Normalized defining polynomial
\( x^{26} + 145 x^{24} + 8786 x^{22} + 297801 x^{20} + 6317962 x^{18} + 88430622 x^{16} + 835400600 x^{14} + \cdots + 3769837201 \)
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: |
\(-3375759861985195519612137229985370185177350891905754190577664\)
\(\medspace = -\,2^{26}\cdot 157^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(212.82\) | 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: | $2\cdot 157^{12/13}\approx 212.82090722547485$ | ||
| Ramified primes: |
\(2\), \(157\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $26$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(628=2^{2}\cdot 157\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{628}(1,·)$, $\chi_{628}(67,·)$, $\chi_{628}(487,·)$, $\chi_{628}(389,·)$, $\chi_{628}(601,·)$, $\chi_{628}(265,·)$, $\chi_{628}(75,·)$, $\chi_{628}(579,·)$, $\chi_{628}(203,·)$, $\chi_{628}(407,·)$, $\chi_{628}(485,·)$, $\chi_{628}(153,·)$, $\chi_{628}(413,·)$, $\chi_{628}(517,·)$, $\chi_{628}(353,·)$, $\chi_{628}(99,·)$, $\chi_{628}(101,·)$, $\chi_{628}(39,·)$, $\chi_{628}(171,·)$, $\chi_{628}(173,·)$, $\chi_{628}(93,·)$, $\chi_{628}(467,·)$, $\chi_{628}(415,·)$, $\chi_{628}(315,·)$, $\chi_{628}(287,·)$, $\chi_{628}(381,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{4096}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13}a^{11}+\frac{1}{13}a^{9}+\frac{1}{13}a^{7}+\frac{1}{13}a^{5}+\frac{1}{13}a^{3}+\frac{1}{13}a$, $\frac{1}{13}a^{12}+\frac{1}{13}a^{10}+\frac{1}{13}a^{8}+\frac{1}{13}a^{6}+\frac{1}{13}a^{4}+\frac{1}{13}a^{2}$, $\frac{1}{13}a^{13}-\frac{1}{13}a$, $\frac{1}{13}a^{14}-\frac{1}{13}a^{2}$, $\frac{1}{13}a^{15}-\frac{1}{13}a^{3}$, $\frac{1}{13}a^{16}-\frac{1}{13}a^{4}$, $\frac{1}{13}a^{17}-\frac{1}{13}a^{5}$, $\frac{1}{169}a^{18}+\frac{3}{169}a^{16}+\frac{5}{169}a^{14}+\frac{6}{169}a^{12}+\frac{45}{169}a^{10}+\frac{58}{169}a^{8}+\frac{31}{169}a^{6}-\frac{10}{169}a^{4}-\frac{25}{169}a^{2}$, $\frac{1}{169}a^{19}+\frac{3}{169}a^{17}+\frac{5}{169}a^{15}+\frac{6}{169}a^{13}+\frac{6}{169}a^{11}+\frac{19}{169}a^{9}-\frac{8}{169}a^{7}-\frac{49}{169}a^{5}-\frac{64}{169}a^{3}-\frac{3}{13}a$, $\frac{1}{169}a^{20}-\frac{4}{169}a^{16}+\frac{4}{169}a^{14}+\frac{1}{169}a^{12}+\frac{66}{169}a^{10}+\frac{40}{169}a^{6}-\frac{21}{169}a^{4}+\frac{36}{169}a^{2}$, $\frac{1}{169}a^{21}-\frac{4}{169}a^{17}+\frac{4}{169}a^{15}+\frac{1}{169}a^{13}+\frac{1}{169}a^{11}-\frac{5}{13}a^{9}-\frac{25}{169}a^{7}+\frac{83}{169}a^{5}-\frac{29}{169}a^{3}-\frac{5}{13}a$, $\frac{1}{687661}a^{22}+\frac{1470}{687661}a^{20}-\frac{1824}{687661}a^{18}+\frac{1624}{52897}a^{16}-\frac{18767}{687661}a^{14}+\frac{15797}{687661}a^{12}+\frac{5994}{687661}a^{10}-\frac{740}{687661}a^{8}+\frac{176598}{687661}a^{6}-\frac{110147}{687661}a^{4}-\frac{207639}{687661}a^{2}+\frac{1849}{4069}$, $\frac{1}{687661}a^{23}+\frac{1470}{687661}a^{21}-\frac{1824}{687661}a^{19}+\frac{1624}{52897}a^{17}-\frac{18767}{687661}a^{15}+\frac{15797}{687661}a^{13}+\frac{5994}{687661}a^{11}-\frac{740}{687661}a^{9}+\frac{176598}{687661}a^{7}-\frac{110147}{687661}a^{5}-\frac{207639}{687661}a^{3}+\frac{1849}{4069}a$, $\frac{1}{10\!\cdots\!09}a^{24}-\frac{59\!\cdots\!30}{10\!\cdots\!09}a^{22}+\frac{35\!\cdots\!15}{81\!\cdots\!93}a^{20}+\frac{29\!\cdots\!66}{10\!\cdots\!09}a^{18}-\frac{47\!\cdots\!55}{25\!\cdots\!29}a^{16}+\frac{19\!\cdots\!10}{10\!\cdots\!09}a^{14}+\frac{34\!\cdots\!74}{10\!\cdots\!09}a^{12}-\frac{77\!\cdots\!12}{10\!\cdots\!09}a^{10}-\frac{14\!\cdots\!98}{10\!\cdots\!09}a^{8}-\frac{21\!\cdots\!69}{81\!\cdots\!93}a^{6}+\frac{19\!\cdots\!98}{10\!\cdots\!09}a^{4}-\frac{53\!\cdots\!39}{10\!\cdots\!09}a^{2}-\frac{19\!\cdots\!36}{63\!\cdots\!61}$, $\frac{1}{50\!\cdots\!07}a^{25}+\frac{64\!\cdots\!12}{50\!\cdots\!07}a^{23}-\frac{41\!\cdots\!32}{29\!\cdots\!03}a^{21}-\frac{58\!\cdots\!31}{50\!\cdots\!07}a^{19}+\frac{34\!\cdots\!73}{11\!\cdots\!67}a^{17}-\frac{16\!\cdots\!87}{50\!\cdots\!07}a^{15}+\frac{58\!\cdots\!86}{50\!\cdots\!07}a^{13}-\frac{12\!\cdots\!92}{50\!\cdots\!07}a^{11}-\frac{16\!\cdots\!90}{50\!\cdots\!07}a^{9}-\frac{13\!\cdots\!42}{38\!\cdots\!39}a^{7}+\frac{11\!\cdots\!67}{50\!\cdots\!07}a^{5}-\frac{45\!\cdots\!87}{50\!\cdots\!07}a^{3}+\frac{12\!\cdots\!43}{29\!\cdots\!03}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $13$ |
Class group and class number
$C_{13}\times C_{4268953}$, which has order $55496389$ (assuming GRH)
Unit group
| Rank: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{3453937810176431842324}{3308700176843220638364245187659} a^{25} - \frac{480128120190588989170507}{3308700176843220638364245187659} a^{23} - \frac{2111766065315231616290615}{254515398218709279874172706743} a^{21} - \frac{861801974554805568236383303}{3308700176843220638364245187659} a^{19} - \frac{39261447139104115264208748}{7859145313166794865473266479} a^{17} - \frac{202441016265507829013163293224}{3308700176843220638364245187659} a^{15} - \frac{1599060069500146769278402748908}{3308700176843220638364245187659} a^{13} - \frac{8003331143846220131054136244513}{3308700176843220638364245187659} a^{11} - \frac{24126290036857196509944964238003}{3308700176843220638364245187659} a^{9} - \frac{3023744864929009222293111227923}{254515398218709279874172706743} a^{7} - \frac{26357660572698679134019679412727}{3308700176843220638364245187659} a^{5} + \frac{1606169599550550988386049093170}{3308700176843220638364245187659} a^{3} + \frac{48771091076762291483583223310}{19578107555285329221090208211} a \)
(order $4$)
| 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: |
$\frac{94\!\cdots\!46}{10\!\cdots\!09}a^{24}+\frac{13\!\cdots\!30}{10\!\cdots\!09}a^{22}+\frac{56\!\cdots\!54}{81\!\cdots\!93}a^{20}+\frac{22\!\cdots\!27}{10\!\cdots\!09}a^{18}+\frac{99\!\cdots\!29}{25\!\cdots\!29}a^{16}+\frac{49\!\cdots\!50}{10\!\cdots\!09}a^{14}+\frac{36\!\cdots\!67}{10\!\cdots\!09}a^{12}+\frac{16\!\cdots\!37}{10\!\cdots\!09}a^{10}+\frac{38\!\cdots\!52}{10\!\cdots\!09}a^{8}+\frac{23\!\cdots\!91}{81\!\cdots\!93}a^{6}-\frac{26\!\cdots\!93}{10\!\cdots\!09}a^{4}-\frac{24\!\cdots\!79}{10\!\cdots\!09}a^{2}-\frac{23\!\cdots\!86}{63\!\cdots\!61}$, $\frac{94\!\cdots\!44}{10\!\cdots\!09}a^{24}+\frac{13\!\cdots\!19}{10\!\cdots\!09}a^{22}+\frac{57\!\cdots\!78}{81\!\cdots\!93}a^{20}+\frac{23\!\cdots\!78}{10\!\cdots\!09}a^{18}+\frac{10\!\cdots\!17}{25\!\cdots\!29}a^{16}+\frac{55\!\cdots\!72}{10\!\cdots\!09}a^{14}+\frac{44\!\cdots\!33}{10\!\cdots\!09}a^{12}+\frac{22\!\cdots\!45}{10\!\cdots\!09}a^{10}+\frac{73\!\cdots\!64}{10\!\cdots\!09}a^{8}+\frac{10\!\cdots\!64}{81\!\cdots\!93}a^{6}+\frac{13\!\cdots\!70}{10\!\cdots\!09}a^{4}+\frac{53\!\cdots\!86}{10\!\cdots\!09}a^{2}+\frac{39\!\cdots\!80}{63\!\cdots\!61}$, $\frac{15\!\cdots\!60}{10\!\cdots\!09}a^{24}+\frac{21\!\cdots\!25}{10\!\cdots\!09}a^{22}+\frac{93\!\cdots\!70}{81\!\cdots\!93}a^{20}+\frac{37\!\cdots\!01}{10\!\cdots\!09}a^{18}+\frac{17\!\cdots\!44}{25\!\cdots\!29}a^{16}+\frac{88\!\cdots\!03}{10\!\cdots\!09}a^{14}+\frac{69\!\cdots\!85}{10\!\cdots\!09}a^{12}+\frac{35\!\cdots\!94}{10\!\cdots\!09}a^{10}+\frac{11\!\cdots\!94}{10\!\cdots\!09}a^{8}+\frac{15\!\cdots\!35}{81\!\cdots\!93}a^{6}+\frac{19\!\cdots\!69}{10\!\cdots\!09}a^{4}+\frac{10\!\cdots\!97}{10\!\cdots\!09}a^{2}+\frac{13\!\cdots\!03}{63\!\cdots\!61}$, $\frac{22\!\cdots\!80}{10\!\cdots\!09}a^{24}+\frac{31\!\cdots\!41}{10\!\cdots\!09}a^{22}+\frac{13\!\cdots\!70}{81\!\cdots\!93}a^{20}+\frac{58\!\cdots\!48}{10\!\cdots\!09}a^{18}+\frac{27\!\cdots\!25}{25\!\cdots\!29}a^{16}+\frac{14\!\cdots\!22}{10\!\cdots\!09}a^{14}+\frac{12\!\cdots\!96}{10\!\cdots\!09}a^{12}+\frac{72\!\cdots\!48}{10\!\cdots\!09}a^{10}+\frac{26\!\cdots\!32}{10\!\cdots\!09}a^{8}+\frac{47\!\cdots\!09}{81\!\cdots\!93}a^{6}+\frac{81\!\cdots\!39}{10\!\cdots\!09}a^{4}+\frac{50\!\cdots\!01}{10\!\cdots\!09}a^{2}+\frac{47\!\cdots\!08}{63\!\cdots\!61}$, $\frac{16\!\cdots\!85}{10\!\cdots\!09}a^{24}+\frac{22\!\cdots\!97}{10\!\cdots\!09}a^{22}+\frac{10\!\cdots\!54}{81\!\cdots\!93}a^{20}+\frac{40\!\cdots\!58}{10\!\cdots\!09}a^{18}+\frac{18\!\cdots\!36}{25\!\cdots\!29}a^{16}+\frac{95\!\cdots\!15}{10\!\cdots\!09}a^{14}+\frac{76\!\cdots\!17}{10\!\cdots\!09}a^{12}+\frac{39\!\cdots\!05}{10\!\cdots\!09}a^{10}+\frac{12\!\cdots\!51}{10\!\cdots\!09}a^{8}+\frac{17\!\cdots\!73}{81\!\cdots\!93}a^{6}+\frac{21\!\cdots\!58}{10\!\cdots\!09}a^{4}+\frac{87\!\cdots\!06}{10\!\cdots\!09}a^{2}+\frac{64\!\cdots\!10}{63\!\cdots\!61}$, $\frac{54\!\cdots\!73}{81\!\cdots\!93}a^{24}+\frac{58\!\cdots\!99}{63\!\cdots\!61}a^{22}+\frac{44\!\cdots\!22}{81\!\cdots\!93}a^{20}+\frac{14\!\cdots\!71}{81\!\cdots\!93}a^{18}+\frac{66\!\cdots\!75}{19\!\cdots\!33}a^{16}+\frac{35\!\cdots\!30}{81\!\cdots\!93}a^{14}+\frac{29\!\cdots\!19}{81\!\cdots\!93}a^{12}+\frac{12\!\cdots\!67}{63\!\cdots\!61}a^{10}+\frac{52\!\cdots\!64}{81\!\cdots\!93}a^{8}+\frac{10\!\cdots\!96}{81\!\cdots\!93}a^{6}+\frac{97\!\cdots\!46}{81\!\cdots\!93}a^{4}+\frac{38\!\cdots\!96}{81\!\cdots\!93}a^{2}+\frac{27\!\cdots\!88}{48\!\cdots\!97}$, $\frac{27\!\cdots\!62}{10\!\cdots\!09}a^{24}+\frac{38\!\cdots\!88}{10\!\cdots\!09}a^{22}+\frac{16\!\cdots\!79}{81\!\cdots\!93}a^{20}+\frac{67\!\cdots\!15}{10\!\cdots\!09}a^{18}+\frac{30\!\cdots\!82}{25\!\cdots\!29}a^{16}+\frac{15\!\cdots\!54}{10\!\cdots\!09}a^{14}+\frac{12\!\cdots\!50}{10\!\cdots\!09}a^{12}+\frac{63\!\cdots\!61}{10\!\cdots\!09}a^{10}+\frac{19\!\cdots\!20}{10\!\cdots\!09}a^{8}+\frac{27\!\cdots\!47}{81\!\cdots\!93}a^{6}+\frac{31\!\cdots\!94}{10\!\cdots\!09}a^{4}+\frac{10\!\cdots\!20}{10\!\cdots\!09}a^{2}+\frac{60\!\cdots\!65}{63\!\cdots\!61}$, $\frac{67\!\cdots\!27}{10\!\cdots\!09}a^{24}+\frac{93\!\cdots\!03}{10\!\cdots\!09}a^{22}+\frac{40\!\cdots\!52}{81\!\cdots\!93}a^{20}+\frac{16\!\cdots\!76}{10\!\cdots\!09}a^{18}+\frac{73\!\cdots\!82}{25\!\cdots\!29}a^{16}+\frac{37\!\cdots\!66}{10\!\cdots\!09}a^{14}+\frac{29\!\cdots\!40}{10\!\cdots\!09}a^{12}+\frac{14\!\cdots\!90}{10\!\cdots\!09}a^{10}+\frac{45\!\cdots\!09}{10\!\cdots\!09}a^{8}+\frac{62\!\cdots\!72}{81\!\cdots\!93}a^{6}+\frac{74\!\cdots\!38}{10\!\cdots\!09}a^{4}+\frac{27\!\cdots\!33}{10\!\cdots\!09}a^{2}+\frac{20\!\cdots\!85}{63\!\cdots\!61}$, $\frac{49\!\cdots\!30}{10\!\cdots\!09}a^{24}+\frac{67\!\cdots\!53}{10\!\cdots\!09}a^{22}+\frac{29\!\cdots\!86}{81\!\cdots\!93}a^{20}+\frac{11\!\cdots\!02}{10\!\cdots\!09}a^{18}+\frac{51\!\cdots\!78}{25\!\cdots\!29}a^{16}+\frac{25\!\cdots\!27}{10\!\cdots\!09}a^{14}+\frac{19\!\cdots\!93}{10\!\cdots\!09}a^{12}+\frac{87\!\cdots\!87}{10\!\cdots\!09}a^{10}+\frac{23\!\cdots\!33}{10\!\cdots\!09}a^{8}+\frac{21\!\cdots\!13}{81\!\cdots\!93}a^{6}-\frac{11\!\cdots\!35}{10\!\cdots\!09}a^{4}-\frac{19\!\cdots\!90}{10\!\cdots\!09}a^{2}-\frac{15\!\cdots\!33}{63\!\cdots\!61}$, $\frac{31\!\cdots\!44}{10\!\cdots\!09}a^{24}+\frac{42\!\cdots\!33}{10\!\cdots\!09}a^{22}+\frac{18\!\cdots\!02}{81\!\cdots\!93}a^{20}+\frac{74\!\cdots\!87}{10\!\cdots\!09}a^{18}+\frac{33\!\cdots\!88}{25\!\cdots\!29}a^{16}+\frac{17\!\cdots\!77}{10\!\cdots\!09}a^{14}+\frac{13\!\cdots\!20}{10\!\cdots\!09}a^{12}+\frac{67\!\cdots\!41}{10\!\cdots\!09}a^{10}+\frac{20\!\cdots\!01}{10\!\cdots\!09}a^{8}+\frac{24\!\cdots\!72}{81\!\cdots\!93}a^{6}+\frac{16\!\cdots\!58}{10\!\cdots\!09}a^{4}-\frac{24\!\cdots\!82}{10\!\cdots\!09}a^{2}-\frac{62\!\cdots\!74}{63\!\cdots\!61}$, $\frac{92\!\cdots\!84}{10\!\cdots\!09}a^{24}+\frac{12\!\cdots\!67}{10\!\cdots\!09}a^{22}+\frac{55\!\cdots\!19}{81\!\cdots\!93}a^{20}+\frac{22\!\cdots\!01}{10\!\cdots\!09}a^{18}+\frac{10\!\cdots\!90}{25\!\cdots\!29}a^{16}+\frac{52\!\cdots\!99}{10\!\cdots\!09}a^{14}+\frac{40\!\cdots\!38}{10\!\cdots\!09}a^{12}+\frac{20\!\cdots\!75}{10\!\cdots\!09}a^{10}+\frac{63\!\cdots\!56}{10\!\cdots\!09}a^{8}+\frac{84\!\cdots\!63}{81\!\cdots\!93}a^{6}+\frac{89\!\cdots\!57}{10\!\cdots\!09}a^{4}+\frac{23\!\cdots\!29}{10\!\cdots\!09}a^{2}+\frac{11\!\cdots\!15}{63\!\cdots\!61}$, $\frac{12\!\cdots\!58}{10\!\cdots\!09}a^{24}+\frac{17\!\cdots\!61}{10\!\cdots\!09}a^{22}+\frac{77\!\cdots\!91}{81\!\cdots\!93}a^{20}+\frac{31\!\cdots\!43}{10\!\cdots\!09}a^{18}+\frac{14\!\cdots\!87}{25\!\cdots\!29}a^{16}+\frac{74\!\cdots\!10}{10\!\cdots\!09}a^{14}+\frac{58\!\cdots\!60}{10\!\cdots\!09}a^{12}+\frac{30\!\cdots\!86}{10\!\cdots\!09}a^{10}+\frac{95\!\cdots\!54}{10\!\cdots\!09}a^{8}+\frac{13\!\cdots\!37}{81\!\cdots\!93}a^{6}+\frac{16\!\cdots\!06}{10\!\cdots\!09}a^{4}+\frac{62\!\cdots\!25}{10\!\cdots\!09}a^{2}+\frac{45\!\cdots\!89}{63\!\cdots\!61}$
| 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: | \( 5466968796671.363 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 5466968796671.363 \cdot 55496389}{4\cdot\sqrt{3375759861985195519612137229985370185177350891905754190577664}}\cr\approx \mathstrut & 0.981982720049626 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 26 |
| The 26 conjugacy class representatives for $C_{26}$ |
| Character table for $C_{26}$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 13.13.224282727500720205065439601.1 |
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 | R | $26$ | ${\href{/padicField/5.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/padicField/13.1.0.1}{1} }^{26}$ | ${\href{/padicField/17.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/padicField/29.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/37.13.0.1}{13} }^{2}$ | ${\href{/padicField/41.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/padicField/53.13.0.1}{13} }^{2}$ | $26$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $26$ | $2$ | $13$ | $26$ | |||
|
\(157\)
| 157.13.12.1 | $x^{13} + 157$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ |
| 157.13.12.1 | $x^{13} + 157$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ |