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Properties

Label 27.3.353...000.1
Degree $27$
Signature $[3, 12]$
Discriminant $3.537\times 10^{42}$
Root discriminant \(37.66\)
Ramified primes $2,3,5$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $\He_3:\GL(2,3)$ (as 27T294)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 36*x^24 + 752*x^21 + 528*x^18 - 2616*x^15 + 44416*x^12 + 37248*x^9 + 11712*x^6 + 1616*x^3 + 64)
 
gp: K = bnfinit(y^27 - 36*y^24 + 752*y^21 + 528*y^18 - 2616*y^15 + 44416*y^12 + 37248*y^9 + 11712*y^6 + 1616*y^3 + 64, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 36*x^24 + 752*x^21 + 528*x^18 - 2616*x^15 + 44416*x^12 + 37248*x^9 + 11712*x^6 + 1616*x^3 + 64);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^27 - 36*x^24 + 752*x^21 + 528*x^18 - 2616*x^15 + 44416*x^12 + 37248*x^9 + 11712*x^6 + 1616*x^3 + 64)
 

\( x^{27} - 36 x^{24} + 752 x^{21} + 528 x^{18} - 2616 x^{15} + 44416 x^{12} + 37248 x^{9} + 11712 x^{6} + \cdots + 64 \) Copy content Toggle raw display

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

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[3, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3537182715531733726396416000000000000000000\) \(\medspace = 2^{52}\cdot 3^{30}\cdot 5^{18}\) 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:  \(37.66\)
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:  $2^{25/12}3^{7/6}5^{3/4}\approx 51.05223872742641$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{24}a^{11}-\frac{1}{24}a^{10}+\frac{1}{24}a^{9}-\frac{5}{12}a^{5}+\frac{5}{12}a^{4}-\frac{5}{12}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{24}a^{12}+\frac{1}{24}a^{9}+\frac{1}{12}a^{6}+\frac{1}{4}a^{3}-\frac{1}{3}$, $\frac{1}{72}a^{13}+\frac{1}{72}a^{12}+\frac{1}{18}a^{10}+\frac{1}{18}a^{9}+\frac{1}{36}a^{7}+\frac{1}{36}a^{6}-\frac{1}{9}a-\frac{1}{9}$, $\frac{1}{144}a^{14}+\frac{1}{72}a^{12}-\frac{1}{72}a^{11}+\frac{1}{24}a^{10}+\frac{1}{72}a^{9}+\frac{1}{72}a^{8}+\frac{1}{36}a^{6}-\frac{1}{12}a^{5}-\frac{5}{12}a^{4}+\frac{5}{12}a^{3}+\frac{5}{18}a^{2}-\frac{1}{3}a+\frac{2}{9}$, $\frac{1}{1728}a^{15}-\frac{1}{432}a^{12}-\frac{7}{144}a^{9}+\frac{2}{27}a^{6}-\frac{119}{432}a^{3}+\frac{13}{108}$, $\frac{1}{1728}a^{16}-\frac{1}{432}a^{13}-\frac{7}{144}a^{10}+\frac{2}{27}a^{7}-\frac{119}{432}a^{4}+\frac{13}{108}a$, $\frac{1}{5184}a^{17}-\frac{1}{5184}a^{16}+\frac{1}{5184}a^{15}-\frac{1}{1296}a^{14}+\frac{1}{1296}a^{13}-\frac{1}{1296}a^{12}-\frac{7}{432}a^{11}+\frac{7}{432}a^{10}-\frac{7}{432}a^{9}+\frac{2}{81}a^{8}-\frac{2}{81}a^{7}+\frac{2}{81}a^{6}-\frac{551}{1296}a^{5}+\frac{551}{1296}a^{4}-\frac{551}{1296}a^{3}-\frac{95}{324}a^{2}+\frac{95}{324}a-\frac{95}{324}$, $\frac{1}{5184}a^{18}-\frac{25}{1296}a^{12}-\frac{13}{324}a^{9}+\frac{25}{144}a^{6}+\frac{1}{162}a^{3}-\frac{14}{81}$, $\frac{1}{15552}a^{19}+\frac{1}{15552}a^{18}-\frac{1}{5184}a^{16}-\frac{1}{5184}a^{15}-\frac{11}{1944}a^{13}-\frac{11}{1944}a^{12}+\frac{11}{3888}a^{10}+\frac{11}{3888}a^{9}+\frac{43}{1296}a^{7}+\frac{43}{1296}a^{6}+\frac{1661}{3888}a^{4}+\frac{1661}{3888}a^{3}-\frac{95}{972}a-\frac{95}{972}$, $\frac{1}{15552}a^{20}-\frac{1}{15552}a^{18}-\frac{1}{5184}a^{16}-\frac{1}{5184}a^{15}+\frac{1}{1944}a^{14}+\frac{1}{1296}a^{13}-\frac{5}{243}a^{12}+\frac{7}{486}a^{11}+\frac{7}{432}a^{10}+\frac{169}{3888}a^{9}+\frac{31}{432}a^{8}-\frac{2}{81}a^{7}-\frac{179}{1296}a^{6}-\frac{121}{243}a^{5}+\frac{551}{1296}a^{4}+\frac{673}{3888}a^{3}-\frac{217}{486}a^{2}+\frac{95}{324}a-\frac{91}{972}$, $\frac{1}{186624}a^{21}+\frac{1}{11664}a^{18}-\frac{5}{93312}a^{15}-\frac{271}{23328}a^{12}+\frac{1391}{46656}a^{9}+\frac{325}{5832}a^{6}-\frac{1369}{7776}a^{3}-\frac{2455}{5832}$, $\frac{1}{1119744}a^{22}-\frac{1}{559872}a^{21}-\frac{5}{279936}a^{19}+\frac{5}{139968}a^{18}+\frac{49}{559872}a^{16}-\frac{49}{279936}a^{15}+\frac{125}{139968}a^{13}-\frac{125}{69984}a^{12}-\frac{16501}{279936}a^{10}-\frac{995}{139968}a^{9}+\frac{11153}{69984}a^{7}+\frac{6343}{34992}a^{6}+\frac{10049}{46656}a^{4}-\frac{4217}{23328}a^{3}-\frac{12409}{34992}a-\frac{5087}{17496}$, $\frac{1}{1119744}a^{23}+\frac{1}{559872}a^{21}-\frac{5}{279936}a^{20}-\frac{5}{139968}a^{18}+\frac{49}{559872}a^{17}+\frac{49}{279936}a^{15}+\frac{125}{139968}a^{14}+\frac{125}{69984}a^{12}-\frac{4837}{279936}a^{11}-\frac{1}{24}a^{10}+\frac{6827}{139968}a^{9}-\frac{6343}{69984}a^{8}-\frac{6343}{34992}a^{6}-\frac{9391}{46656}a^{5}+\frac{5}{12}a^{4}-\frac{5503}{23328}a^{3}-\frac{6577}{34992}a^{2}+\frac{1}{3}a-\frac{745}{17496}$, $\frac{1}{3359232}a^{24}+\frac{1}{419904}a^{21}+\frac{31}{559872}a^{18}-\frac{1}{23328}a^{15}+\frac{4393}{279936}a^{12}+\frac{6305}{104976}a^{9}-\frac{69425}{419904}a^{6}+\frac{23575}{52488}a^{3}-\frac{10237}{26244}$, $\frac{1}{3359232}a^{25}-\frac{1}{3359232}a^{22}-\frac{11}{559872}a^{19}+\frac{1}{15552}a^{18}+\frac{5}{62208}a^{16}-\frac{1}{5184}a^{15}-\frac{965}{279936}a^{13}+\frac{2}{243}a^{12}-\frac{4091}{839808}a^{10}+\frac{227}{3888}a^{9}+\frac{98533}{419904}a^{7}+\frac{79}{1296}a^{6}+\frac{83381}{419904}a^{4}+\frac{1661}{3888}a^{3}+\frac{9605}{104976}a-\frac{203}{972}$, $\frac{1}{10077696}a^{26}-\frac{1}{10077696}a^{25}+\frac{1}{10077696}a^{24}-\frac{1}{10077696}a^{23}+\frac{1}{10077696}a^{22}+\frac{13}{5038848}a^{21}-\frac{47}{1679616}a^{20}+\frac{47}{1679616}a^{19}-\frac{137}{1679616}a^{18}+\frac{17}{186624}a^{17}-\frac{17}{186624}a^{16}-\frac{7}{31104}a^{15}+\frac{2563}{839808}a^{14}+\frac{3269}{839808}a^{13}+\frac{12589}{839808}a^{12}+\frac{16861}{2519424}a^{11}+\frac{53123}{2519424}a^{10}+\frac{39359}{1259712}a^{9}-\frac{119519}{1259712}a^{8}+\frac{137015}{1259712}a^{7}-\frac{222929}{1259712}a^{6}-\frac{305959}{1259712}a^{5}-\frac{323897}{1259712}a^{4}-\frac{44885}{629856}a^{3}+\frac{119009}{314928}a^{2}-\frac{136505}{314928}a+\frac{21745}{157464}$ 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

$C_{3}$, which has order $3$ (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:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{27049}{1259712}a^{26}-\frac{43979}{10077696}a^{25}+\frac{61997}{10077696}a^{24}-\frac{7816073}{10077696}a^{23}+\frac{1594193}{10077696}a^{22}-\frac{1124431}{5038848}a^{21}+\frac{13638859}{839808}a^{20}-\frac{5578319}{1679616}a^{19}+\frac{7873055}{1679616}a^{18}+\frac{5252971}{559872}a^{17}-\frac{825467}{559872}a^{16}+\frac{61009}{31104}a^{15}-\frac{24001307}{419904}a^{14}+\frac{9843115}{839808}a^{13}-\frac{13900591}{839808}a^{12}+\frac{2420173685}{2519424}a^{11}-\frac{495711101}{2519424}a^{10}+\frac{349928443}{1259712}a^{9}+\frac{430869247}{629856}a^{8}-\frac{142806839}{1259712}a^{7}+\frac{192578495}{1259712}a^{6}+\frac{223762705}{1259712}a^{5}-\frac{33368065}{1259712}a^{4}+\frac{21967091}{629856}a^{3}+\frac{5227309}{314928}a^{2}-\frac{404281}{314928}a+\frac{345353}{157464}$, $\frac{3757}{279936}a^{26}+\frac{475}{62208}a^{25}-\frac{7303}{1119744}a^{24}+\frac{545017}{1119744}a^{23}-\frac{154231}{559872}a^{22}+\frac{16523}{69984}a^{21}-\frac{105989}{10368}a^{20}+\frac{1613383}{279936}a^{19}-\frac{2772379}{559872}a^{18}-\frac{811333}{186624}a^{17}+\frac{1003229}{279936}a^{16}-\frac{42901}{17496}a^{15}+\frac{1680677}{46656}a^{14}-\frac{2829865}{139968}a^{13}+\frac{4903795}{279936}a^{12}-\frac{169593181}{279936}a^{11}+\frac{47686771}{139968}a^{10}-\frac{641218}{2187}a^{9}-\frac{23612189}{69984}a^{8}+\frac{18051137}{69984}a^{7}-\frac{8593363}{46656}a^{6}-\frac{11435021}{139968}a^{5}+\frac{1667629}{23328}a^{4}-\frac{362693}{8748}a^{3}-\frac{299153}{34992}a^{2}+\frac{101767}{17496}a-\frac{7429}{2916}$, $\frac{1315}{209952}a^{26}-\frac{1315}{209952}a^{25}+\frac{1315}{209952}a^{24}-\frac{189901}{839808}a^{23}+\frac{189901}{839808}a^{22}-\frac{189901}{839808}a^{21}+\frac{220829}{46656}a^{20}-\frac{220829}{46656}a^{19}+\frac{220829}{46656}a^{18}+\frac{197765}{69984}a^{17}-\frac{197765}{69984}a^{16}+\frac{197765}{69984}a^{15}-\frac{587663}{34992}a^{14}+\frac{587663}{34992}a^{13}-\frac{587663}{34992}a^{12}+\frac{58763875}{209952}a^{11}-\frac{58763875}{209952}a^{10}+\frac{58763875}{209952}a^{9}+\frac{21516001}{104976}a^{8}-\frac{21516001}{104976}a^{7}+\frac{21516001}{104976}a^{6}+\frac{1181363}{26244}a^{5}-\frac{1181363}{26244}a^{4}+\frac{1181363}{26244}a^{3}+\frac{14615}{6561}a^{2}-\frac{14615}{6561}a+\frac{14615}{6561}$, $\frac{9161}{1679616}a^{24}+\frac{167611}{839808}a^{21}-\frac{1180967}{279936}a^{18}-\frac{9809}{23328}a^{15}+\frac{2147815}{139968}a^{12}-\frac{52707229}{209952}a^{9}-\frac{12056111}{209952}a^{6}+\frac{506137}{26244}a^{3}+\frac{47039}{13122}$, $\frac{3863}{629856}a^{26}-\frac{23743}{2519424}a^{25}-\frac{26993}{10077696}a^{24}+\frac{2217443}{10077696}a^{23}+\frac{858205}{2519424}a^{22}+\frac{121255}{1259712}a^{21}-\frac{3849931}{839808}a^{20}-\frac{2996665}{419904}a^{19}-\frac{3372671}{1679616}a^{18}-\frac{2142965}{559872}a^{17}-\frac{183473}{46656}a^{16}-\frac{54103}{34992}a^{15}+\frac{6735053}{419904}a^{14}+\frac{5277143}{209952}a^{13}+\frac{5888791}{839808}a^{12}-\frac{680939615}{2519424}a^{11}-\frac{265966465}{629856}a^{10}-\frac{37329391}{314928}a^{9}-\frac{165999787}{629856}a^{8}-\frac{91076689}{314928}a^{7}-\frac{135573743}{1259712}a^{6}-\frac{96274087}{1259712}a^{5}-\frac{23248481}{314928}a^{4}-\frac{2541361}{78732}a^{3}-\frac{2118691}{314928}a^{2}-\frac{621713}{78732}a-\frac{245953}{78732}$, $\frac{4717}{186624}a^{26}+\frac{17}{3888}a^{25}+\frac{17405}{1679616}a^{24}-\frac{512939}{559872}a^{23}-\frac{3727}{23328}a^{22}-\frac{629903}{1679616}a^{21}+\frac{5384213}{279936}a^{20}+\frac{39355}{11664}a^{19}+\frac{733813}{93312}a^{18}+\frac{2404867}{279936}a^{17}+\frac{23099}{46656}a^{16}+\frac{1112477}{279936}a^{15}-\frac{9571673}{139968}a^{14}-\frac{137969}{11664}a^{13}-\frac{3912485}{139968}a^{12}+\frac{159493931}{139968}a^{11}+\frac{2339999}{11664}a^{10}+\frac{195493463}{419904}a^{9}+\frac{46181275}{69984}a^{8}+\frac{80371}{1458}a^{7}+\frac{62429747}{209952}a^{6}+\frac{2905951}{23328}a^{5}+\frac{18251}{1296}a^{4}+\frac{12493345}{209952}a^{3}+\frac{93257}{17496}a^{2}+\frac{3559}{2916}a+\frac{144433}{52488}$, $\frac{6545}{279936}a^{26}+\frac{8015}{839808}a^{25}-\frac{34325}{3359232}a^{24}+\frac{471889}{559872}a^{23}-\frac{1159547}{3359232}a^{22}+\frac{77843}{209952}a^{21}-\frac{91361}{5184}a^{20}+\frac{675125}{93312}a^{19}-\frac{4361275}{559872}a^{18}-\frac{1070917}{93312}a^{17}+\frac{2137355}{559872}a^{16}-\frac{222053}{69984}a^{15}+\frac{90179}{1458}a^{14}-\frac{3567091}{139968}a^{13}+\frac{858971}{31104}a^{12}-\frac{145771141}{139968}a^{11}+\frac{359634719}{839808}a^{10}-\frac{6059225}{13122}a^{9}-\frac{28680911}{34992}a^{8}+\frac{59425663}{209952}a^{7}-\frac{104475323}{419904}a^{6}-\frac{15835997}{69984}a^{5}+\frac{30045691}{419904}a^{4}-\frac{2619959}{52488}a^{3}-\frac{374261}{17496}a^{2}+\frac{720943}{104976}a-\frac{92071}{26244}$, $\frac{45161}{5038848}a^{26}-\frac{26381}{10077696}a^{25}+\frac{16619}{10077696}a^{24}+\frac{3276349}{10077696}a^{23}+\frac{474535}{5038848}a^{22}-\frac{299995}{5038848}a^{21}-\frac{1433705}{209952}a^{20}-\frac{3302255}{1679616}a^{19}+\frac{2093033}{1679616}a^{18}-\frac{178471}{62208}a^{17}-\frac{402167}{279936}a^{16}+\frac{208831}{279936}a^{15}+\frac{5124905}{209952}a^{14}+\frac{5833639}{839808}a^{13}-\frac{3743977}{839808}a^{12}-\frac{1019626201}{2519424}a^{11}-\frac{146233831}{1259712}a^{10}+\frac{92833723}{1259712}a^{9}-\frac{17599789}{78732}a^{8}-\frac{126916607}{1259712}a^{7}+\frac{68190089}{1259712}a^{6}-\frac{41828813}{1259712}a^{5}-\frac{16020401}{629856}a^{4}+\frac{6093449}{629856}a^{3}-\frac{52601}{314928}a^{2}-\frac{321383}{157464}a+\frac{88259}{157464}$, $\frac{601}{93312}a^{26}-\frac{4469}{124416}a^{25}-\frac{25427}{1679616}a^{24}+\frac{129889}{559872}a^{23}+\frac{30359}{23328}a^{22}+\frac{461479}{839808}a^{21}-\frac{678467}{139968}a^{20}-\frac{5097107}{186624}a^{19}-\frac{3232831}{279936}a^{18}-\frac{933167}{279936}a^{17}-\frac{295205}{23328}a^{16}-\frac{79369}{17496}a^{15}+\frac{153697}{8748}a^{14}+\frac{9016379}{93312}a^{13}+\frac{1899683}{46656}a^{12}-\frac{40078165}{139968}a^{11}-\frac{9434603}{5832}a^{10}-\frac{143732977}{209952}a^{9}-\frac{8273245}{34992}a^{8}-\frac{44973505}{46656}a^{7}-\frac{75336827}{209952}a^{6}-\frac{742175}{23328}a^{5}-\frac{415609}{1944}a^{4}-\frac{4424695}{52488}a^{3}+\frac{12443}{17496}a^{2}-\frac{36821}{2916}a-\frac{33335}{6561}$, $\frac{5771}{10077696}a^{26}-\frac{34999}{10077696}a^{25}+\frac{5089}{10077696}a^{24}+\frac{104491}{5038848}a^{23}+\frac{319963}{2519424}a^{22}-\frac{93611}{5038848}a^{21}-\frac{730889}{1679616}a^{20}-\frac{4506769}{1679616}a^{19}+\frac{662335}{1679616}a^{18}-\frac{57619}{279936}a^{17}-\frac{23117}{69984}a^{16}-\frac{11743}{279936}a^{15}+\frac{1223233}{839808}a^{14}+\frac{8164865}{839808}a^{13}-\frac{1131971}{839808}a^{12}-\frac{32513251}{1259712}a^{11}-\frac{100558429}{629856}a^{10}+\frac{29675807}{1259712}a^{9}-\frac{19568225}{1259712}a^{8}-\frac{50908417}{1259712}a^{7}+\frac{529447}{1259712}a^{6}-\frac{5180189}{629856}a^{5}+\frac{685975}{78732}a^{4}+\frac{887779}{629856}a^{3}-\frac{449039}{157464}a^{2}+\frac{185437}{78732}a+\frac{137353}{157464}$, $\frac{62149}{5038848}a^{26}-\frac{123581}{10077696}a^{25}-\frac{3889}{2519424}a^{24}-\frac{2228323}{5038848}a^{23}+\frac{2235421}{5038848}a^{22}+\frac{140077}{2519424}a^{21}+\frac{7734139}{839808}a^{20}-\frac{15620915}{1679616}a^{19}-\frac{7621}{6561}a^{18}+\frac{2212327}{279936}a^{17}-\frac{1351277}{279936}a^{16}-\frac{37477}{46656}a^{15}-\frac{13515737}{419904}a^{14}+\frac{27703063}{839808}a^{13}+\frac{895439}{209952}a^{12}+\frac{683777059}{1259712}a^{11}-\frac{693531661}{1259712}a^{10}-\frac{43206817}{629856}a^{9}+\frac{341518963}{629856}a^{8}-\frac{452598923}{1259712}a^{7}-\frac{8970269}{157464}a^{6}+\frac{100566497}{629856}a^{5}-\frac{48281243}{629856}a^{4}-\frac{1373333}{314928}a^{3}+\frac{2272913}{157464}a^{2}-\frac{1166969}{157464}a+\frac{110923}{78732}$, $\frac{7927}{10077696}a^{26}+\frac{63691}{5038848}a^{25}-\frac{49693}{5038848}a^{24}-\frac{171089}{5038848}a^{23}-\frac{4601093}{10077696}a^{22}+\frac{225635}{629856}a^{21}+\frac{1338625}{1679616}a^{20}+\frac{4014515}{419904}a^{19}-\frac{6325825}{839808}a^{18}-\frac{1096243}{279936}a^{17}+\frac{1028629}{186624}a^{16}-\frac{386743}{139968}a^{15}-\frac{2592893}{839808}a^{14}-\frac{879275}{26244}a^{13}+\frac{11220635}{419904}a^{12}+\frac{63378185}{1259712}a^{11}+\frac{1424774369}{2519424}a^{10}-\frac{140673109}{314928}a^{9}-\frac{287167943}{1259712}a^{8}+\frac{126583577}{314928}a^{7}-\frac{140094157}{629856}a^{6}-\frac{53678957}{629856}a^{5}+\frac{142365229}{1259712}a^{4}-\frac{13164787}{314928}a^{3}-\frac{2008631}{157464}a^{2}+\frac{3757537}{314928}a-\frac{156289}{78732}$, $\frac{5173}{10077696}a^{26}-\frac{39409}{10077696}a^{25}-\frac{12359}{10077696}a^{24}-\frac{159577}{10077696}a^{23}+\frac{709577}{5038848}a^{22}+\frac{56599}{1259712}a^{21}+\frac{487009}{1679616}a^{20}-\frac{4941811}{1679616}a^{19}-\frac{1596581}{1679616}a^{18}+\frac{1282507}{559872}a^{17}-\frac{569201}{279936}a^{16}-\frac{7325}{139968}a^{15}-\frac{501545}{839808}a^{14}+\frac{8621867}{839808}a^{13}+\frac{2898817}{839808}a^{12}+\frac{39320245}{2519424}a^{11}-\frac{218940305}{1259712}a^{10}-\frac{1114888}{19683}a^{9}+\frac{175118929}{1259712}a^{8}-\frac{181094947}{1259712}a^{7}-\frac{13445165}{1259712}a^{6}+\frac{83038337}{1259712}a^{5}-\frac{26838175}{629856}a^{4}+\frac{1286471}{314928}a^{3}+\frac{2741513}{314928}a^{2}-\frac{662533}{157464}a+\frac{20783}{19683}$, $\frac{64615}{2519424}a^{26}-\frac{28361}{1259712}a^{25}+\frac{25393}{2519424}a^{24}-\frac{2336899}{2519424}a^{23}+\frac{4110155}{5038848}a^{22}-\frac{229987}{629856}a^{21}+\frac{8163259}{419904}a^{20}-\frac{3594095}{209952}a^{19}+\frac{402203}{52488}a^{18}+\frac{1442377}{139968}a^{17}-\frac{2221775}{279936}a^{16}+\frac{499717}{139968}a^{15}-\frac{14449307}{209952}a^{14}+\frac{12743663}{209952}a^{13}-\frac{5701703}{209952}a^{12}+\frac{724721731}{629856}a^{11}-\frac{1277357243}{1259712}a^{10}+\frac{142941833}{314928}a^{9}+\frac{240523987}{314928}a^{8}-\frac{47597743}{78732}a^{7}+\frac{42778721}{157464}a^{6}+\frac{54263999}{314928}a^{5}-\frac{79375657}{629856}a^{4}+\frac{18030989}{314928}a^{3}+\frac{1126835}{78732}a^{2}-\frac{1437397}{157464}a+\frac{314045}{78732}$ 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:  \( 54642144375.730865 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 54642144375.730865 \cdot 3}{2\cdot\sqrt{3537182715531733726396416000000000000000000}}\cr\approx \mathstrut & 1.31989278152437 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^27 - 36*x^24 + 752*x^21 + 528*x^18 - 2616*x^15 + 44416*x^12 + 37248*x^9 + 11712*x^6 + 1616*x^3 + 64)
 
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^27 - 36*x^24 + 752*x^21 + 528*x^18 - 2616*x^15 + 44416*x^12 + 37248*x^9 + 11712*x^6 + 1616*x^3 + 64, 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^27 - 36*x^24 + 752*x^21 + 528*x^18 - 2616*x^15 + 44416*x^12 + 37248*x^9 + 11712*x^6 + 1616*x^3 + 64);
 
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^27 - 36*x^24 + 752*x^21 + 528*x^18 - 2616*x^15 + 44416*x^12 + 37248*x^9 + 11712*x^6 + 1616*x^3 + 64);
 
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

$\He_3:\GL(2,3)$ (as 27T294):

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 1296
The 18 conjugacy class representatives for $\He_3:\GL(2,3)$
Character table for $\He_3:\GL(2,3)$

Intermediate fields

9.3.2239488000000.1

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 27 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 27.3.4852102490441335701504000000000000000000.1

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 R ${\href{/padicField/7.9.0.1}{9} }^{3}$ ${\href{/padicField/11.8.0.1}{8} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.9.0.1}{9} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }^{4}{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ ${\href{/padicField/23.8.0.1}{8} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.8.0.1}{8} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.12.0.1}{12} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.9.0.1}{9} }^{3}$ ${\href{/padicField/41.8.0.1}{8} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.12.0.1}{12} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.2.0.1}{2} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.8.0.1}{8} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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 2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
Deg $24$$24$$1$$50$
\(3\) Copy content Toggle raw display 3.9.9.6$x^{9} - 6 x^{8} + 45 x^{7} + 594 x^{6} + 99 x^{5} + 108 x^{4} - 54 x^{3} + 27 x^{2} + 81 x + 27$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
Deg $18$$6$$3$$21$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.6.4$x^{8} - 20 x^{4} + 50$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
5.8.6.4$x^{8} - 20 x^{4} + 50$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
5.8.6.4$x^{8} - 20 x^{4} + 50$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$