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Properties

Label 36.0.765...984.1
Degree $36$
Signature $[0, 18]$
Discriminant $7.656\times 10^{56}$
Root discriminant \(38.03\)
Ramified primes $2,3,7$
Class number $28$ (GRH)
Class group [2, 14] (GRH)
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 - 38*x^30 + 1315*x^24 - 4900*x^18 + 16603*x^12 - 129*x^6 + 1)
 
gp: K = bnfinit(y^36 - 38*y^30 + 1315*y^24 - 4900*y^18 + 16603*y^12 - 129*y^6 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 38*x^30 + 1315*x^24 - 4900*x^18 + 16603*x^12 - 129*x^6 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 38*x^30 + 1315*x^24 - 4900*x^18 + 16603*x^12 - 129*x^6 + 1)
 

\( x^{36} - 38x^{30} + 1315x^{24} - 4900x^{18} + 16603x^{12} - 129x^{6} + 1 \) 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:   \(765562336274603149526276140236591524202950795854016937984\) \(\medspace = 2^{36}\cdot 3^{54}\cdot 7^{24}\) 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:  \(38.03\)
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:  $2\cdot 3^{3/2}7^{2/3}\approx 38.02862046112015$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(252=2^{2}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(107,·)$, $\chi_{252}(65,·)$, $\chi_{252}(137,·)$, $\chi_{252}(11,·)$, $\chi_{252}(149,·)$, $\chi_{252}(23,·)$, $\chi_{252}(25,·)$, $\chi_{252}(155,·)$, $\chi_{252}(29,·)$, $\chi_{252}(163,·)$, $\chi_{252}(37,·)$, $\chi_{252}(169,·)$, $\chi_{252}(43,·)$, $\chi_{252}(179,·)$, $\chi_{252}(53,·)$, $\chi_{252}(151,·)$, $\chi_{252}(191,·)$, $\chi_{252}(193,·)$, $\chi_{252}(67,·)$, $\chi_{252}(197,·)$, $\chi_{252}(71,·)$, $\chi_{252}(205,·)$, $\chi_{252}(79,·)$, $\chi_{252}(211,·)$, $\chi_{252}(85,·)$, $\chi_{252}(221,·)$, $\chi_{252}(95,·)$, $\chi_{252}(233,·)$, $\chi_{252}(235,·)$, $\chi_{252}(109,·)$, $\chi_{252}(239,·)$, $\chi_{252}(113,·)$, $\chi_{252}(247,·)$, $\chi_{252}(121,·)$, $\chi_{252}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{559}a^{24}-\frac{12}{43}a^{18}+\frac{209}{559}a^{12}-\frac{5}{43}a^{6}+\frac{274}{559}$, $\frac{1}{559}a^{25}-\frac{12}{43}a^{19}+\frac{209}{559}a^{13}-\frac{5}{43}a^{7}+\frac{274}{559}a$, $\frac{1}{559}a^{26}-\frac{12}{43}a^{20}+\frac{209}{559}a^{14}-\frac{5}{43}a^{8}+\frac{274}{559}a^{2}$, $\frac{1}{559}a^{27}-\frac{12}{43}a^{21}+\frac{209}{559}a^{15}-\frac{5}{43}a^{9}+\frac{274}{559}a^{3}$, $\frac{1}{559}a^{28}-\frac{12}{43}a^{22}+\frac{209}{559}a^{16}-\frac{5}{43}a^{10}+\frac{274}{559}a^{4}$, $\frac{1}{559}a^{29}-\frac{12}{43}a^{23}+\frac{209}{559}a^{17}-\frac{5}{43}a^{11}+\frac{274}{559}a^{5}$, $\frac{1}{12201876037}a^{30}-\frac{7467523}{12201876037}a^{24}-\frac{114384399}{12201876037}a^{18}+\frac{5503538510}{12201876037}a^{12}+\frac{1127590751}{12201876037}a^{6}+\frac{5582733509}{12201876037}$, $\frac{1}{12201876037}a^{31}-\frac{7467523}{12201876037}a^{25}-\frac{114384399}{12201876037}a^{19}+\frac{5503538510}{12201876037}a^{13}+\frac{1127590751}{12201876037}a^{7}+\frac{5582733509}{12201876037}a$, $\frac{1}{12201876037}a^{32}-\frac{7467523}{12201876037}a^{26}-\frac{114384399}{12201876037}a^{20}+\frac{5503538510}{12201876037}a^{14}+\frac{1127590751}{12201876037}a^{8}+\frac{5582733509}{12201876037}a^{2}$, $\frac{1}{12201876037}a^{33}-\frac{7467523}{12201876037}a^{27}-\frac{114384399}{12201876037}a^{21}+\frac{5503538510}{12201876037}a^{15}+\frac{1127590751}{12201876037}a^{9}+\frac{5582733509}{12201876037}a^{3}$, $\frac{1}{12201876037}a^{34}-\frac{7467523}{12201876037}a^{28}-\frac{114384399}{12201876037}a^{22}+\frac{5503538510}{12201876037}a^{16}+\frac{1127590751}{12201876037}a^{10}+\frac{5582733509}{12201876037}a^{4}$, $\frac{1}{12201876037}a^{35}-\frac{7467523}{12201876037}a^{29}-\frac{114384399}{12201876037}a^{23}+\frac{5503538510}{12201876037}a^{17}+\frac{1127590751}{12201876037}a^{11}+\frac{5582733509}{12201876037}a^{5}$ Copy content Toggle raw display

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

$C_{2}\times C_{14}$, which has order $28$ (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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{5430336179}{12201876037} a^{35} + \frac{206353143668}{12201876037} a^{29} - \frac{7140904840090}{12201876037} a^{23} + \frac{26609083833276}{12201876037} a^{17} - \frac{90160032745258}{12201876037} a^{11} + \frac{16291037658}{283764559} a^{5} \)  (order $36$) 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:   $\frac{7075524}{12201876037}a^{35}-\frac{244850370}{12201876037}a^{29}+\frac{8391543538}{12201876037}a^{23}-\frac{3091445394}{12201876037}a^{17}+\frac{558594}{283764559}a^{11}+\frac{387866503351}{12201876037}a^{5}$, $\frac{198436}{938605849}a^{34}-\frac{6866930}{938605849}a^{28}+\frac{235334231}{938605849}a^{22}-\frac{86700866}{938605849}a^{16}+\frac{15666}{21828043}a^{10}+\frac{10648303918}{938605849}a^{4}$, $\frac{2416738802}{12201876037}a^{34}-\frac{91835779862}{12201876037}a^{28}+\frac{3178001329435}{12201876037}a^{22}-\frac{11841670194505}{12201876037}a^{16}+\frac{40124985606547}{12201876037}a^{10}-\frac{7250193147}{283764559}a^{4}+1$, $\frac{5453950}{12201876037}a^{34}+\frac{11154294}{283764559}a^{32}-\frac{188735375}{12201876037}a^{28}-\frac{18222348943}{12201876037}a^{26}+\frac{6468625301}{12201876037}a^{22}+\frac{630578266506}{12201876037}a^{20}-\frac{2382945575}{12201876037}a^{16}-\frac{2345320446930}{12201876037}a^{14}+\frac{430575}{283764559}a^{10}+\frac{7947244975065}{12201876037}a^{8}+\frac{311044791177}{12201876037}a^{4}-\frac{141287724}{12201876037}a^{2}$, $\frac{5453950}{12201876037}a^{34}-\frac{4969104}{283764559}a^{31}-\frac{188735375}{12201876037}a^{28}+\frac{8117929470}{12201876037}a^{25}+\frac{6468625301}{12201876037}a^{22}-\frac{280915043696}{12201876037}a^{19}-\frac{2382945575}{12201876037}a^{16}+\frac{1044812088880}{12201876037}a^{13}+\frac{430575}{283764559}a^{10}-\frac{3536864877885}{12201876037}a^{7}+\frac{311044791177}{12201876037}a^{4}+\frac{62941984}{12201876037}a$, $\frac{5453950}{12201876037}a^{34}+\frac{82733471}{938605849}a^{33}-\frac{188735375}{12201876037}a^{28}-\frac{3143905984}{938605849}a^{27}+\frac{6468625301}{12201876037}a^{22}+\frac{108795693920}{938605849}a^{21}-\frac{2382945575}{12201876037}a^{16}-\frac{405434252289}{938605849}a^{15}+\frac{430575}{283764559}a^{10}+\frac{1373638711904}{938605849}a^{9}+\frac{311044791177}{12201876037}a^{4}-\frac{248203104}{21828043}a^{3}$, $\frac{12529474}{12201876037}a^{35}-\frac{433585745}{12201876037}a^{29}+\frac{14860168839}{12201876037}a^{23}-\frac{5474390969}{12201876037}a^{17}+\frac{989169}{283764559}a^{11}+\frac{698911294528}{12201876037}a^{5}-1$, $\frac{9949806}{12201876037}a^{34}+\frac{958094}{12201876037}a^{32}+\frac{294614}{12201876037}a^{30}-\frac{344315655}{12201876037}a^{28}-\frac{33155095}{12201876037}a^{26}-\frac{10195195}{12201876037}a^{24}+\frac{11800823836}{12201876037}a^{22}+\frac{1136426766}{12201876037}a^{20}+\frac{349935295}{12201876037}a^{18}-\frac{4347279711}{12201876037}a^{16}-\frac{418611439}{12201876037}a^{14}-\frac{128723059}{12201876037}a^{12}+\frac{785511}{283764559}a^{10}+\frac{75639}{283764559}a^{8}+\frac{23259}{283764559}a^{6}+\frac{560483343594}{12201876037}a^{4}+\frac{49404362723}{12201876037}a^{2}+\frac{21987013272}{12201876037}$, $\frac{12529474}{12201876037}a^{35}-\frac{198436}{938605849}a^{33}-\frac{169635}{21828043}a^{31}-\frac{433585745}{12201876037}a^{29}+\frac{6866930}{938605849}a^{27}+\frac{6444686}{21828043}a^{25}+\frac{14860168839}{12201876037}a^{23}-\frac{235334231}{938605849}a^{21}-\frac{223020055}{21828043}a^{19}-\frac{5474390969}{12201876037}a^{17}+\frac{86700866}{938605849}a^{15}+\frac{829482275}{21828043}a^{13}+\frac{989169}{283764559}a^{11}-\frac{15666}{21828043}a^{9}-\frac{2815818991}{21828043}a^{7}+\frac{698911294528}{12201876037}a^{5}-\frac{10648303918}{938605849}a^{3}+\frac{21878013}{21828043}a$, $\frac{22479280}{12201876037}a^{35}-\frac{82733471}{938605849}a^{33}+\frac{213007992}{12201876037}a^{31}-\frac{777901400}{12201876037}a^{29}+\frac{3143905984}{938605849}a^{27}-\frac{8094969570}{12201876037}a^{25}+\frac{26660992675}{12201876037}a^{23}-\frac{108795693920}{938605849}a^{21}+\frac{280128552225}{12201876037}a^{19}-\frac{9821670680}{12201876037}a^{17}+\frac{405434252289}{938605849}a^{15}-\frac{1044522200500}{12201876037}a^{13}+\frac{1774680}{283764559}a^{11}-\frac{1373638711904}{938605849}a^{9}+\frac{3536862625545}{12201876037}a^{7}+\frac{1259394638122}{12201876037}a^{5}+\frac{248203104}{21828043}a^{3}-\frac{639076545}{283764559}a$, $\frac{12529474}{12201876037}a^{35}-\frac{82733471}{938605849}a^{33}-\frac{294614}{12201876037}a^{31}-\frac{433585745}{12201876037}a^{29}+\frac{3143905984}{938605849}a^{27}+\frac{10195195}{12201876037}a^{25}+\frac{14860168839}{12201876037}a^{23}-\frac{108795693920}{938605849}a^{21}-\frac{349935295}{12201876037}a^{19}-\frac{5474390969}{12201876037}a^{17}+\frac{405434252289}{938605849}a^{15}+\frac{128723059}{12201876037}a^{13}+\frac{989169}{283764559}a^{11}-\frac{1373638711904}{938605849}a^{9}-\frac{23259}{283764559}a^{7}+\frac{698911294528}{12201876037}a^{5}+\frac{248203104}{21828043}a^{3}-\frac{34188889309}{12201876037}a$, $\frac{12529474}{12201876037}a^{35}+\frac{4354801056}{12201876037}a^{34}-\frac{579}{6553}a^{33}-\frac{433585745}{12201876037}a^{29}-\frac{165482365876}{12201876037}a^{28}+\frac{945894}{281779}a^{27}+\frac{14860168839}{12201876037}a^{23}+\frac{5726560819130}{12201876037}a^{22}-\frac{32732221}{281779}a^{21}-\frac{5474390969}{12201876037}a^{17}-\frac{21338438553519}{12201876037}a^{16}+\frac{121741505}{281779}a^{15}+\frac{989169}{283764559}a^{11}+\frac{72302729490506}{12201876037}a^{10}-\frac{412380482}{281779}a^{9}+\frac{698911294528}{12201876037}a^{5}-\frac{13064397306}{283764559}a^{4}+\frac{7334}{281779}a^{3}$, $\frac{11154294}{283764559}a^{32}-\frac{8949039}{283764559}a^{31}+\frac{169635}{21828043}a^{30}-\frac{18222348943}{12201876037}a^{26}+\frac{14619769469}{12201876037}a^{25}-\frac{6444686}{21828043}a^{24}+\frac{630578266506}{12201876037}a^{20}-\frac{505910055761}{12201876037}a^{19}+\frac{223020055}{21828043}a^{18}-\frac{2345320446930}{12201876037}a^{14}+\frac{1881639855205}{12201876037}a^{13}-\frac{829482275}{21828043}a^{12}+\frac{7947244975065}{12201876037}a^{8}-\frac{6373202159096}{12201876037}a^{7}+\frac{2815818991}{21828043}a^{6}-\frac{141287724}{12201876037}a^{2}+\frac{113354494}{12201876037}a-\frac{49970}{21828043}$, $\frac{12529474}{12201876037}a^{35}-\frac{198436}{938605849}a^{34}+\frac{198436}{938605849}a^{33}-\frac{433585745}{12201876037}a^{29}+\frac{6866930}{938605849}a^{28}-\frac{6866930}{938605849}a^{27}+\frac{14860168839}{12201876037}a^{23}-\frac{235334231}{938605849}a^{22}+\frac{235334231}{938605849}a^{21}-\frac{5474390969}{12201876037}a^{17}+\frac{86700866}{938605849}a^{16}-\frac{86700866}{938605849}a^{15}+\frac{989169}{283764559}a^{11}-\frac{15666}{21828043}a^{10}+\frac{15666}{21828043}a^{9}+\frac{698911294528}{12201876037}a^{5}-\frac{10648303918}{938605849}a^{4}+\frac{10648303918}{938605849}a^{3}$, $\frac{958094}{12201876037}a^{32}-\frac{265668556}{12201876037}a^{31}-\frac{169635}{21828043}a^{30}-\frac{33155095}{12201876037}a^{26}+\frac{10094224278}{12201876037}a^{25}+\frac{6444686}{21828043}a^{24}+\frac{1136426766}{12201876037}a^{20}-\frac{349313287515}{12201876037}a^{19}-\frac{223020055}{21828043}a^{18}-\frac{418611439}{12201876037}a^{14}+\frac{1300379634991}{12201876037}a^{13}+\frac{829482275}{21828043}a^{12}+\frac{75639}{283764559}a^{8}-\frac{4410379097043}{12201876037}a^{7}-\frac{2815818991}{21828043}a^{6}+\frac{61606238760}{12201876037}a^{2}+\frac{796912443}{283764559}a+\frac{49970}{21828043}$, $\frac{157835898}{283764559}a^{35}+\frac{2416738802}{12201876037}a^{34}+\frac{169635}{21828043}a^{30}-\frac{257851196768}{12201876037}a^{29}-\frac{91835779862}{12201876037}a^{28}-\frac{6444686}{21828043}a^{24}+\frac{8922831597702}{12201876037}a^{23}+\frac{3178001329435}{12201876037}a^{22}+\frac{223020055}{21828043}a^{18}-\frac{33186838973310}{12201876037}a^{17}-\frac{11841670194505}{12201876037}a^{16}-\frac{829482275}{21828043}a^{12}+\frac{112427767388751}{12201876037}a^{11}+\frac{40124985606547}{12201876037}a^{10}+\frac{2815818991}{21828043}a^{6}-\frac{1999254708}{12201876037}a^{5}-\frac{7250193147}{283764559}a^{4}-\frac{49970}{21828043}$, $\frac{12529474}{12201876037}a^{35}-\frac{213007992}{12201876037}a^{31}-\frac{368866}{12201876037}a^{30}-\frac{433585745}{12201876037}a^{29}+\frac{8094969570}{12201876037}a^{25}+\frac{12764705}{12201876037}a^{24}+\frac{14860168839}{12201876037}a^{23}-\frac{280128552225}{12201876037}a^{19}-\frac{436556176}{12201876037}a^{18}-\frac{5474390969}{12201876037}a^{17}+\frac{1044522200500}{12201876037}a^{13}+\frac{161165321}{12201876037}a^{12}+\frac{989169}{283764559}a^{11}-\frac{3536862625545}{12201876037}a^{7}-\frac{29121}{283764559}a^{6}+\frac{698911294528}{12201876037}a^{5}+\frac{639076545}{283764559}a-\frac{5430336179}{12201876037}$ Copy content Toggle raw display (assuming GRH)
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:  \( 18271878541587.973 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 18271878541587.973 \cdot 28}{36\cdot\sqrt{765562336274603149526276140236591524202950795854016937984}}\cr\approx \mathstrut & 0.119642665807942 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - 38*x^30 + 1315*x^24 - 4900*x^18 + 16603*x^12 - 129*x^6 + 1)
 
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 - 38*x^30 + 1315*x^24 - 4900*x^18 + 16603*x^12 - 129*x^6 + 1, 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 - 38*x^30 + 1315*x^24 - 4900*x^18 + 16603*x^12 - 129*x^6 + 1);
 
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 - 38*x^30 + 1315*x^24 - 4900*x^18 + 16603*x^12 - 129*x^6 + 1);
 
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{12})\), \(\Q(\zeta_{9})\), 6.0.47258883.1, 6.0.47258883.2, 6.0.64827.1, \(\Q(\zeta_{36})^+\), 6.0.419904.1, 6.6.3024568512.2, 6.0.1008189504.2, 6.6.3024568512.1, 6.0.1008189504.1, 6.6.4148928.1, 6.0.153664.1, 9.9.62523502209.1, \(\Q(\zeta_{36})\), 12.0.9148014683781894144.2, 12.0.9148014683781894144.1, 12.0.17213603549184.1, 18.0.105548084868928352751387.1, 18.18.27668797159880354103659593728.1, 18.0.1024770265180753855691096064.1

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 R ${\href{/padicField/5.6.0.1}{6} }^{6}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.3.0.1}{3} }^{12}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{12}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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 2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
\(3\) Copy content Toggle raw display Deg $36$$6$$6$$54$
\(7\) Copy content Toggle raw display 7.18.12.1$x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$$3$$6$$12$$C_6 \times C_3$$[\ ]_{3}^{6}$
7.18.12.1$x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$$3$$6$$12$$C_6 \times C_3$$[\ ]_{3}^{6}$