Properties

Label 27.1.149...032.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.497\times 10^{41}$
Root discriminant \(33.50\)
Ramified primes $2,563$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 7*x^26 + 21*x^25 - 50*x^24 + 110*x^23 - 244*x^22 + 734*x^21 - 2140*x^20 + 5743*x^19 - 13591*x^18 + 28791*x^17 - 54584*x^16 + 93410*x^15 - 138060*x^14 + 176010*x^13 - 189870*x^12 + 169389*x^11 - 127135*x^10 + 89765*x^9 - 72912*x^8 + 70628*x^7 - 65648*x^6 + 50256*x^5 - 29876*x^4 + 13340*x^3 - 4224*x^2 + 848*x - 80)
 
gp: K = bnfinit(y^27 - 7*y^26 + 21*y^25 - 50*y^24 + 110*y^23 - 244*y^22 + 734*y^21 - 2140*y^20 + 5743*y^19 - 13591*y^18 + 28791*y^17 - 54584*y^16 + 93410*y^15 - 138060*y^14 + 176010*y^13 - 189870*y^12 + 169389*y^11 - 127135*y^10 + 89765*y^9 - 72912*y^8 + 70628*y^7 - 65648*y^6 + 50256*y^5 - 29876*y^4 + 13340*y^3 - 4224*y^2 + 848*y - 80, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 7*x^26 + 21*x^25 - 50*x^24 + 110*x^23 - 244*x^22 + 734*x^21 - 2140*x^20 + 5743*x^19 - 13591*x^18 + 28791*x^17 - 54584*x^16 + 93410*x^15 - 138060*x^14 + 176010*x^13 - 189870*x^12 + 169389*x^11 - 127135*x^10 + 89765*x^9 - 72912*x^8 + 70628*x^7 - 65648*x^6 + 50256*x^5 - 29876*x^4 + 13340*x^3 - 4224*x^2 + 848*x - 80);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 7*x^26 + 21*x^25 - 50*x^24 + 110*x^23 - 244*x^22 + 734*x^21 - 2140*x^20 + 5743*x^19 - 13591*x^18 + 28791*x^17 - 54584*x^16 + 93410*x^15 - 138060*x^14 + 176010*x^13 - 189870*x^12 + 169389*x^11 - 127135*x^10 + 89765*x^9 - 72912*x^8 + 70628*x^7 - 65648*x^6 + 50256*x^5 - 29876*x^4 + 13340*x^3 - 4224*x^2 + 848*x - 80)
 

\( x^{27} - 7 x^{26} + 21 x^{25} - 50 x^{24} + 110 x^{23} - 244 x^{22} + 734 x^{21} - 2140 x^{20} + 5743 x^{19} - 13591 x^{18} + 28791 x^{17} - 54584 x^{16} + 93410 x^{15} - 138060 x^{14} + \cdots - 80 \) 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:  $[1, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-149674927005884133619412112407487159468032\) \(\medspace = -\,2^{18}\cdot 563^{13}\) 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:  \(33.50\)
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:  $2^{2/3}563^{1/2}\approx 37.66525059231157$
Ramified primes:   \(2\), \(563\) 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:  \(\Q(\sqrt{-563}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{20}a^{17}+\frac{1}{10}a^{16}-\frac{1}{20}a^{15}-\frac{1}{10}a^{14}+\frac{1}{5}a^{13}-\frac{1}{10}a^{12}-\frac{1}{20}a^{11}-\frac{1}{5}a^{10}-\frac{1}{10}a^{8}-\frac{1}{4}a^{7}-\frac{1}{20}a^{5}-\frac{2}{5}a^{4}+\frac{1}{10}a^{2}-\frac{2}{5}a$, $\frac{1}{20}a^{18}-\frac{1}{10}a^{14}-\frac{1}{4}a^{13}-\frac{1}{10}a^{12}-\frac{1}{10}a^{11}-\frac{1}{10}a^{10}+\frac{3}{20}a^{9}-\frac{3}{10}a^{8}-\frac{1}{20}a^{6}+\frac{9}{20}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{10}a^{2}-\frac{1}{5}a$, $\frac{1}{20}a^{19}-\frac{1}{10}a^{15}-\frac{1}{4}a^{14}-\frac{1}{10}a^{13}-\frac{1}{10}a^{12}-\frac{1}{10}a^{11}+\frac{3}{20}a^{10}+\frac{1}{5}a^{9}-\frac{1}{2}a^{8}-\frac{1}{20}a^{7}-\frac{1}{20}a^{6}-\frac{1}{5}a^{5}+\frac{1}{10}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{40}a^{20}-\frac{1}{40}a^{19}-\frac{1}{40}a^{17}+\frac{1}{40}a^{16}-\frac{1}{20}a^{15}+\frac{1}{8}a^{14}-\frac{9}{40}a^{13}-\frac{3}{40}a^{12}-\frac{1}{10}a^{11}+\frac{1}{8}a^{10}-\frac{9}{40}a^{9}-\frac{1}{10}a^{8}+\frac{1}{8}a^{7}+\frac{17}{40}a^{6}+\frac{3}{10}a^{5}+\frac{1}{10}a^{4}+\frac{9}{20}a^{3}+\frac{3}{10}a^{2}+\frac{1}{5}a$, $\frac{1}{40}a^{21}-\frac{1}{40}a^{19}-\frac{1}{40}a^{18}-\frac{1}{40}a^{16}+\frac{3}{40}a^{15}-\frac{1}{10}a^{14}+\frac{1}{5}a^{13}-\frac{7}{40}a^{12}+\frac{1}{40}a^{11}-\frac{1}{10}a^{10}+\frac{7}{40}a^{9}+\frac{1}{40}a^{8}-\frac{9}{20}a^{7}-\frac{11}{40}a^{6}-\frac{1}{10}a^{5}-\frac{9}{20}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{5}a$, $\frac{1}{40}a^{22}-\frac{1}{20}a^{16}-\frac{1}{20}a^{15}+\frac{9}{40}a^{14}+\frac{1}{5}a^{13}-\frac{1}{10}a^{11}+\frac{1}{10}a^{8}-\frac{1}{5}a^{7}-\frac{19}{40}a^{6}+\frac{1}{10}a^{5}-\frac{9}{20}a^{4}-\frac{3}{20}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{40}a^{23}+\frac{1}{20}a^{16}-\frac{3}{40}a^{15}-\frac{3}{20}a^{14}-\frac{1}{20}a^{13}-\frac{1}{5}a^{12}+\frac{1}{5}a^{11}+\frac{1}{20}a^{10}-\frac{3}{20}a^{9}+\frac{1}{5}a^{8}+\frac{1}{40}a^{7}-\frac{3}{20}a^{6}-\frac{1}{4}a^{5}-\frac{1}{20}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{40}a^{24}+\frac{3}{40}a^{16}-\frac{1}{10}a^{15}+\frac{1}{20}a^{14}-\frac{3}{20}a^{13}+\frac{1}{20}a^{12}+\frac{1}{10}a^{11}+\frac{1}{20}a^{10}-\frac{1}{20}a^{9}+\frac{3}{8}a^{8}-\frac{2}{5}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{3}{10}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a$, $\frac{1}{973400}a^{25}-\frac{137}{97340}a^{24}-\frac{737}{97340}a^{23}-\frac{4}{785}a^{22}+\frac{489}{97340}a^{21}-\frac{5739}{973400}a^{20}-\frac{14249}{973400}a^{19}+\frac{2012}{121675}a^{18}+\frac{77}{15700}a^{17}+\frac{621}{31400}a^{16}-\frac{4869}{243350}a^{15}-\frac{173167}{973400}a^{14}+\frac{4707}{31400}a^{13}+\frac{160601}{973400}a^{12}+\frac{277}{24335}a^{11}-\frac{209751}{973400}a^{10}+\frac{27823}{243350}a^{9}-\frac{2227}{19468}a^{8}+\frac{120443}{973400}a^{7}-\frac{325971}{973400}a^{6}-\frac{26279}{121675}a^{5}+\frac{2821}{7850}a^{4}+\frac{177613}{486700}a^{3}-\frac{40787}{243350}a^{2}-\frac{56989}{121675}a+\frac{5748}{24335}$, $\frac{1}{36\!\cdots\!00}a^{26}-\frac{71\!\cdots\!12}{14\!\cdots\!75}a^{25}-\frac{55\!\cdots\!87}{73\!\cdots\!00}a^{24}-\frac{69\!\cdots\!11}{36\!\cdots\!00}a^{23}-\frac{79\!\cdots\!91}{14\!\cdots\!80}a^{22}-\frac{50\!\cdots\!68}{45\!\cdots\!25}a^{21}+\frac{20\!\cdots\!09}{73\!\cdots\!00}a^{20}+\frac{12\!\cdots\!01}{73\!\cdots\!00}a^{19}+\frac{31\!\cdots\!23}{36\!\cdots\!00}a^{18}+\frac{11\!\cdots\!87}{11\!\cdots\!00}a^{17}-\frac{22\!\cdots\!01}{18\!\cdots\!00}a^{16}+\frac{18\!\cdots\!77}{18\!\cdots\!00}a^{15}-\frac{56\!\cdots\!29}{91\!\cdots\!50}a^{14}+\frac{42\!\cdots\!19}{36\!\cdots\!00}a^{13}+\frac{50\!\cdots\!99}{36\!\cdots\!00}a^{12}+\frac{74\!\cdots\!81}{91\!\cdots\!50}a^{11}+\frac{94\!\cdots\!26}{45\!\cdots\!25}a^{10}+\frac{47\!\cdots\!63}{36\!\cdots\!00}a^{9}-\frac{95\!\cdots\!57}{36\!\cdots\!00}a^{8}-\frac{44\!\cdots\!79}{36\!\cdots\!00}a^{7}+\frac{21\!\cdots\!77}{18\!\cdots\!00}a^{6}-\frac{62\!\cdots\!87}{18\!\cdots\!00}a^{5}-\frac{43\!\cdots\!69}{18\!\cdots\!00}a^{4}-\frac{12\!\cdots\!51}{91\!\cdots\!50}a^{3}-\frac{31\!\cdots\!71}{91\!\cdots\!50}a^{2}-\frac{21\!\cdots\!16}{45\!\cdots\!25}a-\frac{44\!\cdots\!33}{91\!\cdots\!25}$ 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:  $2$, $5$

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:   $\frac{46\!\cdots\!47}{36\!\cdots\!00}a^{26}-\frac{29\!\cdots\!37}{36\!\cdots\!00}a^{25}+\frac{19\!\cdots\!32}{91\!\cdots\!25}a^{24}-\frac{17\!\cdots\!77}{36\!\cdots\!00}a^{23}+\frac{29\!\cdots\!81}{29\!\cdots\!16}a^{22}-\frac{83\!\cdots\!93}{36\!\cdots\!00}a^{21}+\frac{10\!\cdots\!27}{14\!\cdots\!80}a^{20}-\frac{78\!\cdots\!73}{36\!\cdots\!00}a^{19}+\frac{25\!\cdots\!77}{45\!\cdots\!25}a^{18}-\frac{18\!\cdots\!87}{14\!\cdots\!75}a^{17}+\frac{48\!\cdots\!83}{18\!\cdots\!00}a^{16}-\frac{17\!\cdots\!97}{36\!\cdots\!00}a^{15}+\frac{14\!\cdots\!89}{18\!\cdots\!00}a^{14}-\frac{40\!\cdots\!87}{36\!\cdots\!00}a^{13}+\frac{23\!\cdots\!19}{18\!\cdots\!00}a^{12}-\frac{46\!\cdots\!97}{36\!\cdots\!00}a^{11}+\frac{45\!\cdots\!52}{45\!\cdots\!25}a^{10}-\frac{15\!\cdots\!67}{23\!\cdots\!00}a^{9}+\frac{17\!\cdots\!71}{36\!\cdots\!00}a^{8}-\frac{16\!\cdots\!33}{36\!\cdots\!00}a^{7}+\frac{56\!\cdots\!63}{11\!\cdots\!00}a^{6}-\frac{37\!\cdots\!37}{91\!\cdots\!50}a^{5}+\frac{11\!\cdots\!88}{45\!\cdots\!25}a^{4}-\frac{70\!\cdots\!69}{59\!\cdots\!00}a^{3}+\frac{17\!\cdots\!09}{45\!\cdots\!25}a^{2}-\frac{29\!\cdots\!67}{45\!\cdots\!25}a+\frac{26\!\cdots\!54}{91\!\cdots\!25}$, $\frac{11\!\cdots\!73}{73\!\cdots\!00}a^{26}-\frac{24\!\cdots\!73}{23\!\cdots\!40}a^{25}+\frac{20\!\cdots\!11}{73\!\cdots\!40}a^{24}-\frac{94\!\cdots\!31}{14\!\cdots\!80}a^{23}+\frac{20\!\cdots\!27}{14\!\cdots\!80}a^{22}-\frac{22\!\cdots\!47}{73\!\cdots\!00}a^{21}+\frac{72\!\cdots\!53}{73\!\cdots\!00}a^{20}-\frac{26\!\cdots\!24}{91\!\cdots\!25}a^{19}+\frac{13\!\cdots\!53}{18\!\cdots\!50}a^{18}-\frac{41\!\cdots\!47}{23\!\cdots\!00}a^{17}+\frac{33\!\cdots\!64}{91\!\cdots\!25}a^{16}-\frac{24\!\cdots\!23}{36\!\cdots\!00}a^{15}+\frac{20\!\cdots\!49}{18\!\cdots\!50}a^{14}-\frac{11\!\cdots\!07}{73\!\cdots\!00}a^{13}+\frac{71\!\cdots\!53}{36\!\cdots\!70}a^{12}-\frac{14\!\cdots\!53}{73\!\cdots\!00}a^{11}+\frac{59\!\cdots\!13}{36\!\cdots\!00}a^{10}-\frac{82\!\cdots\!81}{73\!\cdots\!40}a^{9}+\frac{58\!\cdots\!69}{73\!\cdots\!00}a^{8}-\frac{13\!\cdots\!07}{18\!\cdots\!50}a^{7}+\frac{53\!\cdots\!49}{73\!\cdots\!00}a^{6}-\frac{59\!\cdots\!46}{91\!\cdots\!25}a^{5}+\frac{16\!\cdots\!29}{36\!\cdots\!00}a^{4}-\frac{82\!\cdots\!37}{36\!\cdots\!00}a^{3}+\frac{14\!\cdots\!71}{18\!\cdots\!50}a^{2}-\frac{33\!\cdots\!88}{18\!\cdots\!85}a+\frac{64\!\cdots\!20}{36\!\cdots\!77}$, $\frac{56\!\cdots\!77}{36\!\cdots\!00}a^{26}-\frac{18\!\cdots\!21}{18\!\cdots\!00}a^{25}+\frac{25\!\cdots\!97}{91\!\cdots\!25}a^{24}-\frac{23\!\cdots\!27}{36\!\cdots\!00}a^{23}+\frac{41\!\cdots\!37}{29\!\cdots\!16}a^{22}-\frac{11\!\cdots\!63}{36\!\cdots\!00}a^{21}+\frac{14\!\cdots\!87}{14\!\cdots\!80}a^{20}-\frac{10\!\cdots\!73}{36\!\cdots\!00}a^{19}+\frac{13\!\cdots\!03}{18\!\cdots\!00}a^{18}-\frac{20\!\cdots\!11}{11\!\cdots\!00}a^{17}+\frac{33\!\cdots\!89}{91\!\cdots\!50}a^{16}-\frac{24\!\cdots\!27}{36\!\cdots\!00}a^{15}+\frac{20\!\cdots\!49}{18\!\cdots\!00}a^{14}-\frac{59\!\cdots\!67}{36\!\cdots\!00}a^{13}+\frac{58\!\cdots\!17}{29\!\cdots\!50}a^{12}-\frac{74\!\cdots\!27}{36\!\cdots\!00}a^{11}+\frac{30\!\cdots\!03}{18\!\cdots\!00}a^{10}-\frac{21\!\cdots\!27}{18\!\cdots\!00}a^{9}+\frac{30\!\cdots\!61}{36\!\cdots\!00}a^{8}-\frac{26\!\cdots\!03}{36\!\cdots\!00}a^{7}+\frac{27\!\cdots\!73}{36\!\cdots\!00}a^{6}-\frac{30\!\cdots\!71}{45\!\cdots\!25}a^{5}+\frac{21\!\cdots\!58}{45\!\cdots\!25}a^{4}-\frac{43\!\cdots\!99}{18\!\cdots\!00}a^{3}+\frac{40\!\cdots\!69}{45\!\cdots\!25}a^{2}-\frac{96\!\cdots\!97}{45\!\cdots\!25}a+\frac{22\!\cdots\!64}{91\!\cdots\!25}$, $\frac{22\!\cdots\!83}{36\!\cdots\!00}a^{26}-\frac{76\!\cdots\!69}{18\!\cdots\!00}a^{25}+\frac{42\!\cdots\!57}{36\!\cdots\!00}a^{24}-\frac{19\!\cdots\!81}{73\!\cdots\!00}a^{23}+\frac{86\!\cdots\!63}{14\!\cdots\!80}a^{22}-\frac{48\!\cdots\!77}{36\!\cdots\!00}a^{21}+\frac{30\!\cdots\!61}{73\!\cdots\!00}a^{20}-\frac{43\!\cdots\!09}{36\!\cdots\!00}a^{19}+\frac{14\!\cdots\!13}{45\!\cdots\!25}a^{18}-\frac{87\!\cdots\!49}{11\!\cdots\!00}a^{17}+\frac{70\!\cdots\!88}{45\!\cdots\!25}a^{16}-\frac{26\!\cdots\!47}{91\!\cdots\!50}a^{15}+\frac{22\!\cdots\!54}{45\!\cdots\!25}a^{14}-\frac{25\!\cdots\!33}{36\!\cdots\!00}a^{13}+\frac{77\!\cdots\!53}{91\!\cdots\!50}a^{12}-\frac{32\!\cdots\!33}{36\!\cdots\!00}a^{11}+\frac{33\!\cdots\!18}{45\!\cdots\!25}a^{10}-\frac{94\!\cdots\!03}{18\!\cdots\!00}a^{9}+\frac{13\!\cdots\!69}{36\!\cdots\!00}a^{8}-\frac{29\!\cdots\!43}{91\!\cdots\!50}a^{7}+\frac{11\!\cdots\!87}{36\!\cdots\!00}a^{6}-\frac{13\!\cdots\!04}{45\!\cdots\!25}a^{5}+\frac{37\!\cdots\!13}{18\!\cdots\!00}a^{4}-\frac{19\!\cdots\!31}{18\!\cdots\!00}a^{3}+\frac{37\!\cdots\!17}{91\!\cdots\!50}a^{2}-\frac{44\!\cdots\!58}{45\!\cdots\!25}a+\frac{10\!\cdots\!96}{91\!\cdots\!25}$, $\frac{25\!\cdots\!27}{36\!\cdots\!00}a^{26}-\frac{58\!\cdots\!71}{18\!\cdots\!00}a^{25}+\frac{31\!\cdots\!91}{73\!\cdots\!00}a^{24}-\frac{63\!\cdots\!09}{73\!\cdots\!00}a^{23}+\frac{31\!\cdots\!39}{18\!\cdots\!85}a^{22}-\frac{15\!\cdots\!13}{36\!\cdots\!00}a^{21}+\frac{16\!\cdots\!39}{73\!\cdots\!40}a^{20}-\frac{39\!\cdots\!51}{73\!\cdots\!00}a^{19}+\frac{23\!\cdots\!53}{18\!\cdots\!00}a^{18}-\frac{13\!\cdots\!93}{59\!\cdots\!00}a^{17}+\frac{71\!\cdots\!03}{18\!\cdots\!00}a^{16}-\frac{30\!\cdots\!71}{59\!\cdots\!00}a^{15}+\frac{24\!\cdots\!31}{45\!\cdots\!25}a^{14}+\frac{35\!\cdots\!29}{18\!\cdots\!00}a^{13}-\frac{42\!\cdots\!17}{36\!\cdots\!00}a^{12}+\frac{93\!\cdots\!73}{36\!\cdots\!00}a^{11}-\frac{12\!\cdots\!69}{36\!\cdots\!00}a^{10}+\frac{10\!\cdots\!21}{36\!\cdots\!00}a^{9}-\frac{22\!\cdots\!18}{14\!\cdots\!75}a^{8}+\frac{25\!\cdots\!47}{36\!\cdots\!00}a^{7}-\frac{26\!\cdots\!27}{36\!\cdots\!00}a^{6}+\frac{22\!\cdots\!41}{18\!\cdots\!00}a^{5}-\frac{58\!\cdots\!67}{45\!\cdots\!25}a^{4}+\frac{16\!\cdots\!01}{18\!\cdots\!00}a^{3}-\frac{19\!\cdots\!06}{45\!\cdots\!25}a^{2}+\frac{59\!\cdots\!78}{45\!\cdots\!25}a-\frac{16\!\cdots\!36}{91\!\cdots\!25}$, $\frac{44\!\cdots\!97}{59\!\cdots\!00}a^{26}-\frac{23\!\cdots\!23}{45\!\cdots\!25}a^{25}+\frac{13\!\cdots\!59}{91\!\cdots\!25}a^{24}-\frac{23\!\cdots\!93}{73\!\cdots\!00}a^{23}+\frac{41\!\cdots\!64}{59\!\cdots\!35}a^{22}-\frac{57\!\cdots\!41}{36\!\cdots\!00}a^{21}+\frac{45\!\cdots\!69}{91\!\cdots\!25}a^{20}-\frac{21\!\cdots\!47}{14\!\cdots\!80}a^{19}+\frac{13\!\cdots\!27}{36\!\cdots\!00}a^{18}-\frac{13\!\cdots\!89}{14\!\cdots\!75}a^{17}+\frac{21\!\cdots\!67}{11\!\cdots\!00}a^{16}-\frac{31\!\cdots\!81}{91\!\cdots\!50}a^{15}+\frac{10\!\cdots\!33}{18\!\cdots\!00}a^{14}-\frac{11\!\cdots\!13}{14\!\cdots\!75}a^{13}+\frac{35\!\cdots\!41}{36\!\cdots\!00}a^{12}-\frac{35\!\cdots\!89}{36\!\cdots\!00}a^{11}+\frac{71\!\cdots\!83}{91\!\cdots\!50}a^{10}-\frac{19\!\cdots\!83}{36\!\cdots\!00}a^{9}+\frac{13\!\cdots\!77}{36\!\cdots\!00}a^{8}-\frac{12\!\cdots\!91}{36\!\cdots\!00}a^{7}+\frac{13\!\cdots\!01}{36\!\cdots\!00}a^{6}-\frac{29\!\cdots\!49}{91\!\cdots\!50}a^{5}+\frac{30\!\cdots\!06}{14\!\cdots\!75}a^{4}-\frac{18\!\cdots\!63}{18\!\cdots\!00}a^{3}+\frac{15\!\cdots\!48}{45\!\cdots\!25}a^{2}-\frac{32\!\cdots\!44}{45\!\cdots\!25}a+\frac{57\!\cdots\!78}{91\!\cdots\!25}$, $\frac{29\!\cdots\!57}{36\!\cdots\!00}a^{26}-\frac{19\!\cdots\!47}{36\!\cdots\!00}a^{25}+\frac{25\!\cdots\!49}{18\!\cdots\!50}a^{24}-\frac{23\!\cdots\!79}{73\!\cdots\!00}a^{23}+\frac{25\!\cdots\!09}{36\!\cdots\!70}a^{22}-\frac{56\!\cdots\!83}{36\!\cdots\!00}a^{21}+\frac{11\!\cdots\!11}{23\!\cdots\!40}a^{20}-\frac{10\!\cdots\!11}{73\!\cdots\!00}a^{19}+\frac{68\!\cdots\!23}{18\!\cdots\!00}a^{18}-\frac{10\!\cdots\!51}{11\!\cdots\!00}a^{17}+\frac{65\!\cdots\!21}{36\!\cdots\!00}a^{16}-\frac{59\!\cdots\!91}{18\!\cdots\!00}a^{15}+\frac{49\!\cdots\!67}{91\!\cdots\!50}a^{14}-\frac{13\!\cdots\!61}{18\!\cdots\!00}a^{13}+\frac{33\!\cdots\!53}{36\!\cdots\!00}a^{12}-\frac{33\!\cdots\!57}{36\!\cdots\!00}a^{11}+\frac{26\!\cdots\!21}{36\!\cdots\!00}a^{10}-\frac{91\!\cdots\!57}{18\!\cdots\!00}a^{9}+\frac{13\!\cdots\!51}{36\!\cdots\!00}a^{8}-\frac{12\!\cdots\!73}{36\!\cdots\!00}a^{7}+\frac{12\!\cdots\!43}{36\!\cdots\!00}a^{6}-\frac{26\!\cdots\!47}{91\!\cdots\!50}a^{5}+\frac{17\!\cdots\!81}{91\!\cdots\!50}a^{4}-\frac{17\!\cdots\!59}{18\!\cdots\!00}a^{3}+\frac{15\!\cdots\!04}{45\!\cdots\!25}a^{2}-\frac{32\!\cdots\!27}{45\!\cdots\!25}a+\frac{68\!\cdots\!24}{91\!\cdots\!25}$, $\frac{21\!\cdots\!03}{36\!\cdots\!00}a^{26}-\frac{33\!\cdots\!57}{73\!\cdots\!00}a^{25}+\frac{27\!\cdots\!02}{18\!\cdots\!85}a^{24}-\frac{26\!\cdots\!19}{73\!\cdots\!40}a^{23}+\frac{14\!\cdots\!48}{18\!\cdots\!85}a^{22}-\frac{12\!\cdots\!39}{73\!\cdots\!00}a^{21}+\frac{90\!\cdots\!01}{18\!\cdots\!50}a^{20}-\frac{11\!\cdots\!91}{73\!\cdots\!00}a^{19}+\frac{29\!\cdots\!57}{73\!\cdots\!00}a^{18}-\frac{22\!\cdots\!97}{23\!\cdots\!00}a^{17}+\frac{15\!\cdots\!17}{73\!\cdots\!00}a^{16}-\frac{18\!\cdots\!61}{47\!\cdots\!80}a^{15}+\frac{24\!\cdots\!73}{36\!\cdots\!00}a^{14}-\frac{37\!\cdots\!19}{36\!\cdots\!00}a^{13}+\frac{94\!\cdots\!43}{73\!\cdots\!00}a^{12}-\frac{10\!\cdots\!31}{73\!\cdots\!00}a^{11}+\frac{11\!\cdots\!18}{91\!\cdots\!25}a^{10}-\frac{79\!\cdots\!88}{91\!\cdots\!25}a^{9}+\frac{13\!\cdots\!53}{23\!\cdots\!00}a^{8}-\frac{17\!\cdots\!51}{36\!\cdots\!00}a^{7}+\frac{73\!\cdots\!27}{14\!\cdots\!80}a^{6}-\frac{43\!\cdots\!64}{91\!\cdots\!25}a^{5}+\frac{12\!\cdots\!29}{36\!\cdots\!00}a^{4}-\frac{28\!\cdots\!67}{14\!\cdots\!08}a^{3}+\frac{14\!\cdots\!71}{18\!\cdots\!50}a^{2}-\frac{17\!\cdots\!12}{91\!\cdots\!25}a+\frac{41\!\cdots\!84}{18\!\cdots\!85}$, $\frac{51\!\cdots\!23}{21\!\cdots\!50}a^{26}-\frac{12\!\cdots\!57}{85\!\cdots\!00}a^{25}+\frac{59\!\cdots\!41}{17\!\cdots\!00}a^{24}-\frac{13\!\cdots\!99}{17\!\cdots\!00}a^{23}+\frac{59\!\cdots\!71}{34\!\cdots\!60}a^{22}-\frac{82\!\cdots\!87}{21\!\cdots\!50}a^{21}+\frac{90\!\cdots\!19}{68\!\cdots\!12}a^{20}-\frac{20\!\cdots\!41}{55\!\cdots\!00}a^{19}+\frac{41\!\cdots\!63}{42\!\cdots\!00}a^{18}-\frac{30\!\cdots\!03}{13\!\cdots\!00}a^{17}+\frac{18\!\cdots\!63}{42\!\cdots\!00}a^{16}-\frac{68\!\cdots\!67}{85\!\cdots\!00}a^{15}+\frac{56\!\cdots\!79}{42\!\cdots\!00}a^{14}-\frac{15\!\cdots\!57}{85\!\cdots\!00}a^{13}+\frac{17\!\cdots\!93}{85\!\cdots\!00}a^{12}-\frac{42\!\cdots\!23}{21\!\cdots\!50}a^{11}+\frac{13\!\cdots\!51}{85\!\cdots\!00}a^{10}-\frac{44\!\cdots\!17}{42\!\cdots\!00}a^{9}+\frac{66\!\cdots\!31}{85\!\cdots\!00}a^{8}-\frac{80\!\cdots\!36}{10\!\cdots\!75}a^{7}+\frac{81\!\cdots\!51}{10\!\cdots\!75}a^{6}-\frac{66\!\cdots\!41}{10\!\cdots\!75}a^{5}+\frac{42\!\cdots\!68}{10\!\cdots\!75}a^{4}-\frac{19\!\cdots\!01}{10\!\cdots\!75}a^{3}+\frac{12\!\cdots\!73}{21\!\cdots\!50}a^{2}-\frac{12\!\cdots\!12}{10\!\cdots\!75}a+\frac{26\!\cdots\!69}{21\!\cdots\!75}$, $\frac{84\!\cdots\!33}{44\!\cdots\!00}a^{26}-\frac{55\!\cdots\!23}{44\!\cdots\!00}a^{25}+\frac{76\!\cdots\!81}{22\!\cdots\!50}a^{24}-\frac{35\!\cdots\!83}{44\!\cdots\!00}a^{23}+\frac{30\!\cdots\!13}{17\!\cdots\!60}a^{22}-\frac{17\!\cdots\!27}{44\!\cdots\!00}a^{21}+\frac{10\!\cdots\!19}{88\!\cdots\!00}a^{20}-\frac{31\!\cdots\!11}{88\!\cdots\!80}a^{19}+\frac{20\!\cdots\!47}{22\!\cdots\!00}a^{18}-\frac{38\!\cdots\!58}{17\!\cdots\!25}a^{17}+\frac{49\!\cdots\!61}{11\!\cdots\!50}a^{16}-\frac{36\!\cdots\!53}{44\!\cdots\!00}a^{15}+\frac{15\!\cdots\!63}{11\!\cdots\!50}a^{14}-\frac{87\!\cdots\!53}{44\!\cdots\!00}a^{13}+\frac{13\!\cdots\!69}{55\!\cdots\!75}a^{12}-\frac{10\!\cdots\!33}{44\!\cdots\!00}a^{11}+\frac{22\!\cdots\!51}{11\!\cdots\!50}a^{10}-\frac{62\!\cdots\!01}{44\!\cdots\!00}a^{9}+\frac{44\!\cdots\!69}{44\!\cdots\!00}a^{8}-\frac{39\!\cdots\!77}{44\!\cdots\!00}a^{7}+\frac{40\!\cdots\!47}{44\!\cdots\!00}a^{6}-\frac{44\!\cdots\!89}{55\!\cdots\!75}a^{5}+\frac{30\!\cdots\!67}{55\!\cdots\!75}a^{4}-\frac{63\!\cdots\!11}{22\!\cdots\!00}a^{3}+\frac{57\!\cdots\!56}{55\!\cdots\!75}a^{2}-\frac{13\!\cdots\!68}{55\!\cdots\!75}a+\frac{28\!\cdots\!66}{11\!\cdots\!75}$, $\frac{55\!\cdots\!19}{44\!\cdots\!00}a^{26}-\frac{18\!\cdots\!57}{22\!\cdots\!00}a^{25}+\frac{97\!\cdots\!41}{44\!\cdots\!00}a^{24}-\frac{56\!\cdots\!01}{11\!\cdots\!75}a^{23}+\frac{19\!\cdots\!03}{17\!\cdots\!60}a^{22}-\frac{10\!\cdots\!11}{44\!\cdots\!00}a^{21}+\frac{69\!\cdots\!47}{88\!\cdots\!00}a^{20}-\frac{50\!\cdots\!17}{22\!\cdots\!95}a^{19}+\frac{13\!\cdots\!21}{22\!\cdots\!00}a^{18}-\frac{19\!\cdots\!27}{14\!\cdots\!00}a^{17}+\frac{63\!\cdots\!21}{22\!\cdots\!00}a^{16}-\frac{23\!\cdots\!79}{44\!\cdots\!00}a^{15}+\frac{48\!\cdots\!42}{55\!\cdots\!75}a^{14}-\frac{55\!\cdots\!79}{44\!\cdots\!00}a^{13}+\frac{84\!\cdots\!42}{55\!\cdots\!75}a^{12}-\frac{68\!\cdots\!69}{44\!\cdots\!00}a^{11}+\frac{69\!\cdots\!84}{55\!\cdots\!75}a^{10}-\frac{12\!\cdots\!37}{14\!\cdots\!00}a^{9}+\frac{27\!\cdots\!67}{44\!\cdots\!00}a^{8}-\frac{24\!\cdots\!61}{44\!\cdots\!00}a^{7}+\frac{25\!\cdots\!21}{44\!\cdots\!00}a^{6}-\frac{55\!\cdots\!79}{11\!\cdots\!50}a^{5}+\frac{18\!\cdots\!31}{55\!\cdots\!75}a^{4}-\frac{38\!\cdots\!73}{22\!\cdots\!00}a^{3}+\frac{33\!\cdots\!58}{55\!\cdots\!75}a^{2}-\frac{74\!\cdots\!24}{55\!\cdots\!75}a+\frac{15\!\cdots\!88}{11\!\cdots\!75}$, $\frac{57\!\cdots\!37}{22\!\cdots\!00}a^{26}-\frac{13\!\cdots\!81}{11\!\cdots\!50}a^{25}+\frac{17\!\cdots\!19}{11\!\cdots\!75}a^{24}-\frac{26\!\cdots\!63}{88\!\cdots\!00}a^{23}+\frac{10\!\cdots\!11}{17\!\cdots\!60}a^{22}-\frac{30\!\cdots\!03}{22\!\cdots\!00}a^{21}+\frac{70\!\cdots\!21}{88\!\cdots\!00}a^{20}-\frac{17\!\cdots\!31}{88\!\cdots\!00}a^{19}+\frac{10\!\cdots\!51}{22\!\cdots\!00}a^{18}-\frac{11\!\cdots\!87}{14\!\cdots\!00}a^{17}+\frac{55\!\cdots\!27}{44\!\cdots\!00}a^{16}-\frac{61\!\cdots\!29}{44\!\cdots\!00}a^{15}+\frac{19\!\cdots\!33}{22\!\cdots\!00}a^{14}+\frac{12\!\cdots\!31}{44\!\cdots\!00}a^{13}-\frac{35\!\cdots\!49}{44\!\cdots\!00}a^{12}+\frac{32\!\cdots\!13}{22\!\cdots\!00}a^{11}-\frac{84\!\cdots\!83}{44\!\cdots\!00}a^{10}+\frac{74\!\cdots\!37}{44\!\cdots\!00}a^{9}-\frac{22\!\cdots\!09}{22\!\cdots\!00}a^{8}+\frac{11\!\cdots\!27}{22\!\cdots\!00}a^{7}-\frac{97\!\cdots\!77}{22\!\cdots\!00}a^{6}+\frac{73\!\cdots\!31}{11\!\cdots\!50}a^{5}-\frac{79\!\cdots\!53}{11\!\cdots\!50}a^{4}+\frac{29\!\cdots\!88}{55\!\cdots\!75}a^{3}-\frac{14\!\cdots\!02}{55\!\cdots\!75}a^{2}+\frac{46\!\cdots\!66}{55\!\cdots\!75}a-\frac{13\!\cdots\!17}{11\!\cdots\!75}$, $\frac{21\!\cdots\!43}{44\!\cdots\!00}a^{26}-\frac{14\!\cdots\!73}{44\!\cdots\!00}a^{25}+\frac{19\!\cdots\!11}{22\!\cdots\!50}a^{24}-\frac{18\!\cdots\!81}{88\!\cdots\!00}a^{23}+\frac{19\!\cdots\!03}{44\!\cdots\!90}a^{22}-\frac{44\!\cdots\!17}{44\!\cdots\!00}a^{21}+\frac{34\!\cdots\!17}{11\!\cdots\!75}a^{20}-\frac{80\!\cdots\!33}{88\!\cdots\!00}a^{19}+\frac{13\!\cdots\!98}{55\!\cdots\!75}a^{18}-\frac{79\!\cdots\!29}{14\!\cdots\!00}a^{17}+\frac{51\!\cdots\!59}{44\!\cdots\!00}a^{16}-\frac{47\!\cdots\!49}{22\!\cdots\!00}a^{15}+\frac{80\!\cdots\!61}{22\!\cdots\!00}a^{14}-\frac{11\!\cdots\!59}{22\!\cdots\!00}a^{13}+\frac{27\!\cdots\!27}{44\!\cdots\!00}a^{12}-\frac{28\!\cdots\!43}{44\!\cdots\!00}a^{11}+\frac{23\!\cdots\!49}{44\!\cdots\!00}a^{10}-\frac{53\!\cdots\!59}{14\!\cdots\!00}a^{9}+\frac{11\!\cdots\!99}{44\!\cdots\!00}a^{8}-\frac{10\!\cdots\!87}{44\!\cdots\!00}a^{7}+\frac{10\!\cdots\!77}{44\!\cdots\!00}a^{6}-\frac{23\!\cdots\!93}{11\!\cdots\!50}a^{5}+\frac{16\!\cdots\!49}{11\!\cdots\!50}a^{4}-\frac{17\!\cdots\!51}{22\!\cdots\!00}a^{3}+\frac{16\!\cdots\!66}{55\!\cdots\!75}a^{2}-\frac{38\!\cdots\!18}{55\!\cdots\!75}a+\frac{88\!\cdots\!16}{11\!\cdots\!75}$ 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:  \( 14887194175.434155 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 14887194175.434155 \cdot 2}{2\cdot\sqrt{149674927005884133619412112407487159468032}}\cr\approx \mathstrut & 1.83065543691759 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^27 - 7*x^26 + 21*x^25 - 50*x^24 + 110*x^23 - 244*x^22 + 734*x^21 - 2140*x^20 + 5743*x^19 - 13591*x^18 + 28791*x^17 - 54584*x^16 + 93410*x^15 - 138060*x^14 + 176010*x^13 - 189870*x^12 + 169389*x^11 - 127135*x^10 + 89765*x^9 - 72912*x^8 + 70628*x^7 - 65648*x^6 + 50256*x^5 - 29876*x^4 + 13340*x^3 - 4224*x^2 + 848*x - 80)
 
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 - 7*x^26 + 21*x^25 - 50*x^24 + 110*x^23 - 244*x^22 + 734*x^21 - 2140*x^20 + 5743*x^19 - 13591*x^18 + 28791*x^17 - 54584*x^16 + 93410*x^15 - 138060*x^14 + 176010*x^13 - 189870*x^12 + 169389*x^11 - 127135*x^10 + 89765*x^9 - 72912*x^8 + 70628*x^7 - 65648*x^6 + 50256*x^5 - 29876*x^4 + 13340*x^3 - 4224*x^2 + 848*x - 80, 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 - 7*x^26 + 21*x^25 - 50*x^24 + 110*x^23 - 244*x^22 + 734*x^21 - 2140*x^20 + 5743*x^19 - 13591*x^18 + 28791*x^17 - 54584*x^16 + 93410*x^15 - 138060*x^14 + 176010*x^13 - 189870*x^12 + 169389*x^11 - 127135*x^10 + 89765*x^9 - 72912*x^8 + 70628*x^7 - 65648*x^6 + 50256*x^5 - 29876*x^4 + 13340*x^3 - 4224*x^2 + 848*x - 80);
 
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 - 7*x^26 + 21*x^25 - 50*x^24 + 110*x^23 - 244*x^22 + 734*x^21 - 2140*x^20 + 5743*x^19 - 13591*x^18 + 28791*x^17 - 54584*x^16 + 93410*x^15 - 138060*x^14 + 176010*x^13 - 189870*x^12 + 169389*x^11 - 127135*x^10 + 89765*x^9 - 72912*x^8 + 70628*x^7 - 65648*x^6 + 50256*x^5 - 29876*x^4 + 13340*x^3 - 4224*x^2 + 848*x - 80);
 
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

$D_{27}$ (as 27T8):

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 54
The 15 conjugacy class representatives for $D_{27}$
Character table for $D_{27}$

Intermediate fields

3.1.563.1, 9.1.100469346961.1

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 $27$ ${\href{/padicField/5.2.0.1}{2} }^{13}{,}\,{\href{/padicField/5.1.0.1}{1} }$ $27$ ${\href{/padicField/11.9.0.1}{9} }^{3}$ ${\href{/padicField/13.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/padicField/29.2.0.1}{2} }^{13}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{13}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{13}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.2.0.1}{2} }^{13}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }^{13}{,}\,{\href{/padicField/43.1.0.1}{1} }$ $27$ ${\href{/padicField/53.2.0.1}{2} }^{13}{,}\,{\href{/padicField/53.1.0.1}{1} }$ $27$

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$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(563\) Copy content Toggle raw display $\Q_{563}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.563.2t1.a.a$1$ $ 563 $ \(\Q(\sqrt{-563}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.563.3t2.a.a$2$ $ 563 $ 3.1.563.1 $S_3$ (as 3T2) $1$ $0$
* 2.563.9t3.a.c$2$ $ 563 $ 9.1.100469346961.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.563.9t3.a.a$2$ $ 563 $ 9.1.100469346961.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.563.9t3.a.b$2$ $ 563 $ 9.1.100469346961.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2252.27t8.a.c$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.b$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.e$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.i$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.f$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.a$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.h$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.g$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2252.27t8.a.d$2$ $ 2^{2} \cdot 563 $ 27.1.149674927005884133619412112407487159468032.1 $D_{27}$ (as 27T8) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.