+\frac{88\!\cdots\!13}{33\!\cdots\!91}a^{14}+\frac{30\!\cdots\!45}{33\!\cdots\!91}a^{13}+\frac{14\!\cdots\!81}{33\!\cdots\!91}a^{12}-\frac{74\!\cdots\!67}{33\!\cdots\!91}a^{11}+\frac{16\!\cdots\!50}{33\!\cdots\!91}a^{10}+\frac{57\!\cdots\!82}{33\!\cdots\!91}a^{9}+\frac{13\!\cdots\!14}{33\!\cdots\!91}a^{8}+\frac{13\!\cdots\!12}{33\!\cdots\!91}a^{7}+\frac{39\!\cdots\!06}{33\!\cdots\!91}a^{6}+\frac{24\!\cdots\!01}{14\!\cdots\!17}a^{5}+\frac{12\!\cdots\!81}{33\!\cdots\!91}a^{4}+\frac{13\!\cdots\!39}{33\!\cdots\!91}a^{3}+\frac{11\!\cdots\!24}{14\!\cdots\!17}a^{2}+\frac{41\!\cdots\!42}{62\!\cdots\!79}a+\frac{17\!\cdots\!38}{27\!\cdots\!73}$, $\frac{72\!\cdots\!52}{33\!\cdots\!91}a^{25}-\frac{56\!\cdots\!83}{33\!\cdots\!91}a^{24}-\frac{30\!\cdots\!72}{33\!\cdots\!91}a^{23}+\frac{31\!\cdots\!53}{33\!\cdots\!91}a^{22}+\frac{17\!\cdots\!73}{33\!\cdots\!91}a^{21}-\frac{44\!\cdots\!65}{33\!\cdots\!91}a^{20}-\frac{56\!\cdots\!25}{33\!\cdots\!91}a^{19}+\frac{48\!\cdots\!53}{33\!\cdots\!91}a^{18}+\frac{66\!\cdots\!56}{33\!\cdots\!91}a^{17}-\frac{22\!\cdots\!13}{33\!\cdots\!91}a^{16}-\frac{65\!\cdots\!10}{33\!\cdots\!91}a^{15}+\frac{67\!\cdots\!37}{33\!\cdots\!91}a^{14}+\frac{23\!\cdots\!05}{33\!\cdots\!91}a^{13}+\frac{10\!\cdots\!01}{33\!\cdots\!91}a^{12}-\frac{57\!\cdots\!63}{33\!\cdots\!91}a^{11}+\frac{12\!\cdots\!56}{33\!\cdots\!91}a^{10}+\frac{44\!\cdots\!57}{33\!\cdots\!91}a^{9}+\frac{10\!\cdots\!27}{33\!\cdots\!91}a^{8}+\frac{10\!\cdots\!14}{33\!\cdots\!91}a^{7}+\frac{29\!\cdots\!56}{33\!\cdots\!91}a^{6}+\frac{19\!\cdots\!30}{14\!\cdots\!17}a^{5}+\frac{97\!\cdots\!95}{33\!\cdots\!91}a^{4}+\frac{10\!\cdots\!37}{33\!\cdots\!91}a^{3}+\frac{86\!\cdots\!83}{14\!\cdots\!17}a^{2}+\frac{32\!\cdots\!92}{62\!\cdots\!79}a+\frac{13\!\cdots\!55}{27\!\cdots\!73}$, $\frac{22\!\cdots\!40}{33\!\cdots\!91}a^{25}-\frac{20\!\cdots\!94}{33\!\cdots\!91}a^{24}-\frac{44\!\cdots\!94}{33\!\cdots\!91}a^{23}+\frac{82\!\cdots\!00}{33\!\cdots\!91}a^{22}-\frac{33\!\cdots\!30}{33\!\cdots\!91}a^{21}-\frac{11\!\cdots\!71}{33\!\cdots\!91}a^{20}-\frac{72\!\cdots\!04}{33\!\cdots\!91}a^{19}+\frac{12\!\cdots\!41}{33\!\cdots\!91}a^{18}+\frac{12\!\cdots\!68}{33\!\cdots\!91}a^{17}-\frac{65\!\cdots\!38}{33\!\cdots\!91}a^{16}-\frac{14\!\cdots\!70}{33\!\cdots\!91}a^{15}+\frac{21\!\cdots\!39}{33\!\cdots\!91}a^{14}+\frac{56\!\cdots\!57}{33\!\cdots\!91}a^{13}+\frac{67\!\cdots\!33}{33\!\cdots\!91}a^{12}-\frac{12\!\cdots\!45}{33\!\cdots\!91}a^{11}+\frac{33\!\cdots\!99}{33\!\cdots\!91}a^{10}+\frac{10\!\cdots\!41}{33\!\cdots\!91}a^{9}+\frac{23\!\cdots\!95}{33\!\cdots\!91}a^{8}+\frac{24\!\cdots\!99}{33\!\cdots\!91}a^{7}+\frac{67\!\cdots\!09}{33\!\cdots\!91}a^{6}+\frac{43\!\cdots\!76}{14\!\cdots\!17}a^{5}+\frac{21\!\cdots\!76}{33\!\cdots\!91}a^{4}+\frac{24\!\cdots\!62}{33\!\cdots\!91}a^{3}+\frac{19\!\cdots\!55}{14\!\cdots\!17}a^{2}+\frac{72\!\cdots\!82}{62\!\cdots\!79}a+\frac{28\!\cdots\!69}{27\!\cdots\!73}$, $\frac{93\!\cdots\!80}{14\!\cdots\!17}a^{25}-\frac{72\!\cdots\!78}{14\!\cdots\!17}a^{24}-\frac{39\!\cdots\!20}{14\!\cdots\!17}a^{23}+\frac{40\!\cdots\!14}{14\!\cdots\!17}a^{22}+\frac{22\!\cdots\!90}{14\!\cdots\!17}a^{21}-\frac{57\!\cdots\!58}{14\!\cdots\!17}a^{20}-\frac{72\!\cdots\!42}{14\!\cdots\!17}a^{19}+\frac{62\!\cdots\!75}{14\!\cdots\!17}a^{18}+\frac{85\!\cdots\!25}{14\!\cdots\!17}a^{17}-\frac{29\!\cdots\!71}{14\!\cdots\!17}a^{16}-\frac{84\!\cdots\!36}{14\!\cdots\!17}a^{15}+\frac{87\!\cdots\!78}{14\!\cdots\!17}a^{14}+\frac{30\!\cdots\!52}{14\!\cdots\!17}a^{13}+\frac{13\!\cdots\!01}{14\!\cdots\!17}a^{12}-\frac{73\!\cdots\!94}{14\!\cdots\!17}a^{11}+\frac{71\!\cdots\!53}{62\!\cdots\!79}a^{10}+\frac{57\!\cdots\!22}{14\!\cdots\!17}a^{9}+\frac{13\!\cdots\!29}{14\!\cdots\!17}a^{8}+\frac{13\!\cdots\!64}{14\!\cdots\!17}a^{7}+\frac{38\!\cdots\!94}{14\!\cdots\!17}a^{6}+\frac{57\!\cdots\!91}{14\!\cdots\!17}a^{5}+\frac{54\!\cdots\!11}{62\!\cdots\!79}a^{4}+\frac{13\!\cdots\!74}{14\!\cdots\!17}a^{3}+\frac{11\!\cdots\!28}{62\!\cdots\!79}a^{2}+\frac{41\!\cdots\!66}{27\!\cdots\!73}a+\frac{17\!\cdots\!07}{11\!\cdots\!51}$, $\frac{59\!\cdots\!62}{33\!\cdots\!91}a^{25}-\frac{57\!\cdots\!07}{33\!\cdots\!91}a^{24}-\frac{86\!\cdots\!46}{33\!\cdots\!91}a^{23}+\frac{20\!\cdots\!82}{33\!\cdots\!91}a^{22}-\frac{69\!\cdots\!54}{33\!\cdots\!91}a^{21}-\frac{29\!\cdots\!34}{33\!\cdots\!91}a^{20}-\frac{16\!\cdots\!68}{33\!\cdots\!91}a^{19}+\frac{31\!\cdots\!59}{33\!\cdots\!91}a^{18}+\frac{32\!\cdots\!98}{33\!\cdots\!91}a^{17}-\frac{16\!\cdots\!36}{33\!\cdots\!91}a^{16}-\frac{34\!\cdots\!86}{33\!\cdots\!91}a^{15}+\frac{49\!\cdots\!06}{33\!\cdots\!91}a^{14}+\frac{13\!\cdots\!56}{33\!\cdots\!91}a^{13}+\frac{15\!\cdots\!00}{33\!\cdots\!91}a^{12}-\frac{27\!\cdots\!30}{33\!\cdots\!91}a^{11}+\frac{86\!\cdots\!84}{33\!\cdots\!91}a^{10}+\frac{29\!\cdots\!99}{33\!\cdots\!91}a^{9}+\frac{60\!\cdots\!35}{33\!\cdots\!91}a^{8}+\frac{69\!\cdots\!48}{33\!\cdots\!91}a^{7}+\frac{17\!\cdots\!77}{33\!\cdots\!91}a^{6}+\frac{12\!\cdots\!65}{14\!\cdots\!17}a^{5}+\frac{56\!\cdots\!25}{33\!\cdots\!91}a^{4}+\frac{67\!\cdots\!28}{33\!\cdots\!91}a^{3}+\frac{49\!\cdots\!98}{14\!\cdots\!17}a^{2}+\frac{18\!\cdots\!79}{62\!\cdots\!79}a+\frac{70\!\cdots\!39}{27\!\cdots\!73}$, $\frac{78\!\cdots\!22}{33\!\cdots\!91}a^{25}-\frac{60\!\cdots\!16}{33\!\cdots\!91}a^{24}-\frac{32\!\cdots\!10}{33\!\cdots\!91}a^{23}+\frac{33\!\cdots\!31}{33\!\cdots\!91}a^{22}+\frac{18\!\cdots\!31}{33\!\cdots\!91}a^{21}-\frac{47\!\cdots\!81}{33\!\cdots\!91}a^{20}-\frac{60\!\cdots\!43}{33\!\cdots\!91}a^{19}+\frac{52\!\cdots\!88}{33\!\cdots\!91}a^{18}+\frac{71\!\cdots\!26}{33\!\cdots\!91}a^{17}-\frac{24\!\cdots\!16}{33\!\cdots\!91}a^{16}-\frac{70\!\cdots\!84}{33\!\cdots\!91}a^{15}+\frac{71\!\cdots\!08}{33\!\cdots\!91}a^{14}+\frac{25\!\cdots\!82}{33\!\cdots\!91}a^{13}+\frac{11\!\cdots\!33}{33\!\cdots\!91}a^{12}-\frac{61\!\cdots\!78}{33\!\cdots\!91}a^{11}+\frac{13\!\cdots\!60}{33\!\cdots\!91}a^{10}+\frac{48\!\cdots\!79}{33\!\cdots\!91}a^{9}+\frac{11\!\cdots\!59}{33\!\cdots\!91}a^{8}+\frac{11\!\cdots\!13}{33\!\cdots\!91}a^{7}+\frac{32\!\cdots\!36}{33\!\cdots\!91}a^{6}+\frac{21\!\cdots\!63}{14\!\cdots\!17}a^{5}+\frac{10\!\cdots\!09}{33\!\cdots\!91}a^{4}+\frac{11\!\cdots\!67}{33\!\cdots\!91}a^{3}+\frac{93\!\cdots\!37}{14\!\cdots\!17}a^{2}+\frac{34\!\cdots\!06}{62\!\cdots\!79}a+\frac{14\!\cdots\!84}{27\!\cdots\!73}$, $\frac{24\!\cdots\!76}{33\!\cdots\!91}a^{25}-\frac{20\!\cdots\!27}{33\!\cdots\!91}a^{24}-\frac{83\!\cdots\!42}{33\!\cdots\!91}a^{23}+\frac{10\!\cdots\!75}{33\!\cdots\!91}a^{22}+\frac{36\!\cdots\!53}{33\!\cdots\!91}a^{21}-\frac{14\!\cdots\!72}{33\!\cdots\!91}a^{20}-\frac{14\!\cdots\!73}{33\!\cdots\!91}a^{19}+\frac{15\!\cdots\!13}{33\!\cdots\!91}a^{18}+\frac{18\!\cdots\!03}{33\!\cdots\!91}a^{17}-\frac{73\!\cdots\!48}{33\!\cdots\!91}a^{16}-\frac{19\!\cdots\!76}{33\!\cdots\!91}a^{15}+\frac{23\!\cdots\!61}{33\!\cdots\!91}a^{14}+\frac{70\!\cdots\!41}{33\!\cdots\!91}a^{13}+\frac{23\!\cdots\!62}{33\!\cdots\!91}a^{12}-\frac{16\!\cdots\!67}{33\!\cdots\!91}a^{11}+\frac{41\!\cdots\!79}{33\!\cdots\!91}a^{10}+\frac{13\!\cdots\!41}{33\!\cdots\!91}a^{9}+\frac{32\!\cdots\!17}{33\!\cdots\!91}a^{8}+\frac{33\!\cdots\!76}{33\!\cdots\!91}a^{7}+\frac{92\!\cdots\!39}{33\!\cdots\!91}a^{6}+\frac{59\!\cdots\!39}{14\!\cdots\!17}a^{5}+\frac{29\!\cdots\!18}{33\!\cdots\!91}a^{4}+\frac{33\!\cdots\!25}{33\!\cdots\!91}a^{3}+\frac{26\!\cdots\!02}{14\!\cdots\!17}a^{2}+\frac{96\!\cdots\!95}{62\!\cdots\!79}a+\frac{39\!\cdots\!83}{27\!\cdots\!73}$, $\frac{97\!\cdots\!58}{33\!\cdots\!91}a^{25}-\frac{75\!\cdots\!34}{33\!\cdots\!91}a^{24}-\frac{39\!\cdots\!22}{33\!\cdots\!91}a^{23}+\frac{41\!\cdots\!45}{33\!\cdots\!91}a^{22}+\frac{22\!\cdots\!69}{33\!\cdots\!91}a^{21}-\frac{59\!\cdots\!56}{33\!\cdots\!91}a^{20}-\frac{73\!\cdots\!79}{33\!\cdots\!91}a^{19}+\frac{64\!\cdots\!29}{33\!\cdots\!91}a^{18}+\frac{87\!\cdots\!46}{33\!\cdots\!91}a^{17}-\frac{29\!\cdots\!74}{33\!\cdots\!91}a^{16}-\frac{86\!\cdots\!26}{33\!\cdots\!91}a^{15}+\frac{89\!\cdots\!30}{33\!\cdots\!91}a^{14}+\frac{31\!\cdots\!40}{33\!\cdots\!91}a^{13}+\frac{13\!\cdots\!38}{33\!\cdots\!91}a^{12}-\frac{74\!\cdots\!64}{33\!\cdots\!91}a^{11}+\frac{16\!\cdots\!74}{33\!\cdots\!91}a^{10}+\frac{59\!\cdots\!94}{33\!\cdots\!91}a^{9}+\frac{14\!\cdots\!15}{33\!\cdots\!91}a^{8}+\frac{14\!\cdots\!15}{33\!\cdots\!91}a^{7}+\frac{40\!\cdots\!64}{33\!\cdots\!91}a^{6}+\frac{25\!\cdots\!51}{14\!\cdots\!17}a^{5}+\frac{12\!\cdots\!66}{33\!\cdots\!91}a^{4}+\frac{14\!\cdots\!02}{33\!\cdots\!91}a^{3}+\frac{11\!\cdots\!94}{14\!\cdots\!17}a^{2}+\frac{42\!\cdots\!77}{62\!\cdots\!79}a+\frac{17\!\cdots\!57}{27\!\cdots\!73}$, $\frac{24\!\cdots\!58}{33\!\cdots\!91}a^{25}-\frac{22\!\cdots\!60}{33\!\cdots\!91}a^{24}-\frac{59\!\cdots\!80}{33\!\cdots\!91}a^{23}+\frac{96\!\cdots\!53}{33\!\cdots\!91}a^{22}-\frac{60\!\cdots\!69}{33\!\cdots\!91}a^{21}-\frac{13\!\cdots\!25}{33\!\cdots\!91}a^{20}-\frac{74\!\cdots\!99}{33\!\cdots\!91}a^{19}+\frac{14\!\cdots\!61}{33\!\cdots\!91}a^{18}+\frac{11\!\cdots\!71}{33\!\cdots\!91}a^{17}-\frac{75\!\cdots\!99}{33\!\cdots\!91}a^{16}-\frac{14\!\cdots\!54}{33\!\cdots\!91}a^{15}+\frac{26\!\cdots\!73}{33\!\cdots\!91}a^{14}+\frac{54\!\cdots\!65}{33\!\cdots\!91}a^{13}-\frac{41\!\cdots\!90}{33\!\cdots\!91}a^{12}-\frac{12\!\cdots\!17}{33\!\cdots\!91}a^{11}+\frac{44\!\cdots\!11}{33\!\cdots\!91}a^{10}+\frac{11\!\cdots\!50}{33\!\cdots\!91}a^{9}+\frac{25\!\cdots\!72}{33\!\cdots\!91}a^{8}+\frac{27\!\cdots\!46}{33\!\cdots\!91}a^{7}+\frac{80\!\cdots\!65}{33\!\cdots\!91}a^{6}+\frac{49\!\cdots\!36}{14\!\cdots\!17}a^{5}+\frac{24\!\cdots\!19}{33\!\cdots\!91}a^{4}+\frac{26\!\cdots\!89}{33\!\cdots\!91}a^{3}+\frac{20\!\cdots\!50}{14\!\cdots\!17}a^{2}+\frac{72\!\cdots\!24}{62\!\cdots\!79}a+\frac{27\!\cdots\!30}{27\!\cdots\!73}$
| 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
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 57529828940.82975 \cdot 8511191}{2\cdot\sqrt{338317693370822771423393222562696496647525569958371057767}}\cr\approx \mathstrut & 0.316613447558080 \end{aligned}\] (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
| \(\Q(\sqrt{-7}) \), 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 | ${\href{/padicField/2.13.0.1}{13} }^{2}$ | $26$ | $26$ | R | ${\href{/padicField/11.13.0.1}{13} }^{2}$ | $26$ | $26$ | $26$ | ${\href{/padicField/23.1.0.1}{1} }^{26}$ | ${\href{/padicField/29.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/37.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/43.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/53.13.0.1}{13} }^{2}$ | $26$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| Deg $26$ | $2$ | $13$ | $13$ | |||
|
\(79\)
| 79.13.12.1 | $x^{13} + 79$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ |
| 79.13.12.1 | $x^{13} + 79$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ |