Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 8*x^25 + 14*x^24 + 18*x^23 - 92*x^22 + 514*x^21 - 1212*x^20 - 596*x^19 + 954*x^18 - 2053*x^17 + 29744*x^16 - 4491*x^15 + 71256*x^14 - 216720*x^13 - 344608*x^12 + 50717*x^11 + 1942970*x^10 + 2307612*x^9 + 264538*x^8 - 5680405*x^7 - 6094469*x^6 - 2337568*x^5 + 3586683*x^4 + 5626157*x^3 + 5313221*x^2 + 843733*x + 496261)
gp: K = bnfinit(y^26 - 8*y^25 + 14*y^24 + 18*y^23 - 92*y^22 + 514*y^21 - 1212*y^20 - 596*y^19 + 954*y^18 - 2053*y^17 + 29744*y^16 - 4491*y^15 + 71256*y^14 - 216720*y^13 - 344608*y^12 + 50717*y^11 + 1942970*y^10 + 2307612*y^9 + 264538*y^8 - 5680405*y^7 - 6094469*y^6 - 2337568*y^5 + 3586683*y^4 + 5626157*y^3 + 5313221*y^2 + 843733*y + 496261, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 8*x^25 + 14*x^24 + 18*x^23 - 92*x^22 + 514*x^21 - 1212*x^20 - 596*x^19 + 954*x^18 - 2053*x^17 + 29744*x^16 - 4491*x^15 + 71256*x^14 - 216720*x^13 - 344608*x^12 + 50717*x^11 + 1942970*x^10 + 2307612*x^9 + 264538*x^8 - 5680405*x^7 - 6094469*x^6 - 2337568*x^5 + 3586683*x^4 + 5626157*x^3 + 5313221*x^2 + 843733*x + 496261);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 8*x^25 + 14*x^24 + 18*x^23 - 92*x^22 + 514*x^21 - 1212*x^20 - 596*x^19 + 954*x^18 - 2053*x^17 + 29744*x^16 - 4491*x^15 + 71256*x^14 - 216720*x^13 - 344608*x^12 + 50717*x^11 + 1942970*x^10 + 2307612*x^9 + 264538*x^8 - 5680405*x^7 - 6094469*x^6 - 2337568*x^5 + 3586683*x^4 + 5626157*x^3 + 5313221*x^2 + 843733*x + 496261)
\( x^{26} - 8 x^{25} + 14 x^{24} + 18 x^{23} - 92 x^{22} + 514 x^{21} - 1212 x^{20} - 596 x^{19} + \cdots + 496261 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
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: |
\(-526721251926767614625217802292605164794921875\)
\(\medspace = -\,5^{13}\cdot 19^{13}\cdot 29^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.49\) | 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: | $5^{1/2}19^{1/2}29^{1/2}\approx 52.48809388804284$ | ||
| Ramified primes: |
\(5\), \(19\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-2755}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19}a^{18}+\frac{5}{19}a^{17}+\frac{4}{19}a^{16}+\frac{9}{19}a^{15}-\frac{7}{19}a^{14}+\frac{6}{19}a^{13}+\frac{2}{19}a^{12}-\frac{4}{19}a^{11}+\frac{9}{19}a^{10}-\frac{7}{19}a^{9}-\frac{3}{19}a^{8}-\frac{8}{19}a^{7}-\frac{9}{19}a^{6}-\frac{9}{19}a^{5}+\frac{6}{19}a^{4}+\frac{8}{19}a^{3}-\frac{3}{19}a^{2}$, $\frac{1}{19}a^{19}-\frac{2}{19}a^{17}+\frac{8}{19}a^{16}+\frac{5}{19}a^{15}+\frac{3}{19}a^{14}-\frac{9}{19}a^{13}+\frac{5}{19}a^{12}-\frac{9}{19}a^{11}+\frac{5}{19}a^{10}-\frac{6}{19}a^{9}+\frac{7}{19}a^{8}-\frac{7}{19}a^{7}-\frac{2}{19}a^{6}-\frac{6}{19}a^{5}-\frac{3}{19}a^{4}-\frac{5}{19}a^{3}-\frac{4}{19}a^{2}$, $\frac{1}{19}a^{20}-\frac{1}{19}a^{17}-\frac{6}{19}a^{16}+\frac{2}{19}a^{15}-\frac{4}{19}a^{14}-\frac{2}{19}a^{13}-\frac{5}{19}a^{12}-\frac{3}{19}a^{11}-\frac{7}{19}a^{10}-\frac{7}{19}a^{9}+\frac{6}{19}a^{8}+\frac{1}{19}a^{7}-\frac{5}{19}a^{6}-\frac{2}{19}a^{5}+\frac{7}{19}a^{4}-\frac{7}{19}a^{3}-\frac{6}{19}a^{2}$, $\frac{1}{19}a^{21}-\frac{1}{19}a^{17}+\frac{6}{19}a^{16}+\frac{5}{19}a^{15}-\frac{9}{19}a^{14}+\frac{1}{19}a^{13}-\frac{1}{19}a^{12}+\frac{8}{19}a^{11}+\frac{2}{19}a^{10}-\frac{1}{19}a^{9}-\frac{2}{19}a^{8}+\frac{6}{19}a^{7}+\frac{8}{19}a^{6}-\frac{2}{19}a^{5}-\frac{1}{19}a^{4}+\frac{2}{19}a^{3}-\frac{3}{19}a^{2}$, $\frac{1}{551}a^{22}-\frac{5}{551}a^{21}-\frac{6}{551}a^{20}+\frac{11}{551}a^{19}-\frac{13}{551}a^{18}+\frac{49}{551}a^{17}+\frac{127}{551}a^{16}-\frac{118}{551}a^{15}-\frac{193}{551}a^{14}-\frac{260}{551}a^{13}+\frac{131}{551}a^{12}+\frac{157}{551}a^{11}-\frac{155}{551}a^{10}-\frac{241}{551}a^{9}-\frac{173}{551}a^{8}+\frac{1}{19}a^{7}-\frac{268}{551}a^{6}+\frac{272}{551}a^{5}-\frac{102}{551}a^{4}+\frac{258}{551}a^{3}-\frac{90}{551}a^{2}+\frac{14}{29}a-\frac{12}{29}$, $\frac{1}{7163}a^{23}+\frac{2}{7163}a^{22}+\frac{46}{7163}a^{21}+\frac{27}{7163}a^{20}-\frac{168}{7163}a^{19}+\frac{45}{7163}a^{18}+\frac{673}{7163}a^{17}+\frac{1090}{7163}a^{16}-\frac{2498}{7163}a^{15}-\frac{74}{7163}a^{14}-\frac{2414}{7163}a^{13}-\frac{840}{7163}a^{12}+\frac{3206}{7163}a^{11}+\frac{1922}{7163}a^{10}-\frac{2121}{7163}a^{9}-\frac{1240}{7163}a^{8}+\frac{892}{7163}a^{7}+\frac{1238}{7163}a^{6}-\frac{1736}{7163}a^{5}-\frac{2225}{7163}a^{4}-\frac{2170}{7163}a^{3}+\frac{3000}{7163}a^{2}-\frac{88}{377}a-\frac{113}{377}$, $\frac{1}{436943}a^{24}+\frac{20}{436943}a^{23}+\frac{30}{436943}a^{22}+\frac{8278}{436943}a^{21}+\frac{1384}{436943}a^{20}-\frac{535}{436943}a^{19}+\frac{159}{7163}a^{18}-\frac{94527}{436943}a^{17}+\frac{160564}{436943}a^{16}-\frac{205159}{436943}a^{15}+\frac{203084}{436943}a^{14}-\frac{157444}{436943}a^{13}-\frac{171411}{436943}a^{12}-\frac{36752}{436943}a^{11}+\frac{3589}{436943}a^{10}-\frac{60062}{436943}a^{9}-\frac{209980}{436943}a^{8}-\frac{29077}{436943}a^{7}+\frac{128734}{436943}a^{6}-\frac{106429}{436943}a^{5}-\frac{124380}{436943}a^{4}-\frac{109796}{436943}a^{3}-\frac{82482}{436943}a^{2}-\frac{3179}{22997}a-\frac{2918}{22997}$, $\frac{1}{93\!\cdots\!01}a^{25}-\frac{18\!\cdots\!80}{93\!\cdots\!01}a^{24}+\frac{10\!\cdots\!21}{93\!\cdots\!01}a^{23}+\frac{17\!\cdots\!20}{93\!\cdots\!01}a^{22}+\frac{52\!\cdots\!91}{32\!\cdots\!69}a^{21}-\frac{12\!\cdots\!50}{93\!\cdots\!01}a^{20}+\frac{21\!\cdots\!85}{93\!\cdots\!01}a^{19}+\frac{80\!\cdots\!65}{93\!\cdots\!01}a^{18}+\frac{19\!\cdots\!83}{93\!\cdots\!01}a^{17}+\frac{65\!\cdots\!56}{93\!\cdots\!01}a^{16}-\frac{17\!\cdots\!36}{93\!\cdots\!01}a^{15}+\frac{41\!\cdots\!12}{93\!\cdots\!01}a^{14}-\frac{10\!\cdots\!25}{93\!\cdots\!01}a^{13}+\frac{37\!\cdots\!02}{93\!\cdots\!01}a^{12}-\frac{75\!\cdots\!77}{93\!\cdots\!01}a^{11}+\frac{27\!\cdots\!18}{93\!\cdots\!01}a^{10}-\frac{35\!\cdots\!59}{93\!\cdots\!01}a^{9}-\frac{34\!\cdots\!68}{93\!\cdots\!01}a^{8}+\frac{41\!\cdots\!03}{93\!\cdots\!01}a^{7}+\frac{19\!\cdots\!34}{93\!\cdots\!01}a^{6}-\frac{38\!\cdots\!79}{93\!\cdots\!01}a^{5}-\frac{40\!\cdots\!25}{93\!\cdots\!01}a^{4}-\frac{16\!\cdots\!86}{93\!\cdots\!01}a^{3}-\frac{39\!\cdots\!17}{93\!\cdots\!01}a^{2}-\frac{12\!\cdots\!67}{37\!\cdots\!83}a-\frac{27\!\cdots\!65}{37\!\cdots\!83}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{2}\times C_{4}$, which has order $8$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| 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: |
$\frac{85\!\cdots\!29}{93\!\cdots\!01}a^{25}-\frac{66\!\cdots\!56}{93\!\cdots\!01}a^{24}+\frac{10\!\cdots\!74}{93\!\cdots\!01}a^{23}+\frac{17\!\cdots\!47}{93\!\cdots\!01}a^{22}-\frac{26\!\cdots\!94}{32\!\cdots\!69}a^{21}+\frac{42\!\cdots\!56}{93\!\cdots\!01}a^{20}-\frac{73\!\cdots\!59}{72\!\cdots\!77}a^{19}-\frac{72\!\cdots\!68}{93\!\cdots\!01}a^{18}+\frac{83\!\cdots\!12}{93\!\cdots\!01}a^{17}-\frac{16\!\cdots\!53}{93\!\cdots\!01}a^{16}+\frac{25\!\cdots\!34}{93\!\cdots\!01}a^{15}+\frac{63\!\cdots\!55}{93\!\cdots\!01}a^{14}+\frac{58\!\cdots\!32}{93\!\cdots\!01}a^{13}-\frac{13\!\cdots\!38}{72\!\cdots\!77}a^{12}-\frac{33\!\cdots\!40}{93\!\cdots\!01}a^{11}+\frac{14\!\cdots\!12}{93\!\cdots\!01}a^{10}+\frac{16\!\cdots\!73}{93\!\cdots\!01}a^{9}+\frac{22\!\cdots\!34}{93\!\cdots\!01}a^{8}+\frac{43\!\cdots\!69}{93\!\cdots\!01}a^{7}-\frac{47\!\cdots\!68}{93\!\cdots\!01}a^{6}-\frac{62\!\cdots\!83}{93\!\cdots\!01}a^{5}-\frac{27\!\cdots\!36}{93\!\cdots\!01}a^{4}+\frac{31\!\cdots\!85}{93\!\cdots\!01}a^{3}+\frac{64\!\cdots\!67}{93\!\cdots\!01}a^{2}+\frac{28\!\cdots\!14}{49\!\cdots\!79}a+\frac{71\!\cdots\!81}{49\!\cdots\!79}$, $\frac{95\!\cdots\!67}{93\!\cdots\!01}a^{25}-\frac{71\!\cdots\!88}{93\!\cdots\!01}a^{24}+\frac{96\!\cdots\!17}{93\!\cdots\!01}a^{23}+\frac{23\!\cdots\!44}{93\!\cdots\!01}a^{22}-\frac{91\!\cdots\!45}{11\!\cdots\!61}a^{21}+\frac{44\!\cdots\!35}{93\!\cdots\!01}a^{20}-\frac{91\!\cdots\!30}{93\!\cdots\!01}a^{19}-\frac{85\!\cdots\!42}{72\!\cdots\!77}a^{18}+\frac{45\!\cdots\!20}{93\!\cdots\!01}a^{17}-\frac{11\!\cdots\!40}{93\!\cdots\!01}a^{16}+\frac{26\!\cdots\!77}{93\!\cdots\!01}a^{15}+\frac{16\!\cdots\!40}{15\!\cdots\!41}a^{14}+\frac{68\!\cdots\!10}{93\!\cdots\!01}a^{13}-\frac{18\!\cdots\!41}{93\!\cdots\!01}a^{12}-\frac{41\!\cdots\!37}{93\!\cdots\!01}a^{11}-\frac{15\!\cdots\!53}{93\!\cdots\!01}a^{10}+\frac{19\!\cdots\!46}{93\!\cdots\!01}a^{9}+\frac{31\!\cdots\!88}{93\!\cdots\!01}a^{8}+\frac{14\!\cdots\!29}{93\!\cdots\!01}a^{7}-\frac{52\!\cdots\!58}{93\!\cdots\!01}a^{6}-\frac{88\!\cdots\!95}{93\!\cdots\!01}a^{5}-\frac{59\!\cdots\!03}{93\!\cdots\!01}a^{4}+\frac{29\!\cdots\!25}{93\!\cdots\!01}a^{3}+\frac{79\!\cdots\!10}{93\!\cdots\!01}a^{2}+\frac{45\!\cdots\!82}{49\!\cdots\!79}a+\frac{17\!\cdots\!07}{49\!\cdots\!79}$, $\frac{22\!\cdots\!34}{93\!\cdots\!01}a^{25}-\frac{18\!\cdots\!78}{93\!\cdots\!01}a^{24}+\frac{33\!\cdots\!10}{93\!\cdots\!01}a^{23}+\frac{39\!\cdots\!59}{93\!\cdots\!01}a^{22}-\frac{74\!\cdots\!17}{32\!\cdots\!69}a^{21}+\frac{89\!\cdots\!54}{72\!\cdots\!77}a^{20}-\frac{28\!\cdots\!61}{93\!\cdots\!01}a^{19}-\frac{11\!\cdots\!59}{93\!\cdots\!01}a^{18}+\frac{28\!\cdots\!21}{93\!\cdots\!01}a^{17}-\frac{39\!\cdots\!41}{93\!\cdots\!01}a^{16}+\frac{65\!\cdots\!31}{93\!\cdots\!01}a^{15}-\frac{19\!\cdots\!15}{93\!\cdots\!01}a^{14}+\frac{14\!\cdots\!42}{93\!\cdots\!01}a^{13}-\frac{39\!\cdots\!97}{72\!\cdots\!77}a^{12}-\frac{72\!\cdots\!07}{93\!\cdots\!01}a^{11}+\frac{31\!\cdots\!85}{93\!\cdots\!01}a^{10}+\frac{46\!\cdots\!31}{93\!\cdots\!01}a^{9}+\frac{44\!\cdots\!02}{93\!\cdots\!01}a^{8}-\frac{10\!\cdots\!66}{72\!\cdots\!77}a^{7}-\frac{14\!\cdots\!42}{93\!\cdots\!01}a^{6}-\frac{11\!\cdots\!27}{93\!\cdots\!01}a^{5}+\frac{54\!\cdots\!68}{93\!\cdots\!01}a^{4}+\frac{11\!\cdots\!55}{93\!\cdots\!01}a^{3}+\frac{11\!\cdots\!98}{93\!\cdots\!01}a^{2}+\frac{40\!\cdots\!62}{49\!\cdots\!79}a-\frac{10\!\cdots\!96}{49\!\cdots\!79}$, $\frac{21\!\cdots\!93}{49\!\cdots\!79}a^{25}-\frac{18\!\cdots\!15}{49\!\cdots\!79}a^{24}+\frac{41\!\cdots\!08}{49\!\cdots\!79}a^{23}+\frac{11\!\cdots\!47}{25\!\cdots\!41}a^{22}-\frac{81\!\cdots\!09}{16\!\cdots\!51}a^{21}+\frac{12\!\cdots\!05}{49\!\cdots\!79}a^{20}-\frac{32\!\cdots\!60}{49\!\cdots\!79}a^{19}+\frac{26\!\cdots\!11}{49\!\cdots\!79}a^{18}+\frac{40\!\cdots\!47}{49\!\cdots\!79}a^{17}-\frac{60\!\cdots\!25}{49\!\cdots\!79}a^{16}+\frac{64\!\cdots\!93}{49\!\cdots\!79}a^{15}-\frac{54\!\cdots\!73}{49\!\cdots\!79}a^{14}+\frac{13\!\cdots\!82}{49\!\cdots\!79}a^{13}-\frac{54\!\cdots\!97}{49\!\cdots\!79}a^{12}-\frac{42\!\cdots\!60}{49\!\cdots\!79}a^{11}+\frac{85\!\cdots\!45}{49\!\cdots\!79}a^{10}+\frac{41\!\cdots\!26}{49\!\cdots\!79}a^{9}+\frac{13\!\cdots\!85}{49\!\cdots\!79}a^{8}-\frac{43\!\cdots\!50}{49\!\cdots\!79}a^{7}-\frac{12\!\cdots\!90}{49\!\cdots\!79}a^{6}-\frac{11\!\cdots\!61}{49\!\cdots\!79}a^{5}+\frac{78\!\cdots\!76}{49\!\cdots\!79}a^{4}+\frac{10\!\cdots\!35}{49\!\cdots\!79}a^{3}+\frac{51\!\cdots\!26}{80\!\cdots\!39}a^{2}+\frac{16\!\cdots\!44}{25\!\cdots\!41}a-\frac{39\!\cdots\!91}{25\!\cdots\!41}$, $\frac{56\!\cdots\!47}{93\!\cdots\!01}a^{25}-\frac{50\!\cdots\!63}{93\!\cdots\!01}a^{24}+\frac{12\!\cdots\!02}{93\!\cdots\!01}a^{23}-\frac{29\!\cdots\!82}{93\!\cdots\!01}a^{22}-\frac{17\!\cdots\!63}{32\!\cdots\!69}a^{21}+\frac{35\!\cdots\!34}{93\!\cdots\!01}a^{20}-\frac{10\!\cdots\!24}{93\!\cdots\!01}a^{19}+\frac{69\!\cdots\!55}{93\!\cdots\!01}a^{18}-\frac{11\!\cdots\!56}{93\!\cdots\!01}a^{17}-\frac{19\!\cdots\!72}{93\!\cdots\!01}a^{16}+\frac{19\!\cdots\!12}{93\!\cdots\!01}a^{15}-\frac{21\!\cdots\!08}{93\!\cdots\!01}a^{14}+\frac{62\!\cdots\!90}{93\!\cdots\!01}a^{13}-\frac{17\!\cdots\!73}{93\!\cdots\!01}a^{12}-\frac{38\!\cdots\!56}{93\!\cdots\!01}a^{11}+\frac{84\!\cdots\!38}{93\!\cdots\!01}a^{10}+\frac{84\!\cdots\!62}{93\!\cdots\!01}a^{9}+\frac{50\!\cdots\!99}{93\!\cdots\!01}a^{8}-\frac{50\!\cdots\!40}{93\!\cdots\!01}a^{7}-\frac{22\!\cdots\!93}{93\!\cdots\!01}a^{6}-\frac{14\!\cdots\!99}{93\!\cdots\!01}a^{5}-\frac{16\!\cdots\!70}{93\!\cdots\!01}a^{4}+\frac{85\!\cdots\!82}{93\!\cdots\!01}a^{3}+\frac{29\!\cdots\!59}{93\!\cdots\!01}a^{2}+\frac{11\!\cdots\!32}{49\!\cdots\!79}a+\frac{77\!\cdots\!14}{49\!\cdots\!79}$, $\frac{79\!\cdots\!10}{72\!\cdots\!77}a^{25}-\frac{81\!\cdots\!49}{93\!\cdots\!01}a^{24}+\frac{12\!\cdots\!22}{93\!\cdots\!01}a^{23}+\frac{20\!\cdots\!09}{72\!\cdots\!77}a^{22}-\frac{36\!\cdots\!36}{32\!\cdots\!69}a^{21}+\frac{50\!\cdots\!20}{93\!\cdots\!01}a^{20}-\frac{11\!\cdots\!76}{93\!\cdots\!01}a^{19}-\frac{11\!\cdots\!80}{93\!\cdots\!01}a^{18}+\frac{20\!\cdots\!43}{93\!\cdots\!01}a^{17}-\frac{17\!\cdots\!35}{93\!\cdots\!01}a^{16}+\frac{28\!\cdots\!00}{93\!\cdots\!01}a^{15}+\frac{81\!\cdots\!24}{93\!\cdots\!01}a^{14}+\frac{47\!\cdots\!71}{93\!\cdots\!01}a^{13}-\frac{20\!\cdots\!25}{93\!\cdots\!01}a^{12}-\frac{43\!\cdots\!32}{93\!\cdots\!01}a^{11}+\frac{20\!\cdots\!65}{93\!\cdots\!01}a^{10}+\frac{22\!\cdots\!77}{93\!\cdots\!01}a^{9}+\frac{22\!\cdots\!76}{93\!\cdots\!01}a^{8}-\frac{10\!\cdots\!72}{93\!\cdots\!01}a^{7}-\frac{71\!\cdots\!56}{93\!\cdots\!01}a^{6}-\frac{56\!\cdots\!78}{93\!\cdots\!01}a^{5}+\frac{12\!\cdots\!41}{93\!\cdots\!01}a^{4}+\frac{41\!\cdots\!39}{72\!\cdots\!77}a^{3}+\frac{41\!\cdots\!54}{93\!\cdots\!01}a^{2}+\frac{10\!\cdots\!57}{37\!\cdots\!83}a+\frac{96\!\cdots\!81}{49\!\cdots\!79}$, $\frac{19\!\cdots\!19}{93\!\cdots\!01}a^{25}-\frac{18\!\cdots\!11}{93\!\cdots\!01}a^{24}+\frac{50\!\cdots\!17}{93\!\cdots\!01}a^{23}+\frac{25\!\cdots\!73}{93\!\cdots\!01}a^{22}-\frac{15\!\cdots\!99}{52\!\cdots\!29}a^{21}+\frac{13\!\cdots\!20}{93\!\cdots\!01}a^{20}-\frac{37\!\cdots\!94}{93\!\cdots\!01}a^{19}+\frac{17\!\cdots\!48}{93\!\cdots\!01}a^{18}+\frac{59\!\cdots\!25}{93\!\cdots\!01}a^{17}-\frac{74\!\cdots\!41}{72\!\cdots\!77}a^{16}+\frac{62\!\cdots\!06}{93\!\cdots\!01}a^{15}-\frac{69\!\cdots\!79}{72\!\cdots\!77}a^{14}+\frac{12\!\cdots\!75}{93\!\cdots\!01}a^{13}-\frac{55\!\cdots\!80}{93\!\cdots\!01}a^{12}-\frac{10\!\cdots\!21}{93\!\cdots\!01}a^{11}+\frac{14\!\cdots\!32}{93\!\cdots\!01}a^{10}+\frac{33\!\cdots\!71}{93\!\cdots\!01}a^{9}-\frac{22\!\cdots\!41}{93\!\cdots\!01}a^{8}-\frac{72\!\cdots\!19}{93\!\cdots\!01}a^{7}-\frac{83\!\cdots\!85}{93\!\cdots\!01}a^{6}+\frac{96\!\cdots\!91}{93\!\cdots\!01}a^{5}+\frac{15\!\cdots\!95}{93\!\cdots\!01}a^{4}+\frac{88\!\cdots\!71}{93\!\cdots\!01}a^{3}-\frac{83\!\cdots\!99}{72\!\cdots\!77}a^{2}-\frac{48\!\cdots\!60}{49\!\cdots\!79}a-\frac{76\!\cdots\!86}{49\!\cdots\!79}$, $\frac{17\!\cdots\!65}{93\!\cdots\!01}a^{25}-\frac{11\!\cdots\!98}{93\!\cdots\!01}a^{24}+\frac{62\!\cdots\!30}{93\!\cdots\!01}a^{23}+\frac{39\!\cdots\!24}{72\!\cdots\!77}a^{22}-\frac{17\!\cdots\!52}{32\!\cdots\!69}a^{21}+\frac{58\!\cdots\!48}{93\!\cdots\!01}a^{20}-\frac{10\!\cdots\!42}{93\!\cdots\!01}a^{19}-\frac{27\!\cdots\!46}{93\!\cdots\!01}a^{18}-\frac{45\!\cdots\!82}{93\!\cdots\!01}a^{17}+\frac{69\!\cdots\!73}{93\!\cdots\!01}a^{16}+\frac{50\!\cdots\!16}{93\!\cdots\!01}a^{15}+\frac{45\!\cdots\!78}{93\!\cdots\!01}a^{14}+\frac{12\!\cdots\!24}{72\!\cdots\!77}a^{13}-\frac{30\!\cdots\!68}{72\!\cdots\!77}a^{12}-\frac{72\!\cdots\!07}{72\!\cdots\!77}a^{11}-\frac{13\!\cdots\!02}{93\!\cdots\!01}a^{10}+\frac{45\!\cdots\!78}{93\!\cdots\!01}a^{9}+\frac{10\!\cdots\!01}{93\!\cdots\!01}a^{8}+\frac{52\!\cdots\!99}{93\!\cdots\!01}a^{7}-\frac{14\!\cdots\!82}{72\!\cdots\!77}a^{6}-\frac{22\!\cdots\!19}{72\!\cdots\!77}a^{5}-\frac{11\!\cdots\!39}{93\!\cdots\!01}a^{4}+\frac{25\!\cdots\!57}{93\!\cdots\!01}a^{3}+\frac{25\!\cdots\!24}{93\!\cdots\!01}a^{2}+\frac{23\!\cdots\!62}{49\!\cdots\!79}a+\frac{14\!\cdots\!51}{49\!\cdots\!79}$, $\frac{37\!\cdots\!56}{93\!\cdots\!01}a^{25}-\frac{35\!\cdots\!53}{93\!\cdots\!01}a^{24}+\frac{10\!\cdots\!26}{93\!\cdots\!01}a^{23}-\frac{56\!\cdots\!73}{93\!\cdots\!01}a^{22}-\frac{41\!\cdots\!30}{11\!\cdots\!61}a^{21}+\frac{25\!\cdots\!77}{93\!\cdots\!01}a^{20}-\frac{61\!\cdots\!60}{72\!\cdots\!77}a^{19}+\frac{75\!\cdots\!35}{93\!\cdots\!01}a^{18}-\frac{76\!\cdots\!22}{93\!\cdots\!01}a^{17}-\frac{15\!\cdots\!68}{93\!\cdots\!01}a^{16}+\frac{13\!\cdots\!08}{93\!\cdots\!01}a^{15}-\frac{20\!\cdots\!40}{93\!\cdots\!01}a^{14}+\frac{45\!\cdots\!67}{93\!\cdots\!01}a^{13}-\frac{12\!\cdots\!94}{93\!\cdots\!01}a^{12}+\frac{17\!\cdots\!83}{72\!\cdots\!77}a^{11}+\frac{93\!\cdots\!61}{93\!\cdots\!01}a^{10}+\frac{52\!\cdots\!06}{93\!\cdots\!01}a^{9}+\frac{11\!\cdots\!79}{15\!\cdots\!41}a^{8}-\frac{23\!\cdots\!09}{93\!\cdots\!01}a^{7}-\frac{15\!\cdots\!21}{93\!\cdots\!01}a^{6}+\frac{91\!\cdots\!78}{93\!\cdots\!01}a^{5}-\frac{10\!\cdots\!41}{93\!\cdots\!01}a^{4}+\frac{86\!\cdots\!73}{93\!\cdots\!01}a^{3}+\frac{93\!\cdots\!61}{93\!\cdots\!01}a^{2}+\frac{62\!\cdots\!50}{49\!\cdots\!79}a+\frac{52\!\cdots\!17}{49\!\cdots\!79}$, $\frac{99\!\cdots\!12}{92\!\cdots\!01}a^{25}-\frac{89\!\cdots\!24}{92\!\cdots\!01}a^{24}+\frac{21\!\cdots\!66}{92\!\cdots\!01}a^{23}+\frac{74\!\cdots\!92}{92\!\cdots\!01}a^{22}-\frac{44\!\cdots\!49}{31\!\cdots\!69}a^{21}+\frac{62\!\cdots\!90}{92\!\cdots\!01}a^{20}-\frac{16\!\cdots\!19}{92\!\cdots\!01}a^{19}+\frac{30\!\cdots\!70}{92\!\cdots\!01}a^{18}+\frac{28\!\cdots\!55}{92\!\cdots\!01}a^{17}-\frac{49\!\cdots\!41}{92\!\cdots\!01}a^{16}+\frac{23\!\cdots\!53}{71\!\cdots\!77}a^{15}-\frac{29\!\cdots\!30}{92\!\cdots\!01}a^{14}+\frac{57\!\cdots\!30}{92\!\cdots\!01}a^{13}-\frac{24\!\cdots\!33}{92\!\cdots\!01}a^{12}-\frac{15\!\cdots\!35}{92\!\cdots\!01}a^{11}+\frac{55\!\cdots\!23}{92\!\cdots\!01}a^{10}+\frac{16\!\cdots\!64}{92\!\cdots\!01}a^{9}-\frac{12\!\cdots\!62}{92\!\cdots\!01}a^{8}-\frac{21\!\cdots\!98}{92\!\cdots\!01}a^{7}-\frac{41\!\cdots\!25}{92\!\cdots\!01}a^{6}-\frac{20\!\cdots\!24}{92\!\cdots\!01}a^{5}+\frac{38\!\cdots\!76}{92\!\cdots\!01}a^{4}+\frac{32\!\cdots\!38}{92\!\cdots\!01}a^{3}-\frac{70\!\cdots\!96}{92\!\cdots\!01}a^{2}+\frac{13\!\cdots\!41}{48\!\cdots\!79}a-\frac{68\!\cdots\!02}{48\!\cdots\!79}$, $\frac{21\!\cdots\!78}{93\!\cdots\!01}a^{25}-\frac{13\!\cdots\!41}{93\!\cdots\!01}a^{24}-\frac{17\!\cdots\!15}{93\!\cdots\!01}a^{23}+\frac{18\!\cdots\!91}{72\!\cdots\!77}a^{22}-\frac{18\!\cdots\!19}{32\!\cdots\!69}a^{21}+\frac{11\!\cdots\!76}{93\!\cdots\!01}a^{20}+\frac{37\!\cdots\!84}{72\!\cdots\!77}a^{19}-\frac{15\!\cdots\!65}{93\!\cdots\!01}a^{18}+\frac{32\!\cdots\!62}{93\!\cdots\!01}a^{17}-\frac{43\!\cdots\!36}{93\!\cdots\!01}a^{16}+\frac{93\!\cdots\!83}{93\!\cdots\!01}a^{15}+\frac{13\!\cdots\!82}{93\!\cdots\!01}a^{14}-\frac{38\!\cdots\!11}{93\!\cdots\!01}a^{13}+\frac{82\!\cdots\!93}{93\!\cdots\!01}a^{12}-\frac{41\!\cdots\!95}{93\!\cdots\!01}a^{11}+\frac{49\!\cdots\!34}{93\!\cdots\!01}a^{10}+\frac{66\!\cdots\!96}{93\!\cdots\!01}a^{9}+\frac{13\!\cdots\!49}{93\!\cdots\!01}a^{8}-\frac{81\!\cdots\!74}{93\!\cdots\!01}a^{7}-\frac{14\!\cdots\!91}{93\!\cdots\!01}a^{6}-\frac{21\!\cdots\!18}{93\!\cdots\!01}a^{5}+\frac{51\!\cdots\!73}{93\!\cdots\!01}a^{4}+\frac{43\!\cdots\!45}{93\!\cdots\!01}a^{3}+\frac{22\!\cdots\!81}{93\!\cdots\!01}a^{2}+\frac{33\!\cdots\!91}{49\!\cdots\!79}a+\frac{11\!\cdots\!77}{49\!\cdots\!79}$, $\frac{23\!\cdots\!92}{72\!\cdots\!77}a^{25}-\frac{23\!\cdots\!67}{93\!\cdots\!01}a^{24}+\frac{37\!\cdots\!30}{93\!\cdots\!01}a^{23}+\frac{57\!\cdots\!79}{93\!\cdots\!01}a^{22}-\frac{33\!\cdots\!96}{11\!\cdots\!61}a^{21}+\frac{16\!\cdots\!80}{93\!\cdots\!01}a^{20}-\frac{34\!\cdots\!92}{93\!\cdots\!01}a^{19}-\frac{25\!\cdots\!97}{93\!\cdots\!01}a^{18}+\frac{37\!\cdots\!02}{93\!\cdots\!01}a^{17}-\frac{15\!\cdots\!56}{93\!\cdots\!01}a^{16}+\frac{95\!\cdots\!55}{93\!\cdots\!01}a^{15}+\frac{18\!\cdots\!81}{93\!\cdots\!01}a^{14}+\frac{22\!\cdots\!66}{93\!\cdots\!01}a^{13}-\frac{36\!\cdots\!17}{72\!\cdots\!77}a^{12}-\frac{13\!\cdots\!88}{93\!\cdots\!01}a^{11}-\frac{17\!\cdots\!43}{93\!\cdots\!01}a^{10}+\frac{38\!\cdots\!95}{93\!\cdots\!01}a^{9}+\frac{81\!\cdots\!51}{93\!\cdots\!01}a^{8}+\frac{78\!\cdots\!97}{93\!\cdots\!01}a^{7}-\frac{73\!\cdots\!57}{93\!\cdots\!01}a^{6}-\frac{28\!\cdots\!75}{93\!\cdots\!01}a^{5}-\frac{36\!\cdots\!20}{93\!\cdots\!01}a^{4}-\frac{34\!\cdots\!86}{93\!\cdots\!01}a^{3}-\frac{13\!\cdots\!09}{93\!\cdots\!01}a^{2}-\frac{19\!\cdots\!78}{49\!\cdots\!79}a-\frac{46\!\cdots\!41}{49\!\cdots\!79}$
| 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: | \( 154406339448.51657 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 154406339448.51657 \cdot 8}{2\cdot\sqrt{526721251926767614625217802292605164794921875}}\cr\approx \mathstrut & 0.640137082872717 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 8*x^25 + 14*x^24 + 18*x^23 - 92*x^22 + 514*x^21 - 1212*x^20 - 596*x^19 + 954*x^18 - 2053*x^17 + 29744*x^16 - 4491*x^15 + 71256*x^14 - 216720*x^13 - 344608*x^12 + 50717*x^11 + 1942970*x^10 + 2307612*x^9 + 264538*x^8 - 5680405*x^7 - 6094469*x^6 - 2337568*x^5 + 3586683*x^4 + 5626157*x^3 + 5313221*x^2 + 843733*x + 496261)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^26 - 8*x^25 + 14*x^24 + 18*x^23 - 92*x^22 + 514*x^21 - 1212*x^20 - 596*x^19 + 954*x^18 - 2053*x^17 + 29744*x^16 - 4491*x^15 + 71256*x^14 - 216720*x^13 - 344608*x^12 + 50717*x^11 + 1942970*x^10 + 2307612*x^9 + 264538*x^8 - 5680405*x^7 - 6094469*x^6 - 2337568*x^5 + 3586683*x^4 + 5626157*x^3 + 5313221*x^2 + 843733*x + 496261, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^26 - 8*x^25 + 14*x^24 + 18*x^23 - 92*x^22 + 514*x^21 - 1212*x^20 - 596*x^19 + 954*x^18 - 2053*x^17 + 29744*x^16 - 4491*x^15 + 71256*x^14 - 216720*x^13 - 344608*x^12 + 50717*x^11 + 1942970*x^10 + 2307612*x^9 + 264538*x^8 - 5680405*x^7 - 6094469*x^6 - 2337568*x^5 + 3586683*x^4 + 5626157*x^3 + 5313221*x^2 + 843733*x + 496261);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 8*x^25 + 14*x^24 + 18*x^23 - 92*x^22 + 514*x^21 - 1212*x^20 - 596*x^19 + 954*x^18 - 2053*x^17 + 29744*x^16 - 4491*x^15 + 71256*x^14 - 216720*x^13 - 344608*x^12 + 50717*x^11 + 1942970*x^10 + 2307612*x^9 + 264538*x^8 - 5680405*x^7 - 6094469*x^6 - 2337568*x^5 + 3586683*x^4 + 5626157*x^3 + 5313221*x^2 + 843733*x + 496261);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 52 |
| The 16 conjugacy class representatives for $D_{26}$ |
| Character table for $D_{26}$ |
Intermediate fields
| \(\Q(\sqrt{-2755}) \), 13.1.27983987175790801.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 26 sibling: | 26.2.955936936346220716198217427028321533203125.1 |
| Minimal sibling: | 26.2.955936936346220716198217427028321533203125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $26$ | $26$ | R | $26$ | ${\href{/padicField/11.2.0.1}{2} }^{13}$ | ${\href{/padicField/13.2.0.1}{2} }^{12}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{12}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | $26$ | R | ${\href{/padicField/31.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/41.13.0.1}{13} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{13}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $26$ | $2$ | $13$ | $13$ | |||
|
\(19\)
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(29\)
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.2755.2t1.a.a | $1$ | $ 5 \cdot 19 \cdot 29 $ | \(\Q(\sqrt{-2755}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.551.2t1.a.a | $1$ | $ 19 \cdot 29 $ | \(\Q(\sqrt{-551}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 2.13775.26t3.a.d | $2$ | $ 5^{2} \cdot 19 \cdot 29 $ | 26.0.526721251926767614625217802292605164794921875.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
| * | 2.551.13t2.a.f | $2$ | $ 19 \cdot 29 $ | 13.1.27983987175790801.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
| * | 2.551.13t2.a.e | $2$ | $ 19 \cdot 29 $ | 13.1.27983987175790801.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
| * | 2.13775.26t3.a.e | $2$ | $ 5^{2} \cdot 19 \cdot 29 $ | 26.0.526721251926767614625217802292605164794921875.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
| * | 2.13775.26t3.a.f | $2$ | $ 5^{2} \cdot 19 \cdot 29 $ | 26.0.526721251926767614625217802292605164794921875.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
| * | 2.551.13t2.a.b | $2$ | $ 19 \cdot 29 $ | 13.1.27983987175790801.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
| * | 2.551.13t2.a.c | $2$ | $ 19 \cdot 29 $ | 13.1.27983987175790801.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
| * | 2.551.13t2.a.d | $2$ | $ 19 \cdot 29 $ | 13.1.27983987175790801.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
| * | 2.13775.26t3.a.a | $2$ | $ 5^{2} \cdot 19 \cdot 29 $ | 26.0.526721251926767614625217802292605164794921875.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
| * | 2.13775.26t3.a.c | $2$ | $ 5^{2} \cdot 19 \cdot 29 $ | 26.0.526721251926767614625217802292605164794921875.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
| * | 2.551.13t2.a.a | $2$ | $ 19 \cdot 29 $ | 13.1.27983987175790801.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
| * | 2.13775.26t3.a.b | $2$ | $ 5^{2} \cdot 19 \cdot 29 $ | 26.0.526721251926767614625217802292605164794921875.1 | $D_{26}$ (as 26T3) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.