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Properties

Label 36.0.758...144.1
Degree $36$
Signature $[0, 18]$
Discriminant $7.588\times 10^{66}$
Root discriminant \(72.07\)
Ramified primes $2,7,19$
Class number not computed
Class group not computed
Galois group $C_6^2$ (as 36T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 13*x^34 + 119*x^32 - 946*x^30 + 6985*x^28 - 49336*x^26 + 338472*x^24 - 1782270*x^22 + 8663374*x^20 - 40095490*x^18 + 175403900*x^16 - 699981526*x^14 + 2294243848*x^12 - 3420884824*x^10 + 5086203969*x^8 - 7494358949*x^6 + 10739824263*x^4 - 14123762450*x^2 + 13841287201)
 
gp: K = bnfinit(y^36 - 13*y^34 + 119*y^32 - 946*y^30 + 6985*y^28 - 49336*y^26 + 338472*y^24 - 1782270*y^22 + 8663374*y^20 - 40095490*y^18 + 175403900*y^16 - 699981526*y^14 + 2294243848*y^12 - 3420884824*y^10 + 5086203969*y^8 - 7494358949*y^6 + 10739824263*y^4 - 14123762450*y^2 + 13841287201, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 13*x^34 + 119*x^32 - 946*x^30 + 6985*x^28 - 49336*x^26 + 338472*x^24 - 1782270*x^22 + 8663374*x^20 - 40095490*x^18 + 175403900*x^16 - 699981526*x^14 + 2294243848*x^12 - 3420884824*x^10 + 5086203969*x^8 - 7494358949*x^6 + 10739824263*x^4 - 14123762450*x^2 + 13841287201);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 13*x^34 + 119*x^32 - 946*x^30 + 6985*x^28 - 49336*x^26 + 338472*x^24 - 1782270*x^22 + 8663374*x^20 - 40095490*x^18 + 175403900*x^16 - 699981526*x^14 + 2294243848*x^12 - 3420884824*x^10 + 5086203969*x^8 - 7494358949*x^6 + 10739824263*x^4 - 14123762450*x^2 + 13841287201)
 

\( x^{36} - 13 x^{34} + 119 x^{32} - 946 x^{30} + 6985 x^{28} - 49336 x^{26} + 338472 x^{24} + \cdots + 13841287201 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 18]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(7587653100749080800127745255338750713360512873262289573077418246144\) \(\medspace = 2^{36}\cdot 7^{30}\cdot 19^{24}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(72.07\)
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:  $2\cdot 7^{5/6}19^{2/3}\approx 72.07435474130278$
Ramified primes:   \(2\), \(7\), \(19\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(532=2^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{532}(1,·)$, $\chi_{532}(387,·)$, $\chi_{532}(391,·)$, $\chi_{532}(11,·)$, $\chi_{532}(45,·)$, $\chi_{532}(121,·)$, $\chi_{532}(277,·)$, $\chi_{532}(153,·)$, $\chi_{532}(201,·)$, $\chi_{532}(239,·)$, $\chi_{532}(159,·)$, $\chi_{532}(163,·)$, $\chi_{532}(39,·)$, $\chi_{532}(425,·)$, $\chi_{532}(429,·)$, $\chi_{532}(125,·)$, $\chi_{532}(305,·)$, $\chi_{532}(115,·)$, $\chi_{532}(311,·)$, $\chi_{532}(191,·)$, $\chi_{532}(267,·)$, $\chi_{532}(197,·)$, $\chi_{532}(457,·)$, $\chi_{532}(463,·)$, $\chi_{532}(83,·)$, $\chi_{532}(87,·)$, $\chi_{532}(349,·)$, $\chi_{532}(353,·)$, $\chi_{532}(419,·)$, $\chi_{532}(229,·)$, $\chi_{532}(235,·)$, $\chi_{532}(495,·)$, $\chi_{532}(467,·)$, $\chi_{532}(501,·)$, $\chi_{532}(505,·)$, $\chi_{532}(381,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{8}a^{14}+\frac{1}{8}$, $\frac{1}{8}a^{15}+\frac{1}{8}a$, $\frac{1}{8}a^{16}+\frac{1}{8}a^{2}$, $\frac{1}{8}a^{17}+\frac{1}{8}a^{3}$, $\frac{1}{8}a^{18}+\frac{1}{8}a^{4}$, $\frac{1}{8}a^{19}+\frac{1}{8}a^{5}$, $\frac{1}{8}a^{20}+\frac{1}{8}a^{6}$, $\frac{1}{8}a^{21}+\frac{1}{8}a^{7}$, $\frac{1}{8}a^{22}+\frac{1}{8}a^{8}$, $\frac{1}{8}a^{23}+\frac{1}{8}a^{9}$, $\frac{1}{8}a^{24}+\frac{1}{8}a^{10}$, $\frac{1}{56}a^{25}+\frac{1}{56}a^{23}-\frac{1}{56}a^{19}-\frac{1}{56}a^{17}+\frac{1}{7}a^{13}-\frac{1}{8}a^{11}-\frac{15}{56}a^{9}-\frac{1}{7}a^{7}+\frac{1}{8}a^{5}+\frac{15}{56}a^{3}+\frac{1}{7}a$, $\frac{1}{465319718314472}a^{26}-\frac{3243115003540}{58164964789309}a^{24}-\frac{2505762039743}{66474245473496}a^{22}+\frac{6490631255265}{232659859157236}a^{20}+\frac{976289912167}{232659859157236}a^{18}+\frac{188121821805}{8309280684187}a^{16}+\frac{1810931332925}{465319718314472}a^{14}-\frac{16463970362265}{66474245473496}a^{12}+\frac{5935032075111}{58164964789309}a^{10}+\frac{102997627485423}{465319718314472}a^{8}-\frac{3103496852465}{33237122736748}a^{6}-\frac{24055837713785}{232659859157236}a^{4}-\frac{12110059037173}{58164964789309}a^{2}+\frac{3102595254541}{9496320781928}$, $\frac{1}{32\!\cdots\!04}a^{27}-\frac{3243115003540}{407154753525163}a^{25}+\frac{1450879661111}{116329929578618}a^{23}-\frac{45183702278779}{32\!\cdots\!04}a^{21}-\frac{172542314543593}{32\!\cdots\!04}a^{19}+\frac{9814255258627}{465319718314472}a^{17}+\frac{44076456425213}{814309507050326}a^{15}-\frac{82938215835761}{465319718314472}a^{13}+\frac{180429926443038}{407154753525163}a^{11}+\frac{272950507225919}{814309507050326}a^{9}-\frac{14516274389117}{465319718314472}a^{7}-\frac{222606569795497}{32\!\cdots\!04}a^{5}+\frac{891923929120869}{32\!\cdots\!04}a^{3}+\frac{6414089277905}{16618561368374}a$, $\frac{1}{18\!\cdots\!24}a^{28}-\frac{5}{57\!\cdots\!82}a^{26}+\frac{50243014059881}{814309507050326}a^{24}-\frac{198792879690063}{22\!\cdots\!28}a^{22}+\frac{646086344365783}{22\!\cdots\!28}a^{20}+\frac{144157829974089}{32\!\cdots\!04}a^{18}+\frac{13\!\cdots\!71}{22\!\cdots\!28}a^{16}-\frac{211887109294689}{13\!\cdots\!16}a^{14}-\frac{606052974670761}{57\!\cdots\!82}a^{12}+\frac{16\!\cdots\!83}{57\!\cdots\!82}a^{10}-\frac{18716055839905}{32\!\cdots\!04}a^{8}-\frac{73\!\cdots\!37}{22\!\cdots\!28}a^{6}+\frac{41\!\cdots\!43}{22\!\cdots\!28}a^{4}-\frac{130736640838241}{465319718314472}a^{2}+\frac{27909818601233}{75970566255424}$, $\frac{1}{12\!\cdots\!68}a^{29}-\frac{5}{39\!\cdots\!74}a^{27}+\frac{50243014059881}{57\!\cdots\!82}a^{25}+\frac{10\!\cdots\!95}{19\!\cdots\!87}a^{23}+\frac{646086344365783}{15\!\cdots\!96}a^{21}+\frac{137828145874813}{57\!\cdots\!82}a^{19}+\frac{70\!\cdots\!53}{15\!\cdots\!96}a^{17}-\frac{34\!\cdots\!93}{91\!\cdots\!12}a^{15}+\frac{16\!\cdots\!85}{39\!\cdots\!74}a^{13}+\frac{13\!\cdots\!47}{39\!\cdots\!74}a^{11}+\frac{150343525591948}{28\!\cdots\!41}a^{9}-\frac{30\!\cdots\!65}{15\!\cdots\!96}a^{7}-\frac{39\!\cdots\!11}{39\!\cdots\!74}a^{5}+\frac{916232725369321}{32\!\cdots\!04}a^{3}+\frac{84887743292801}{531793963787968}a$, $\frac{1}{89\!\cdots\!76}a^{30}-\frac{13}{89\!\cdots\!76}a^{28}+\frac{45}{15\!\cdots\!96}a^{26}+\frac{57\!\cdots\!59}{11\!\cdots\!72}a^{24}+\frac{47\!\cdots\!47}{11\!\cdots\!72}a^{22}+\frac{248224465313147}{15\!\cdots\!96}a^{20}-\frac{41\!\cdots\!85}{11\!\cdots\!72}a^{18}+\frac{70\!\cdots\!39}{63\!\cdots\!84}a^{16}-\frac{15\!\cdots\!01}{44\!\cdots\!88}a^{14}-\frac{45\!\cdots\!85}{11\!\cdots\!72}a^{12}-\frac{53\!\cdots\!55}{15\!\cdots\!96}a^{10}-\frac{23\!\cdots\!69}{11\!\cdots\!72}a^{8}-\frac{43\!\cdots\!59}{11\!\cdots\!72}a^{6}-\frac{45\!\cdots\!01}{22\!\cdots\!28}a^{4}-\frac{818778848072407}{37\!\cdots\!76}a^{2}+\frac{28072101076347}{75970566255424}$, $\frac{1}{62\!\cdots\!32}a^{31}-\frac{13}{62\!\cdots\!32}a^{29}+\frac{45}{11\!\cdots\!72}a^{27}+\frac{57\!\cdots\!59}{78\!\cdots\!04}a^{25}-\frac{23\!\cdots\!71}{78\!\cdots\!04}a^{23}+\frac{10\!\cdots\!67}{55\!\cdots\!36}a^{21}+\frac{12\!\cdots\!03}{97\!\cdots\!63}a^{19}-\frac{23\!\cdots\!05}{44\!\cdots\!88}a^{17}+\frac{11\!\cdots\!71}{31\!\cdots\!16}a^{15}+\frac{66\!\cdots\!87}{78\!\cdots\!04}a^{13}+\frac{26\!\cdots\!37}{11\!\cdots\!72}a^{11}+\frac{17\!\cdots\!57}{78\!\cdots\!04}a^{9}+\frac{41\!\cdots\!61}{39\!\cdots\!52}a^{7}+\frac{26\!\cdots\!21}{19\!\cdots\!87}a^{5}-\frac{59\!\cdots\!99}{26\!\cdots\!32}a^{3}+\frac{47064742640203}{531793963787968}a$, $\frac{1}{43\!\cdots\!24}a^{32}-\frac{13}{43\!\cdots\!24}a^{30}+\frac{17}{62\!\cdots\!32}a^{28}+\frac{14647}{27\!\cdots\!64}a^{26}+\frac{19\!\cdots\!49}{54\!\cdots\!28}a^{24}+\frac{23\!\cdots\!33}{97\!\cdots\!63}a^{22}+\frac{29\!\cdots\!17}{54\!\cdots\!28}a^{20}+\frac{97\!\cdots\!11}{31\!\cdots\!16}a^{18}-\frac{12\!\cdots\!33}{21\!\cdots\!12}a^{16}+\frac{17\!\cdots\!39}{21\!\cdots\!12}a^{14}-\frac{15\!\cdots\!99}{39\!\cdots\!52}a^{12}+\frac{30\!\cdots\!13}{54\!\cdots\!28}a^{10}-\frac{33\!\cdots\!27}{68\!\cdots\!41}a^{8}-\frac{36\!\cdots\!19}{11\!\cdots\!72}a^{6}-\frac{42\!\cdots\!95}{18\!\cdots\!24}a^{4}+\frac{12\!\cdots\!19}{37\!\cdots\!76}a^{2}-\frac{10699939378865}{75970566255424}$, $\frac{1}{30\!\cdots\!68}a^{33}-\frac{13}{30\!\cdots\!68}a^{31}+\frac{17}{43\!\cdots\!24}a^{29}+\frac{14647}{19\!\cdots\!48}a^{27}+\frac{19\!\cdots\!49}{38\!\cdots\!96}a^{25}+\frac{10\!\cdots\!95}{27\!\cdots\!64}a^{23}-\frac{48\!\cdots\!78}{47\!\cdots\!87}a^{21}+\frac{12\!\cdots\!67}{21\!\cdots\!12}a^{19}-\frac{66\!\cdots\!61}{15\!\cdots\!84}a^{17}+\frac{56\!\cdots\!67}{15\!\cdots\!84}a^{15}+\frac{63\!\cdots\!05}{27\!\cdots\!64}a^{13}+\frac{16\!\cdots\!97}{38\!\cdots\!96}a^{11}+\frac{75\!\cdots\!25}{19\!\cdots\!48}a^{9}-\frac{48\!\cdots\!43}{97\!\cdots\!63}a^{7}-\frac{15\!\cdots\!35}{12\!\cdots\!68}a^{5}+\frac{275447714267475}{26\!\cdots\!32}a^{3}+\frac{1184671740713}{75970566255424}a$, $\frac{1}{21\!\cdots\!76}a^{34}-\frac{13}{21\!\cdots\!76}a^{32}+\frac{17}{30\!\cdots\!68}a^{30}-\frac{473}{10\!\cdots\!88}a^{28}-\frac{719727}{26\!\cdots\!72}a^{26}+\frac{16\!\cdots\!59}{38\!\cdots\!96}a^{24}-\frac{11\!\cdots\!67}{26\!\cdots\!72}a^{22}+\frac{23\!\cdots\!23}{15\!\cdots\!84}a^{20}+\frac{36\!\cdots\!95}{10\!\cdots\!88}a^{18}+\frac{38\!\cdots\!95}{10\!\cdots\!88}a^{16}-\frac{36\!\cdots\!89}{76\!\cdots\!92}a^{14}-\frac{67\!\cdots\!55}{26\!\cdots\!72}a^{12}+\frac{10\!\cdots\!93}{26\!\cdots\!72}a^{10}+\frac{21\!\cdots\!25}{54\!\cdots\!28}a^{8}+\frac{35\!\cdots\!73}{89\!\cdots\!76}a^{6}-\frac{50\!\cdots\!17}{18\!\cdots\!24}a^{4}+\frac{16409972761735}{75970566255424}a^{2}-\frac{907430289301}{37985283127712}$, $\frac{1}{15\!\cdots\!32}a^{35}-\frac{13}{15\!\cdots\!32}a^{33}+\frac{17}{21\!\cdots\!76}a^{31}-\frac{473}{75\!\cdots\!16}a^{29}-\frac{719727}{18\!\cdots\!04}a^{27}+\frac{16\!\cdots\!59}{26\!\cdots\!72}a^{25}-\frac{11\!\cdots\!67}{18\!\cdots\!04}a^{23}-\frac{16\!\cdots\!25}{10\!\cdots\!88}a^{21}-\frac{23\!\cdots\!77}{75\!\cdots\!16}a^{19}-\frac{95\!\cdots\!41}{75\!\cdots\!16}a^{17}-\frac{13\!\cdots\!63}{53\!\cdots\!44}a^{15}+\frac{53\!\cdots\!89}{18\!\cdots\!04}a^{13}-\frac{53\!\cdots\!51}{18\!\cdots\!04}a^{11}-\frac{87\!\cdots\!31}{38\!\cdots\!96}a^{9}+\frac{29\!\cdots\!29}{62\!\cdots\!32}a^{7}+\frac{13\!\cdots\!51}{12\!\cdots\!68}a^{5}+\frac{6913651979807}{531793963787968}a^{3}+\frac{10044996510737}{37985283127712}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

not computed

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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{10625579}{1276837307054911168} a^{33} - \frac{2680158372175}{638418653527455584} a^{19} - \frac{468901578337858483}{1276837307054911168} a^{5} \)  (order $28$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - 13*x^34 + 119*x^32 - 946*x^30 + 6985*x^28 - 49336*x^26 + 338472*x^24 - 1782270*x^22 + 8663374*x^20 - 40095490*x^18 + 175403900*x^16 - 699981526*x^14 + 2294243848*x^12 - 3420884824*x^10 + 5086203969*x^8 - 7494358949*x^6 + 10739824263*x^4 - 14123762450*x^2 + 13841287201)
 
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^36 - 13*x^34 + 119*x^32 - 946*x^30 + 6985*x^28 - 49336*x^26 + 338472*x^24 - 1782270*x^22 + 8663374*x^20 - 40095490*x^18 + 175403900*x^16 - 699981526*x^14 + 2294243848*x^12 - 3420884824*x^10 + 5086203969*x^8 - 7494358949*x^6 + 10739824263*x^4 - 14123762450*x^2 + 13841287201, 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^36 - 13*x^34 + 119*x^32 - 946*x^30 + 6985*x^28 - 49336*x^26 + 338472*x^24 - 1782270*x^22 + 8663374*x^20 - 40095490*x^18 + 175403900*x^16 - 699981526*x^14 + 2294243848*x^12 - 3420884824*x^10 + 5086203969*x^8 - 7494358949*x^6 + 10739824263*x^4 - 14123762450*x^2 + 13841287201);
 
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^36 - 13*x^34 + 119*x^32 - 946*x^30 + 6985*x^28 - 49336*x^26 + 338472*x^24 - 1782270*x^22 + 8663374*x^20 - 40095490*x^18 + 175403900*x^16 - 699981526*x^14 + 2294243848*x^12 - 3420884824*x^10 + 5086203969*x^8 - 7494358949*x^6 + 10739824263*x^4 - 14123762450*x^2 + 13841287201);
 
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

$C_6^2$ (as 36T4):

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)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), 3.3.17689.1, \(\Q(\zeta_{7})^+\), 3.3.361.1, 3.3.17689.2, \(\Q(i, \sqrt{7})\), 6.0.20025646144.2, 6.0.153664.1, 6.0.8340544.1, 6.0.20025646144.1, 6.6.140179523008.1, 6.0.2190305047.1, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{7})\), 6.6.2860806592.1, 6.0.44700103.1, 6.6.140179523008.2, 6.0.2190305047.2, 9.9.5534900853769.1, 12.0.19650298670750401368064.1, \(\Q(\zeta_{28})\), 12.0.8184214356830654464.1, 12.0.19650298670750401368064.2, 18.0.8030814853150226545771901353984.1, 18.18.2754569494630527705199762164416512.1, 18.0.10507848719141112156676338823.1

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 ${\href{/padicField/3.6.0.1}{6} }^{6}$ ${\href{/padicField/5.6.0.1}{6} }^{6}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ R ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.3.0.1}{3} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{12}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{6}$ ${\href{/padicField/53.3.0.1}{3} }^{12}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
\(7\) Copy content Toggle raw display 7.12.10.1$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
\(19\) Copy content Toggle raw display 19.18.12.1$x^{18} + 1938 x^{16} + 165 x^{15} + 1564986 x^{14} + 217074 x^{13} + 673956770 x^{12} + 124220751 x^{11} + 163260541175 x^{10} + 40698676942 x^{9} + 21101810097561 x^{8} + 7872565858164 x^{7} + 1139107203476720 x^{6} + 760256762812749 x^{5} + 441034593923007 x^{4} + 19443033036221259 x^{3} + 22753074450170540 x^{2} + 9283251966890258 x + 2743830934025437$$3$$6$$12$$C_6 \times C_3$$[\ ]_{3}^{6}$
19.18.12.1$x^{18} + 1938 x^{16} + 165 x^{15} + 1564986 x^{14} + 217074 x^{13} + 673956770 x^{12} + 124220751 x^{11} + 163260541175 x^{10} + 40698676942 x^{9} + 21101810097561 x^{8} + 7872565858164 x^{7} + 1139107203476720 x^{6} + 760256762812749 x^{5} + 441034593923007 x^{4} + 19443033036221259 x^{3} + 22753074450170540 x^{2} + 9283251966890258 x + 2743830934025437$$3$$6$$12$$C_6 \times C_3$$[\ ]_{3}^{6}$