Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 21*x^24 + 38*x^23 + 294*x^22 - 464*x^21 - 1357*x^20 + 1930*x^19 + 9504*x^18 - 8228*x^17 + 25716*x^16 - 28630*x^15 + 286963*x^14 - 378044*x^13 + 2103312*x^12 - 1678782*x^11 + 11411452*x^10 - 5574160*x^9 + 50609745*x^8 - 14301644*x^7 + 160353701*x^6 - 26227056*x^5 + 361035384*x^4 - 44819066*x^3 + 515469360*x^2 - 43859864*x + 329025401)
gp: K = bnfinit(y^26 - 2*y^25 - 21*y^24 + 38*y^23 + 294*y^22 - 464*y^21 - 1357*y^20 + 1930*y^19 + 9504*y^18 - 8228*y^17 + 25716*y^16 - 28630*y^15 + 286963*y^14 - 378044*y^13 + 2103312*y^12 - 1678782*y^11 + 11411452*y^10 - 5574160*y^9 + 50609745*y^8 - 14301644*y^7 + 160353701*y^6 - 26227056*y^5 + 361035384*y^4 - 44819066*y^3 + 515469360*y^2 - 43859864*y + 329025401, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 2*x^25 - 21*x^24 + 38*x^23 + 294*x^22 - 464*x^21 - 1357*x^20 + 1930*x^19 + 9504*x^18 - 8228*x^17 + 25716*x^16 - 28630*x^15 + 286963*x^14 - 378044*x^13 + 2103312*x^12 - 1678782*x^11 + 11411452*x^10 - 5574160*x^9 + 50609745*x^8 - 14301644*x^7 + 160353701*x^6 - 26227056*x^5 + 361035384*x^4 - 44819066*x^3 + 515469360*x^2 - 43859864*x + 329025401);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 2*x^25 - 21*x^24 + 38*x^23 + 294*x^22 - 464*x^21 - 1357*x^20 + 1930*x^19 + 9504*x^18 - 8228*x^17 + 25716*x^16 - 28630*x^15 + 286963*x^14 - 378044*x^13 + 2103312*x^12 - 1678782*x^11 + 11411452*x^10 - 5574160*x^9 + 50609745*x^8 - 14301644*x^7 + 160353701*x^6 - 26227056*x^5 + 361035384*x^4 - 44819066*x^3 + 515469360*x^2 - 43859864*x + 329025401)
\( x^{26} - 2 x^{25} - 21 x^{24} + 38 x^{23} + 294 x^{22} - 464 x^{21} - 1357 x^{20} + 1930 x^{19} + \cdots + 329025401 \)
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: |
\(-132675490325051365459874809737094631268103216670179328\)
\(\medspace = -\,2^{39}\cdot 53^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(110.45\) | 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: | $2^{3/2}53^{12/13}\approx 110.45471457632615$ | ||
| Ramified primes: |
\(2\), \(53\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $26$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(424=2^{3}\cdot 53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{424}(1,·)$, $\chi_{424}(259,·)$, $\chi_{424}(227,·)$, $\chi_{424}(225,·)$, $\chi_{424}(201,·)$, $\chi_{424}(395,·)$, $\chi_{424}(417,·)$, $\chi_{424}(81,·)$, $\chi_{424}(275,·)$, $\chi_{424}(203,·)$, $\chi_{424}(281,·)$, $\chi_{424}(153,·)$, $\chi_{424}(331,·)$, $\chi_{424}(155,·)$, $\chi_{424}(289,·)$, $\chi_{424}(99,·)$, $\chi_{424}(195,·)$, $\chi_{424}(97,·)$, $\chi_{424}(169,·)$, $\chi_{424}(107,·)$, $\chi_{424}(387,·)$, $\chi_{424}(49,·)$, $\chi_{424}(307,·)$, $\chi_{424}(89,·)$, $\chi_{424}(121,·)$, $\chi_{424}(187,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{4096}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23}a^{22}-\frac{6}{23}a^{21}-\frac{4}{23}a^{20}-\frac{2}{23}a^{19}+\frac{1}{23}a^{18}-\frac{11}{23}a^{17}-\frac{9}{23}a^{16}+\frac{7}{23}a^{15}-\frac{6}{23}a^{14}+\frac{2}{23}a^{13}-\frac{10}{23}a^{12}-\frac{4}{23}a^{11}+\frac{2}{23}a^{10}+\frac{1}{23}a^{9}+\frac{7}{23}a^{8}-\frac{2}{23}a^{7}+\frac{7}{23}a^{6}-\frac{2}{23}a^{5}+\frac{8}{23}a^{4}-\frac{11}{23}a^{3}-\frac{1}{23}a^{2}+\frac{9}{23}a+\frac{6}{23}$, $\frac{1}{1909}a^{23}+\frac{17}{1909}a^{22}-\frac{648}{1909}a^{21}+\frac{504}{1909}a^{20}-\frac{183}{1909}a^{19}-\frac{471}{1909}a^{18}+\frac{796}{1909}a^{17}-\frac{131}{1909}a^{16}-\frac{167}{1909}a^{15}+\frac{416}{1909}a^{14}-\frac{493}{1909}a^{13}+\frac{19}{1909}a^{12}-\frac{412}{1909}a^{11}+\frac{93}{1909}a^{10}+\frac{260}{1909}a^{9}+\frac{113}{1909}a^{8}+\frac{53}{1909}a^{7}-\frac{163}{1909}a^{6}+\frac{77}{1909}a^{5}+\frac{35}{1909}a^{4}+\frac{712}{1909}a^{3}-\frac{865}{1909}a^{2}+\frac{834}{1909}a+\frac{8}{83}$, $\frac{1}{46556235749}a^{24}-\frac{2538442}{46556235749}a^{23}+\frac{667381408}{46556235749}a^{22}-\frac{13730156511}{46556235749}a^{21}-\frac{15492865346}{46556235749}a^{20}+\frac{5592677835}{46556235749}a^{19}-\frac{148864578}{435105007}a^{18}+\frac{15578016692}{46556235749}a^{17}-\frac{22108418631}{46556235749}a^{16}+\frac{4943024733}{46556235749}a^{15}+\frac{8075717252}{46556235749}a^{14}-\frac{17464510569}{46556235749}a^{13}+\frac{6765095360}{46556235749}a^{12}+\frac{21109855762}{46556235749}a^{11}-\frac{206278182}{46556235749}a^{10}+\frac{12511024429}{46556235749}a^{9}-\frac{3536278384}{46556235749}a^{8}-\frac{6025674974}{46556235749}a^{7}-\frac{8847845368}{46556235749}a^{6}-\frac{1099413561}{46556235749}a^{5}-\frac{12396291691}{46556235749}a^{4}-\frac{2957496828}{46556235749}a^{3}-\frac{12654458003}{46556235749}a^{2}+\frac{626375174}{46556235749}a-\frac{4908460323}{46556235749}$, $\frac{1}{25\!\cdots\!27}a^{25}-\frac{73\!\cdots\!15}{25\!\cdots\!27}a^{24}-\frac{12\!\cdots\!11}{25\!\cdots\!27}a^{23}+\frac{16\!\cdots\!34}{25\!\cdots\!27}a^{22}-\frac{11\!\cdots\!04}{25\!\cdots\!27}a^{21}+\frac{15\!\cdots\!74}{25\!\cdots\!27}a^{20}-\frac{79\!\cdots\!68}{25\!\cdots\!27}a^{19}+\frac{25\!\cdots\!61}{25\!\cdots\!27}a^{18}-\frac{94\!\cdots\!77}{25\!\cdots\!27}a^{17}+\frac{83\!\cdots\!09}{25\!\cdots\!27}a^{16}+\frac{24\!\cdots\!24}{25\!\cdots\!27}a^{15}-\frac{51\!\cdots\!67}{25\!\cdots\!27}a^{14}+\frac{71\!\cdots\!51}{10\!\cdots\!49}a^{13}+\frac{93\!\cdots\!92}{25\!\cdots\!27}a^{12}+\frac{94\!\cdots\!14}{25\!\cdots\!27}a^{11}-\frac{30\!\cdots\!57}{25\!\cdots\!27}a^{10}+\frac{92\!\cdots\!56}{25\!\cdots\!27}a^{9}+\frac{46\!\cdots\!78}{25\!\cdots\!27}a^{8}+\frac{12\!\cdots\!41}{25\!\cdots\!27}a^{7}+\frac{56\!\cdots\!05}{25\!\cdots\!27}a^{6}+\frac{34\!\cdots\!97}{25\!\cdots\!27}a^{5}-\frac{91\!\cdots\!15}{25\!\cdots\!27}a^{4}+\frac{51\!\cdots\!64}{25\!\cdots\!27}a^{3}-\frac{92\!\cdots\!11}{25\!\cdots\!27}a^{2}-\frac{31\!\cdots\!83}{25\!\cdots\!27}a-\frac{23\!\cdots\!66}{10\!\cdots\!49}$
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
$C_{1064051}$, which has order $1064051$ (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: |
$\frac{61\!\cdots\!24}{25\!\cdots\!27}a^{25}-\frac{43\!\cdots\!34}{25\!\cdots\!27}a^{24}-\frac{65\!\cdots\!84}{25\!\cdots\!27}a^{23}+\frac{10\!\cdots\!16}{25\!\cdots\!27}a^{22}+\frac{26\!\cdots\!20}{25\!\cdots\!27}a^{21}-\frac{16\!\cdots\!12}{25\!\cdots\!27}a^{20}+\frac{13\!\cdots\!52}{25\!\cdots\!27}a^{19}+\frac{13\!\cdots\!88}{25\!\cdots\!27}a^{18}-\frac{98\!\cdots\!28}{25\!\cdots\!27}a^{17}-\frac{91\!\cdots\!53}{25\!\cdots\!27}a^{16}+\frac{10\!\cdots\!24}{25\!\cdots\!27}a^{15}+\frac{26\!\cdots\!38}{25\!\cdots\!27}a^{14}+\frac{25\!\cdots\!92}{25\!\cdots\!27}a^{13}-\frac{16\!\cdots\!46}{25\!\cdots\!27}a^{12}+\frac{26\!\cdots\!92}{25\!\cdots\!27}a^{11}-\frac{27\!\cdots\!70}{25\!\cdots\!27}a^{10}+\frac{82\!\cdots\!48}{25\!\cdots\!27}a^{9}-\frac{16\!\cdots\!08}{25\!\cdots\!27}a^{8}+\frac{47\!\cdots\!00}{25\!\cdots\!27}a^{7}-\frac{70\!\cdots\!50}{25\!\cdots\!27}a^{6}+\frac{14\!\cdots\!76}{25\!\cdots\!27}a^{5}-\frac{11\!\cdots\!50}{25\!\cdots\!27}a^{4}+\frac{34\!\cdots\!04}{25\!\cdots\!27}a^{3}-\frac{30\!\cdots\!88}{25\!\cdots\!27}a^{2}+\frac{43\!\cdots\!00}{25\!\cdots\!27}a-\frac{15\!\cdots\!04}{10\!\cdots\!49}$, $\frac{54\!\cdots\!40}{25\!\cdots\!27}a^{25}+\frac{19\!\cdots\!96}{25\!\cdots\!27}a^{24}-\frac{17\!\cdots\!44}{25\!\cdots\!27}a^{23}-\frac{49\!\cdots\!64}{25\!\cdots\!27}a^{22}+\frac{29\!\cdots\!48}{25\!\cdots\!27}a^{21}+\frac{79\!\cdots\!86}{25\!\cdots\!27}a^{20}-\frac{24\!\cdots\!12}{25\!\cdots\!27}a^{19}-\frac{53\!\cdots\!66}{25\!\cdots\!27}a^{18}+\frac{14\!\cdots\!56}{25\!\cdots\!27}a^{17}+\frac{17\!\cdots\!86}{10\!\cdots\!49}a^{16}-\frac{28\!\cdots\!68}{25\!\cdots\!27}a^{15}-\frac{15\!\cdots\!22}{10\!\cdots\!49}a^{14}+\frac{15\!\cdots\!76}{30\!\cdots\!69}a^{13}+\frac{72\!\cdots\!28}{25\!\cdots\!27}a^{12}-\frac{24\!\cdots\!92}{25\!\cdots\!27}a^{11}+\frac{36\!\cdots\!42}{25\!\cdots\!27}a^{10}+\frac{29\!\cdots\!32}{30\!\cdots\!69}a^{9}+\frac{22\!\cdots\!05}{25\!\cdots\!27}a^{8}+\frac{12\!\cdots\!20}{25\!\cdots\!27}a^{7}+\frac{94\!\cdots\!04}{25\!\cdots\!27}a^{6}+\frac{41\!\cdots\!44}{25\!\cdots\!27}a^{5}+\frac{26\!\cdots\!94}{25\!\cdots\!27}a^{4}+\frac{81\!\cdots\!24}{25\!\cdots\!27}a^{3}+\frac{48\!\cdots\!00}{25\!\cdots\!27}a^{2}+\frac{56\!\cdots\!72}{25\!\cdots\!27}a+\frac{43\!\cdots\!24}{25\!\cdots\!27}$, $\frac{21\!\cdots\!88}{25\!\cdots\!27}a^{25}-\frac{17\!\cdots\!76}{25\!\cdots\!27}a^{24}-\frac{16\!\cdots\!56}{25\!\cdots\!27}a^{23}+\frac{39\!\cdots\!29}{25\!\cdots\!27}a^{22}+\frac{24\!\cdots\!02}{25\!\cdots\!27}a^{21}-\frac{55\!\cdots\!91}{25\!\cdots\!27}a^{20}+\frac{50\!\cdots\!72}{25\!\cdots\!27}a^{19}+\frac{32\!\cdots\!71}{25\!\cdots\!27}a^{18}-\frac{10\!\cdots\!42}{10\!\cdots\!49}a^{17}-\frac{20\!\cdots\!83}{25\!\cdots\!27}a^{16}+\frac{25\!\cdots\!08}{25\!\cdots\!27}a^{15}-\frac{57\!\cdots\!89}{25\!\cdots\!27}a^{14}+\frac{68\!\cdots\!32}{25\!\cdots\!27}a^{13}-\frac{47\!\cdots\!63}{25\!\cdots\!27}a^{12}+\frac{90\!\cdots\!92}{25\!\cdots\!27}a^{11}-\frac{22\!\cdots\!92}{25\!\cdots\!27}a^{10}+\frac{37\!\cdots\!54}{25\!\cdots\!27}a^{9}-\frac{11\!\cdots\!10}{25\!\cdots\!27}a^{8}+\frac{15\!\cdots\!20}{25\!\cdots\!27}a^{7}-\frac{43\!\cdots\!29}{25\!\cdots\!27}a^{6}+\frac{39\!\cdots\!64}{25\!\cdots\!27}a^{5}-\frac{11\!\cdots\!84}{25\!\cdots\!27}a^{4}+\frac{79\!\cdots\!86}{25\!\cdots\!27}a^{3}-\frac{20\!\cdots\!25}{25\!\cdots\!27}a^{2}+\frac{83\!\cdots\!98}{25\!\cdots\!27}a-\frac{14\!\cdots\!24}{25\!\cdots\!27}$, $\frac{82\!\cdots\!56}{25\!\cdots\!27}a^{25}-\frac{14\!\cdots\!72}{25\!\cdots\!27}a^{24}+\frac{19\!\cdots\!68}{25\!\cdots\!27}a^{23}+\frac{28\!\cdots\!39}{25\!\cdots\!27}a^{22}-\frac{52\!\cdots\!02}{25\!\cdots\!27}a^{21}-\frac{37\!\cdots\!10}{25\!\cdots\!27}a^{20}+\frac{89\!\cdots\!18}{25\!\cdots\!27}a^{19}+\frac{14\!\cdots\!70}{25\!\cdots\!27}a^{18}-\frac{42\!\cdots\!48}{25\!\cdots\!27}a^{17}-\frac{96\!\cdots\!45}{25\!\cdots\!27}a^{16}+\frac{25\!\cdots\!96}{25\!\cdots\!27}a^{15}-\frac{43\!\cdots\!99}{25\!\cdots\!27}a^{14}+\frac{76\!\cdots\!20}{25\!\cdots\!27}a^{13}-\frac{36\!\cdots\!54}{25\!\cdots\!27}a^{12}+\frac{78\!\cdots\!48}{25\!\cdots\!27}a^{11}-\frac{27\!\cdots\!55}{25\!\cdots\!27}a^{10}+\frac{44\!\cdots\!52}{25\!\cdots\!27}a^{9}-\frac{13\!\cdots\!23}{25\!\cdots\!27}a^{8}+\frac{16\!\cdots\!34}{25\!\cdots\!27}a^{7}-\frac{51\!\cdots\!16}{25\!\cdots\!27}a^{6}+\frac{50\!\cdots\!82}{25\!\cdots\!27}a^{5}-\frac{14\!\cdots\!94}{25\!\cdots\!27}a^{4}+\frac{11\!\cdots\!18}{25\!\cdots\!27}a^{3}-\frac{26\!\cdots\!59}{25\!\cdots\!27}a^{2}+\frac{10\!\cdots\!48}{25\!\cdots\!27}a-\frac{19\!\cdots\!12}{25\!\cdots\!27}$, $\frac{62\!\cdots\!90}{25\!\cdots\!27}a^{25}-\frac{10\!\cdots\!80}{25\!\cdots\!27}a^{24}-\frac{14\!\cdots\!74}{25\!\cdots\!27}a^{23}+\frac{20\!\cdots\!72}{25\!\cdots\!27}a^{22}+\frac{21\!\cdots\!86}{25\!\cdots\!27}a^{21}-\frac{25\!\cdots\!04}{25\!\cdots\!27}a^{20}-\frac{13\!\cdots\!96}{25\!\cdots\!27}a^{19}+\frac{12\!\cdots\!20}{25\!\cdots\!27}a^{18}+\frac{84\!\cdots\!90}{25\!\cdots\!27}a^{17}-\frac{45\!\cdots\!67}{25\!\cdots\!27}a^{16}+\frac{19\!\cdots\!80}{25\!\cdots\!27}a^{15}-\frac{83\!\cdots\!00}{25\!\cdots\!27}a^{14}+\frac{15\!\cdots\!34}{25\!\cdots\!27}a^{13}-\frac{14\!\cdots\!41}{25\!\cdots\!27}a^{12}+\frac{10\!\cdots\!58}{25\!\cdots\!27}a^{11}-\frac{37\!\cdots\!36}{25\!\cdots\!27}a^{10}+\frac{48\!\cdots\!26}{25\!\cdots\!27}a^{9}-\frac{34\!\cdots\!60}{25\!\cdots\!27}a^{8}+\frac{21\!\cdots\!96}{25\!\cdots\!27}a^{7}+\frac{13\!\cdots\!67}{25\!\cdots\!27}a^{6}+\frac{57\!\cdots\!84}{25\!\cdots\!27}a^{5}+\frac{99\!\cdots\!61}{25\!\cdots\!27}a^{4}+\frac{11\!\cdots\!84}{25\!\cdots\!27}a^{3}+\frac{12\!\cdots\!44}{25\!\cdots\!27}a^{2}+\frac{10\!\cdots\!98}{25\!\cdots\!27}a-\frac{77\!\cdots\!91}{25\!\cdots\!27}$, $\frac{12\!\cdots\!88}{25\!\cdots\!27}a^{25}-\frac{14\!\cdots\!74}{25\!\cdots\!27}a^{24}-\frac{29\!\cdots\!30}{25\!\cdots\!27}a^{23}+\frac{24\!\cdots\!09}{25\!\cdots\!27}a^{22}+\frac{43\!\cdots\!82}{25\!\cdots\!27}a^{21}-\frac{24\!\cdots\!37}{25\!\cdots\!27}a^{20}-\frac{26\!\cdots\!68}{25\!\cdots\!27}a^{19}+\frac{51\!\cdots\!50}{25\!\cdots\!27}a^{18}+\frac{16\!\cdots\!40}{25\!\cdots\!27}a^{17}+\frac{31\!\cdots\!38}{25\!\cdots\!27}a^{16}+\frac{53\!\cdots\!80}{25\!\cdots\!27}a^{15}-\frac{37\!\cdots\!17}{25\!\cdots\!27}a^{14}+\frac{32\!\cdots\!34}{25\!\cdots\!27}a^{13}-\frac{13\!\cdots\!45}{25\!\cdots\!27}a^{12}+\frac{19\!\cdots\!86}{25\!\cdots\!27}a^{11}-\frac{35\!\cdots\!67}{25\!\cdots\!27}a^{10}+\frac{10\!\cdots\!54}{25\!\cdots\!27}a^{9}+\frac{22\!\cdots\!80}{25\!\cdots\!27}a^{8}+\frac{48\!\cdots\!48}{25\!\cdots\!27}a^{7}+\frac{15\!\cdots\!96}{25\!\cdots\!27}a^{6}+\frac{13\!\cdots\!94}{25\!\cdots\!27}a^{5}+\frac{49\!\cdots\!65}{25\!\cdots\!27}a^{4}+\frac{28\!\cdots\!20}{25\!\cdots\!27}a^{3}+\frac{84\!\cdots\!48}{25\!\cdots\!27}a^{2}+\frac{26\!\cdots\!00}{25\!\cdots\!27}a+\frac{42\!\cdots\!63}{25\!\cdots\!27}$, $\frac{59\!\cdots\!46}{25\!\cdots\!27}a^{25}-\frac{26\!\cdots\!85}{25\!\cdots\!27}a^{24}-\frac{10\!\cdots\!62}{25\!\cdots\!27}a^{23}+\frac{57\!\cdots\!12}{25\!\cdots\!27}a^{22}+\frac{13\!\cdots\!56}{25\!\cdots\!27}a^{21}-\frac{78\!\cdots\!04}{25\!\cdots\!27}a^{20}-\frac{38\!\cdots\!38}{25\!\cdots\!27}a^{19}+\frac{40\!\cdots\!00}{25\!\cdots\!27}a^{18}+\frac{15\!\cdots\!50}{10\!\cdots\!49}a^{17}-\frac{22\!\cdots\!47}{25\!\cdots\!27}a^{16}+\frac{19\!\cdots\!36}{25\!\cdots\!27}a^{15}-\frac{36\!\cdots\!97}{25\!\cdots\!27}a^{14}+\frac{17\!\cdots\!62}{25\!\cdots\!27}a^{13}-\frac{56\!\cdots\!79}{25\!\cdots\!27}a^{12}+\frac{14\!\cdots\!78}{25\!\cdots\!27}a^{11}-\frac{31\!\cdots\!39}{25\!\cdots\!27}a^{10}+\frac{64\!\cdots\!64}{25\!\cdots\!27}a^{9}-\frac{15\!\cdots\!90}{25\!\cdots\!27}a^{8}+\frac{25\!\cdots\!50}{25\!\cdots\!27}a^{7}-\frac{56\!\cdots\!15}{25\!\cdots\!27}a^{6}+\frac{61\!\cdots\!62}{25\!\cdots\!27}a^{5}-\frac{15\!\cdots\!21}{25\!\cdots\!27}a^{4}+\frac{10\!\cdots\!00}{25\!\cdots\!27}a^{3}-\frac{27\!\cdots\!90}{25\!\cdots\!27}a^{2}+\frac{10\!\cdots\!90}{25\!\cdots\!27}a-\frac{25\!\cdots\!55}{25\!\cdots\!27}$, $\frac{71\!\cdots\!20}{25\!\cdots\!27}a^{25}-\frac{46\!\cdots\!60}{25\!\cdots\!27}a^{24}-\frac{17\!\cdots\!06}{25\!\cdots\!27}a^{23}+\frac{53\!\cdots\!22}{25\!\cdots\!27}a^{22}+\frac{27\!\cdots\!14}{25\!\cdots\!27}a^{21}-\frac{33\!\cdots\!56}{25\!\cdots\!27}a^{20}-\frac{19\!\cdots\!00}{25\!\cdots\!27}a^{19}-\frac{71\!\cdots\!76}{25\!\cdots\!27}a^{18}+\frac{12\!\cdots\!46}{25\!\cdots\!27}a^{17}+\frac{93\!\cdots\!73}{25\!\cdots\!27}a^{16}-\frac{10\!\cdots\!38}{25\!\cdots\!27}a^{15}-\frac{42\!\cdots\!41}{25\!\cdots\!27}a^{14}+\frac{19\!\cdots\!70}{25\!\cdots\!27}a^{13}+\frac{72\!\cdots\!59}{25\!\cdots\!27}a^{12}+\frac{93\!\cdots\!70}{25\!\cdots\!27}a^{11}+\frac{14\!\cdots\!30}{25\!\cdots\!27}a^{10}+\frac{53\!\cdots\!80}{25\!\cdots\!27}a^{9}+\frac{31\!\cdots\!22}{25\!\cdots\!27}a^{8}+\frac{23\!\cdots\!64}{25\!\cdots\!27}a^{7}+\frac{15\!\cdots\!61}{25\!\cdots\!27}a^{6}+\frac{64\!\cdots\!64}{25\!\cdots\!27}a^{5}+\frac{34\!\cdots\!06}{25\!\cdots\!27}a^{4}+\frac{12\!\cdots\!16}{25\!\cdots\!27}a^{3}+\frac{58\!\cdots\!53}{25\!\cdots\!27}a^{2}+\frac{10\!\cdots\!06}{25\!\cdots\!27}a+\frac{25\!\cdots\!01}{25\!\cdots\!27}$, $\frac{95\!\cdots\!88}{47\!\cdots\!63}a^{25}-\frac{18\!\cdots\!46}{47\!\cdots\!63}a^{24}-\frac{19\!\cdots\!96}{47\!\cdots\!63}a^{23}+\frac{34\!\cdots\!29}{47\!\cdots\!63}a^{22}+\frac{27\!\cdots\!32}{47\!\cdots\!63}a^{21}-\frac{42\!\cdots\!66}{47\!\cdots\!63}a^{20}-\frac{12\!\cdots\!98}{47\!\cdots\!63}a^{19}+\frac{16\!\cdots\!31}{47\!\cdots\!63}a^{18}+\frac{87\!\cdots\!54}{47\!\cdots\!63}a^{17}-\frac{70\!\cdots\!28}{47\!\cdots\!63}a^{16}+\frac{27\!\cdots\!88}{47\!\cdots\!63}a^{15}-\frac{28\!\cdots\!94}{47\!\cdots\!63}a^{14}+\frac{28\!\cdots\!92}{47\!\cdots\!63}a^{13}-\frac{36\!\cdots\!23}{47\!\cdots\!63}a^{12}+\frac{21\!\cdots\!72}{47\!\cdots\!63}a^{11}-\frac{16\!\cdots\!17}{47\!\cdots\!63}a^{10}+\frac{11\!\cdots\!24}{47\!\cdots\!63}a^{9}-\frac{55\!\cdots\!05}{47\!\cdots\!63}a^{8}+\frac{53\!\cdots\!50}{47\!\cdots\!63}a^{7}-\frac{14\!\cdots\!49}{47\!\cdots\!63}a^{6}+\frac{17\!\cdots\!66}{47\!\cdots\!63}a^{5}-\frac{25\!\cdots\!44}{47\!\cdots\!63}a^{4}+\frac{44\!\cdots\!86}{47\!\cdots\!63}a^{3}-\frac{39\!\cdots\!15}{47\!\cdots\!63}a^{2}+\frac{46\!\cdots\!10}{47\!\cdots\!63}a-\frac{37\!\cdots\!54}{47\!\cdots\!63}$, $\frac{54\!\cdots\!04}{25\!\cdots\!27}a^{25}-\frac{39\!\cdots\!96}{25\!\cdots\!27}a^{24}-\frac{66\!\cdots\!28}{25\!\cdots\!27}a^{23}+\frac{88\!\cdots\!74}{25\!\cdots\!27}a^{22}+\frac{60\!\cdots\!80}{25\!\cdots\!27}a^{21}-\frac{12\!\cdots\!62}{25\!\cdots\!27}a^{20}+\frac{52\!\cdots\!16}{25\!\cdots\!27}a^{19}+\frac{72\!\cdots\!78}{25\!\cdots\!27}a^{18}-\frac{15\!\cdots\!00}{25\!\cdots\!27}a^{17}-\frac{45\!\cdots\!26}{25\!\cdots\!27}a^{16}+\frac{41\!\cdots\!36}{25\!\cdots\!27}a^{15}-\frac{66\!\cdots\!10}{25\!\cdots\!27}a^{14}+\frac{16\!\cdots\!00}{25\!\cdots\!27}a^{13}-\frac{10\!\cdots\!74}{25\!\cdots\!27}a^{12}+\frac{19\!\cdots\!76}{25\!\cdots\!27}a^{11}-\frac{50\!\cdots\!08}{25\!\cdots\!27}a^{10}+\frac{76\!\cdots\!76}{25\!\cdots\!27}a^{9}-\frac{26\!\cdots\!76}{25\!\cdots\!27}a^{8}+\frac{30\!\cdots\!72}{25\!\cdots\!27}a^{7}-\frac{10\!\cdots\!18}{25\!\cdots\!27}a^{6}+\frac{75\!\cdots\!88}{25\!\cdots\!27}a^{5}-\frac{28\!\cdots\!25}{25\!\cdots\!27}a^{4}+\frac{14\!\cdots\!96}{25\!\cdots\!27}a^{3}-\frac{53\!\cdots\!66}{25\!\cdots\!27}a^{2}+\frac{15\!\cdots\!24}{25\!\cdots\!27}a-\frac{48\!\cdots\!68}{25\!\cdots\!27}$, $\frac{13\!\cdots\!36}{25\!\cdots\!27}a^{25}-\frac{27\!\cdots\!16}{25\!\cdots\!27}a^{24}+\frac{13\!\cdots\!60}{25\!\cdots\!27}a^{23}+\frac{27\!\cdots\!94}{10\!\cdots\!49}a^{22}-\frac{46\!\cdots\!02}{25\!\cdots\!27}a^{21}-\frac{94\!\cdots\!35}{25\!\cdots\!27}a^{20}+\frac{89\!\cdots\!84}{25\!\cdots\!27}a^{19}+\frac{57\!\cdots\!47}{25\!\cdots\!27}a^{18}-\frac{44\!\cdots\!40}{25\!\cdots\!27}a^{17}-\frac{37\!\cdots\!97}{25\!\cdots\!27}a^{16}+\frac{27\!\cdots\!82}{25\!\cdots\!27}a^{15}+\frac{98\!\cdots\!75}{25\!\cdots\!27}a^{14}+\frac{42\!\cdots\!72}{25\!\cdots\!27}a^{13}-\frac{78\!\cdots\!70}{25\!\cdots\!27}a^{12}+\frac{99\!\cdots\!32}{25\!\cdots\!27}a^{11}-\frac{37\!\cdots\!27}{25\!\cdots\!27}a^{10}+\frac{27\!\cdots\!56}{25\!\cdots\!27}a^{9}-\frac{21\!\cdots\!41}{25\!\cdots\!27}a^{8}+\frac{97\!\cdots\!96}{25\!\cdots\!27}a^{7}-\frac{86\!\cdots\!28}{25\!\cdots\!27}a^{6}+\frac{60\!\cdots\!78}{10\!\cdots\!49}a^{5}-\frac{24\!\cdots\!83}{25\!\cdots\!27}a^{4}+\frac{96\!\cdots\!18}{25\!\cdots\!27}a^{3}-\frac{46\!\cdots\!34}{25\!\cdots\!27}a^{2}+\frac{20\!\cdots\!60}{25\!\cdots\!27}a-\frac{43\!\cdots\!74}{25\!\cdots\!27}$, $\frac{73\!\cdots\!24}{25\!\cdots\!27}a^{25}+\frac{14\!\cdots\!25}{25\!\cdots\!27}a^{24}-\frac{21\!\cdots\!60}{25\!\cdots\!27}a^{23}-\frac{39\!\cdots\!09}{25\!\cdots\!27}a^{22}+\frac{34\!\cdots\!22}{25\!\cdots\!27}a^{21}+\frac{66\!\cdots\!35}{25\!\cdots\!27}a^{20}-\frac{26\!\cdots\!40}{25\!\cdots\!27}a^{19}-\frac{46\!\cdots\!71}{25\!\cdots\!27}a^{18}+\frac{15\!\cdots\!42}{25\!\cdots\!27}a^{17}+\frac{35\!\cdots\!69}{25\!\cdots\!27}a^{16}-\frac{20\!\cdots\!28}{25\!\cdots\!27}a^{15}-\frac{30\!\cdots\!33}{25\!\cdots\!27}a^{14}+\frac{17\!\cdots\!44}{25\!\cdots\!27}a^{13}+\frac{61\!\cdots\!36}{25\!\cdots\!27}a^{12}+\frac{42\!\cdots\!96}{25\!\cdots\!27}a^{11}+\frac{32\!\cdots\!76}{25\!\cdots\!27}a^{10}+\frac{46\!\cdots\!14}{25\!\cdots\!27}a^{9}+\frac{21\!\cdots\!87}{25\!\cdots\!27}a^{8}+\frac{22\!\cdots\!28}{25\!\cdots\!27}a^{7}+\frac{39\!\cdots\!57}{10\!\cdots\!49}a^{6}+\frac{74\!\cdots\!08}{25\!\cdots\!27}a^{5}+\frac{25\!\cdots\!15}{25\!\cdots\!27}a^{4}+\frac{71\!\cdots\!94}{10\!\cdots\!49}a^{3}+\frac{47\!\cdots\!47}{25\!\cdots\!27}a^{2}+\frac{14\!\cdots\!38}{25\!\cdots\!27}a+\frac{41\!\cdots\!88}{25\!\cdots\!27}$
| 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: | \( 5382739421.971964 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 5382739421.971964 \cdot 1064051}{2\cdot\sqrt{132675490325051365459874809737094631268103216670179328}}\cr\approx \mathstrut & 0.187015990900233 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 21*x^24 + 38*x^23 + 294*x^22 - 464*x^21 - 1357*x^20 + 1930*x^19 + 9504*x^18 - 8228*x^17 + 25716*x^16 - 28630*x^15 + 286963*x^14 - 378044*x^13 + 2103312*x^12 - 1678782*x^11 + 11411452*x^10 - 5574160*x^9 + 50609745*x^8 - 14301644*x^7 + 160353701*x^6 - 26227056*x^5 + 361035384*x^4 - 44819066*x^3 + 515469360*x^2 - 43859864*x + 329025401)
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 - 2*x^25 - 21*x^24 + 38*x^23 + 294*x^22 - 464*x^21 - 1357*x^20 + 1930*x^19 + 9504*x^18 - 8228*x^17 + 25716*x^16 - 28630*x^15 + 286963*x^14 - 378044*x^13 + 2103312*x^12 - 1678782*x^11 + 11411452*x^10 - 5574160*x^9 + 50609745*x^8 - 14301644*x^7 + 160353701*x^6 - 26227056*x^5 + 361035384*x^4 - 44819066*x^3 + 515469360*x^2 - 43859864*x + 329025401, 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 - 2*x^25 - 21*x^24 + 38*x^23 + 294*x^22 - 464*x^21 - 1357*x^20 + 1930*x^19 + 9504*x^18 - 8228*x^17 + 25716*x^16 - 28630*x^15 + 286963*x^14 - 378044*x^13 + 2103312*x^12 - 1678782*x^11 + 11411452*x^10 - 5574160*x^9 + 50609745*x^8 - 14301644*x^7 + 160353701*x^6 - 26227056*x^5 + 361035384*x^4 - 44819066*x^3 + 515469360*x^2 - 43859864*x + 329025401);
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 - 2*x^25 - 21*x^24 + 38*x^23 + 294*x^22 - 464*x^21 - 1357*x^20 + 1930*x^19 + 9504*x^18 - 8228*x^17 + 25716*x^16 - 28630*x^15 + 286963*x^14 - 378044*x^13 + 2103312*x^12 - 1678782*x^11 + 11411452*x^10 - 5574160*x^9 + 50609745*x^8 - 14301644*x^7 + 160353701*x^6 - 26227056*x^5 + 361035384*x^4 - 44819066*x^3 + 515469360*x^2 - 43859864*x + 329025401);
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 cyclic group of order 26 |
| The 26 conjugacy class representatives for $C_{26}$ |
| Character table for $C_{26}$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 13.13.491258904256726154641.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/padicField/11.13.0.1}{13} }^{2}$ | $26$ | ${\href{/padicField/17.13.0.1}{13} }^{2}$ | ${\href{/padicField/19.13.0.1}{13} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{13}$ | $26$ | $26$ | $26$ | ${\href{/padicField/41.13.0.1}{13} }^{2}$ | ${\href{/padicField/43.13.0.1}{13} }^{2}$ | $26$ | R | ${\href{/padicField/59.13.0.1}{13} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $26$ | $2$ | $13$ | $39$ | |||
|
\(53\)
| Deg $26$ | $13$ | $2$ | $24$ |