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Properties

Label 27.1.128...351.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.287\times 10^{45}$
Root discriminant \(46.85\)
Ramified primes $13,227$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 4*x^26 + 28*x^25 - 42*x^24 + 71*x^23 + 267*x^22 - 142*x^21 + 385*x^20 + 2210*x^19 + 2329*x^18 - 3315*x^17 + 15117*x^16 + 19014*x^15 - 23157*x^14 + 44479*x^13 + 77004*x^12 - 41903*x^11 + 19829*x^10 + 208952*x^9 - 35302*x^8 - 53453*x^7 + 258838*x^6 - 35701*x^5 + 60990*x^4 - 5393*x^3 + 76227*x^2 + 19031*x + 2209)
 
gp: K = bnfinit(y^27 - 4*y^26 + 28*y^25 - 42*y^24 + 71*y^23 + 267*y^22 - 142*y^21 + 385*y^20 + 2210*y^19 + 2329*y^18 - 3315*y^17 + 15117*y^16 + 19014*y^15 - 23157*y^14 + 44479*y^13 + 77004*y^12 - 41903*y^11 + 19829*y^10 + 208952*y^9 - 35302*y^8 - 53453*y^7 + 258838*y^6 - 35701*y^5 + 60990*y^4 - 5393*y^3 + 76227*y^2 + 19031*y + 2209, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 4*x^26 + 28*x^25 - 42*x^24 + 71*x^23 + 267*x^22 - 142*x^21 + 385*x^20 + 2210*x^19 + 2329*x^18 - 3315*x^17 + 15117*x^16 + 19014*x^15 - 23157*x^14 + 44479*x^13 + 77004*x^12 - 41903*x^11 + 19829*x^10 + 208952*x^9 - 35302*x^8 - 53453*x^7 + 258838*x^6 - 35701*x^5 + 60990*x^4 - 5393*x^3 + 76227*x^2 + 19031*x + 2209);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 4*x^26 + 28*x^25 - 42*x^24 + 71*x^23 + 267*x^22 - 142*x^21 + 385*x^20 + 2210*x^19 + 2329*x^18 - 3315*x^17 + 15117*x^16 + 19014*x^15 - 23157*x^14 + 44479*x^13 + 77004*x^12 - 41903*x^11 + 19829*x^10 + 208952*x^9 - 35302*x^8 - 53453*x^7 + 258838*x^6 - 35701*x^5 + 60990*x^4 - 5393*x^3 + 76227*x^2 + 19031*x + 2209)
 

\( x^{27} - 4 x^{26} + 28 x^{25} - 42 x^{24} + 71 x^{23} + 267 x^{22} - 142 x^{21} + 385 x^{20} + \cdots + 2209 \) 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:   \(-1287062756823986424273208421285283554009333351\) \(\medspace = -\,13^{13}\cdot 227^{13}\) 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:  \(46.85\)
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:  $13^{1/2}227^{1/2}\approx 54.323107422164284$
Ramified primes:   \(13\), \(227\) 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:  \(\Q(\sqrt{-2951}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}+\frac{1}{13}a^{9}-\frac{6}{13}a^{6}+\frac{5}{13}a^{3}-\frac{1}{13}$, $\frac{1}{13}a^{13}+\frac{1}{13}a^{10}-\frac{6}{13}a^{7}+\frac{5}{13}a^{4}-\frac{1}{13}a$, $\frac{1}{13}a^{14}+\frac{1}{13}a^{11}-\frac{6}{13}a^{8}+\frac{5}{13}a^{5}-\frac{1}{13}a^{2}$, $\frac{1}{13}a^{15}+\frac{6}{13}a^{9}-\frac{2}{13}a^{6}-\frac{6}{13}a^{3}+\frac{1}{13}$, $\frac{1}{13}a^{16}+\frac{6}{13}a^{10}-\frac{2}{13}a^{7}-\frac{6}{13}a^{4}+\frac{1}{13}a$, $\frac{1}{13}a^{17}+\frac{6}{13}a^{11}-\frac{2}{13}a^{8}-\frac{6}{13}a^{5}+\frac{1}{13}a^{2}$, $\frac{1}{13}a^{18}+\frac{5}{13}a^{9}+\frac{4}{13}a^{6}-\frac{3}{13}a^{3}+\frac{6}{13}$, $\frac{1}{13}a^{19}+\frac{5}{13}a^{10}+\frac{4}{13}a^{7}-\frac{3}{13}a^{4}+\frac{6}{13}a$, $\frac{1}{91}a^{20}+\frac{2}{91}a^{19}-\frac{2}{91}a^{18}-\frac{2}{91}a^{17}-\frac{1}{91}a^{16}+\frac{3}{91}a^{15}-\frac{1}{91}a^{14}-\frac{2}{91}a^{13}-\frac{3}{91}a^{12}+\frac{18}{91}a^{11}+\frac{41}{91}a^{10}-\frac{8}{91}a^{9}-\frac{12}{91}a^{8}+\frac{5}{13}a^{7}-\frac{9}{91}a^{6}+\frac{30}{91}a^{5}-\frac{10}{91}a^{4}-\frac{2}{13}a^{3}+\frac{18}{91}a^{2}+\frac{2}{7}a-\frac{32}{91}$, $\frac{1}{13013}a^{21}-\frac{10}{1859}a^{20}+\frac{50}{13013}a^{19}-\frac{194}{13013}a^{18}+\frac{311}{13013}a^{17}-\frac{471}{13013}a^{16}-\frac{61}{1859}a^{15}-\frac{34}{1859}a^{14}+\frac{456}{13013}a^{13}+\frac{17}{13013}a^{12}-\frac{5833}{13013}a^{11}+\frac{1975}{13013}a^{10}-\frac{2502}{13013}a^{9}+\frac{3321}{13013}a^{8}+\frac{1006}{13013}a^{7}+\frac{4150}{13013}a^{6}+\frac{106}{1859}a^{5}+\frac{4241}{13013}a^{4}+\frac{753}{13013}a^{3}-\frac{6345}{13013}a^{2}+\frac{267}{1859}a+\frac{1751}{13013}$, $\frac{1}{13013}a^{22}+\frac{12}{13013}a^{20}+\frac{17}{13013}a^{19}+\frac{30}{13013}a^{18}-\frac{437}{13013}a^{17}-\frac{17}{1001}a^{16}+\frac{474}{13013}a^{15}-\frac{45}{13013}a^{14}+\frac{191}{13013}a^{13}-\frac{30}{1859}a^{12}-\frac{5506}{13013}a^{11}-\frac{750}{1859}a^{10}+\frac{6502}{13013}a^{9}-\frac{5048}{13013}a^{8}+\frac{1497}{13013}a^{7}-\frac{1765}{13013}a^{6}+\frac{263}{1859}a^{5}+\frac{1756}{13013}a^{4}-\frac{62}{1183}a^{3}+\frac{2592}{13013}a^{2}-\frac{896}{1859}a-\frac{1983}{13013}$, $\frac{1}{32831187389}a^{23}+\frac{38078}{2525475953}a^{22}+\frac{263255}{32831187389}a^{21}+\frac{13352125}{2525475953}a^{20}+\frac{138742609}{32831187389}a^{19}-\frac{11719439}{4690169627}a^{18}+\frac{812060659}{32831187389}a^{17}-\frac{1031178641}{32831187389}a^{16}-\frac{832232967}{32831187389}a^{15}+\frac{567648079}{32831187389}a^{14}-\frac{101919060}{32831187389}a^{13}-\frac{694048000}{32831187389}a^{12}+\frac{262516510}{2525475953}a^{11}+\frac{15794305718}{32831187389}a^{10}+\frac{14796286543}{32831187389}a^{9}+\frac{234234939}{1931246317}a^{8}+\frac{4791672685}{32831187389}a^{7}+\frac{658026669}{2525475953}a^{6}-\frac{7454500618}{32831187389}a^{5}-\frac{799295}{25081121}a^{4}+\frac{2217150401}{4690169627}a^{3}-\frac{1261962947}{4690169627}a^{2}-\frac{972239964}{2984653399}a+\frac{4744434294}{32831187389}$, $\frac{1}{32831187389}a^{24}+\frac{690231}{32831187389}a^{22}+\frac{151654}{32831187389}a^{21}-\frac{166775}{64248899}a^{20}+\frac{422606453}{32831187389}a^{19}-\frac{222635368}{32831187389}a^{18}-\frac{796535078}{32831187389}a^{17}+\frac{1086552017}{32831187389}a^{16}-\frac{389182806}{32831187389}a^{15}+\frac{375307630}{32831187389}a^{14}+\frac{123101492}{4690169627}a^{13}-\frac{1170284438}{32831187389}a^{12}-\frac{87556412}{1727957231}a^{11}-\frac{5022633202}{32831187389}a^{10}+\frac{10797141997}{32831187389}a^{9}+\frac{317546521}{32831187389}a^{8}-\frac{4351310079}{32831187389}a^{7}+\frac{7741336298}{32831187389}a^{6}+\frac{4320077516}{32831187389}a^{5}+\frac{1167687818}{2984653399}a^{4}-\frac{13482856159}{32831187389}a^{3}-\frac{15554742529}{32831187389}a^{2}+\frac{3126599635}{32831187389}a-\frac{9178792284}{32831187389}$, $\frac{1}{523033646294159}a^{25}+\frac{18}{74719092327737}a^{24}+\frac{193}{523033646294159}a^{23}+\frac{16601461330}{523033646294159}a^{22}+\frac{4244798319}{523033646294159}a^{21}-\frac{822304632991}{523033646294159}a^{20}+\frac{7056405286235}{523033646294159}a^{19}+\frac{13248912255927}{523033646294159}a^{18}-\frac{2541159895485}{523033646294159}a^{17}+\frac{6164195203032}{523033646294159}a^{16}+\frac{16860351078640}{523033646294159}a^{15}+\frac{1141904505874}{47548513299469}a^{14}-\frac{16574990243257}{523033646294159}a^{13}-\frac{1564029979894}{74719092327737}a^{12}-\frac{1771508114132}{27528086647061}a^{11}-\frac{23360182650449}{523033646294159}a^{10}-\frac{1400742565977}{4888164918637}a^{9}+\frac{103184134443274}{523033646294159}a^{8}-\frac{542936003431}{3932583806723}a^{7}-\frac{18384955880722}{74719092327737}a^{6}+\frac{30208588644495}{523033646294159}a^{5}+\frac{257883409950519}{523033646294159}a^{4}-\frac{229668110010253}{523033646294159}a^{3}+\frac{187960243724587}{523033646294159}a^{2}-\frac{220610074041613}{523033646294159}a+\frac{6090731561546}{40233357407243}$, $\frac{1}{33\!\cdots\!99}a^{26}-\frac{5}{10\!\cdots\!13}a^{25}+\frac{657032}{33\!\cdots\!99}a^{24}+\frac{34574003}{33\!\cdots\!99}a^{23}-\frac{74177854409982}{33\!\cdots\!99}a^{22}+\frac{107367355540727}{33\!\cdots\!99}a^{21}+\frac{1307487188814}{42\!\cdots\!33}a^{20}-\frac{47\!\cdots\!70}{33\!\cdots\!99}a^{19}+\frac{21\!\cdots\!52}{33\!\cdots\!99}a^{18}-\frac{20\!\cdots\!38}{17\!\cdots\!21}a^{17}-\frac{67\!\cdots\!37}{17\!\cdots\!21}a^{16}+\frac{10\!\cdots\!31}{33\!\cdots\!99}a^{15}-\frac{87\!\cdots\!23}{25\!\cdots\!23}a^{14}+\frac{58\!\cdots\!54}{30\!\cdots\!09}a^{13}+\frac{90\!\cdots\!41}{30\!\cdots\!09}a^{12}+\frac{64\!\cdots\!66}{33\!\cdots\!99}a^{11}-\frac{75\!\cdots\!77}{33\!\cdots\!99}a^{10}-\frac{54\!\cdots\!91}{17\!\cdots\!21}a^{9}+\frac{11\!\cdots\!61}{33\!\cdots\!99}a^{8}-\frac{92\!\cdots\!17}{33\!\cdots\!99}a^{7}+\frac{52\!\cdots\!52}{17\!\cdots\!21}a^{6}+\frac{68\!\cdots\!14}{33\!\cdots\!99}a^{5}-\frac{21\!\cdots\!26}{33\!\cdots\!99}a^{4}+\frac{23\!\cdots\!66}{30\!\cdots\!09}a^{3}+\frac{77\!\cdots\!81}{33\!\cdots\!99}a^{2}+\frac{29\!\cdots\!86}{33\!\cdots\!99}a-\frac{15\!\cdots\!54}{33\!\cdots\!99}$ 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:  $13$

Class group and class number

Trivial group, which has order $1$ (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:   $\frac{987135386502549}{33\!\cdots\!99}a^{26}-\frac{43\!\cdots\!92}{33\!\cdots\!99}a^{25}+\frac{28\!\cdots\!79}{33\!\cdots\!99}a^{24}-\frac{45\!\cdots\!20}{30\!\cdots\!09}a^{23}+\frac{33\!\cdots\!96}{17\!\cdots\!21}a^{22}+\frac{25\!\cdots\!13}{33\!\cdots\!99}a^{21}-\frac{29\!\cdots\!68}{37\!\cdots\!91}a^{20}+\frac{13\!\cdots\!09}{33\!\cdots\!99}a^{19}+\frac{19\!\cdots\!88}{33\!\cdots\!99}a^{18}+\frac{78\!\cdots\!69}{17\!\cdots\!21}a^{17}-\frac{59\!\cdots\!04}{33\!\cdots\!99}a^{16}+\frac{12\!\cdots\!89}{33\!\cdots\!99}a^{15}+\frac{12\!\cdots\!50}{25\!\cdots\!23}a^{14}-\frac{34\!\cdots\!59}{33\!\cdots\!99}a^{13}+\frac{29\!\cdots\!53}{33\!\cdots\!99}a^{12}+\frac{67\!\cdots\!27}{33\!\cdots\!99}a^{11}-\frac{57\!\cdots\!62}{31\!\cdots\!57}a^{10}-\frac{24\!\cdots\!75}{33\!\cdots\!99}a^{9}+\frac{17\!\cdots\!34}{33\!\cdots\!99}a^{8}-\frac{53\!\cdots\!25}{33\!\cdots\!99}a^{7}-\frac{98\!\cdots\!22}{33\!\cdots\!99}a^{6}+\frac{19\!\cdots\!53}{33\!\cdots\!99}a^{5}-\frac{66\!\cdots\!59}{33\!\cdots\!99}a^{4}+\frac{44\!\cdots\!09}{30\!\cdots\!09}a^{3}-\frac{41\!\cdots\!87}{33\!\cdots\!99}a^{2}+\frac{56\!\cdots\!09}{33\!\cdots\!99}a-\frac{53\!\cdots\!20}{33\!\cdots\!99}$, $\frac{94696815416167}{33\!\cdots\!99}a^{26}-\frac{427384275284521}{33\!\cdots\!99}a^{25}+\frac{250353328553570}{30\!\cdots\!09}a^{24}-\frac{46\!\cdots\!70}{33\!\cdots\!99}a^{23}+\frac{60\!\cdots\!45}{33\!\cdots\!99}a^{22}+\frac{31\!\cdots\!89}{33\!\cdots\!99}a^{21}-\frac{21\!\cdots\!41}{33\!\cdots\!99}a^{20}+\frac{28\!\cdots\!97}{33\!\cdots\!99}a^{19}+\frac{27\!\cdots\!14}{30\!\cdots\!09}a^{18}+\frac{32\!\cdots\!31}{33\!\cdots\!99}a^{17}-\frac{35\!\cdots\!39}{33\!\cdots\!99}a^{16}+\frac{20\!\cdots\!19}{33\!\cdots\!99}a^{15}+\frac{27\!\cdots\!25}{25\!\cdots\!23}a^{14}-\frac{18\!\cdots\!32}{33\!\cdots\!99}a^{13}+\frac{54\!\cdots\!58}{33\!\cdots\!99}a^{12}+\frac{13\!\cdots\!53}{30\!\cdots\!09}a^{11}+\frac{67\!\cdots\!98}{33\!\cdots\!99}a^{10}+\frac{89\!\cdots\!24}{33\!\cdots\!99}a^{9}+\frac{31\!\cdots\!67}{33\!\cdots\!99}a^{8}+\frac{12\!\cdots\!64}{33\!\cdots\!99}a^{7}-\frac{92\!\cdots\!63}{33\!\cdots\!99}a^{6}+\frac{29\!\cdots\!10}{33\!\cdots\!99}a^{5}+\frac{12\!\cdots\!95}{30\!\cdots\!09}a^{4}+\frac{46\!\cdots\!57}{33\!\cdots\!99}a^{3}+\frac{36\!\cdots\!73}{33\!\cdots\!99}a^{2}+\frac{63\!\cdots\!41}{33\!\cdots\!99}a+\frac{49\!\cdots\!37}{33\!\cdots\!99}$, $\frac{9078678706421}{19\!\cdots\!47}a^{26}-\frac{348224073156948}{33\!\cdots\!99}a^{25}+\frac{39\!\cdots\!77}{33\!\cdots\!99}a^{24}-\frac{13\!\cdots\!60}{33\!\cdots\!99}a^{23}+\frac{16\!\cdots\!93}{33\!\cdots\!99}a^{22}+\frac{43\!\cdots\!44}{33\!\cdots\!99}a^{21}+\frac{66\!\cdots\!73}{33\!\cdots\!99}a^{20}+\frac{21\!\cdots\!91}{33\!\cdots\!99}a^{19}+\frac{22\!\cdots\!37}{17\!\cdots\!21}a^{18}+\frac{89\!\cdots\!88}{33\!\cdots\!99}a^{17}+\frac{12\!\cdots\!21}{33\!\cdots\!99}a^{16}+\frac{33\!\cdots\!40}{33\!\cdots\!99}a^{15}+\frac{17\!\cdots\!52}{15\!\cdots\!19}a^{14}+\frac{48\!\cdots\!03}{33\!\cdots\!99}a^{13}+\frac{13\!\cdots\!61}{33\!\cdots\!99}a^{12}+\frac{49\!\cdots\!65}{17\!\cdots\!21}a^{11}+\frac{10\!\cdots\!03}{33\!\cdots\!99}a^{10}+\frac{17\!\cdots\!71}{19\!\cdots\!47}a^{9}+\frac{22\!\cdots\!31}{33\!\cdots\!99}a^{8}+\frac{78\!\cdots\!26}{33\!\cdots\!99}a^{7}+\frac{42\!\cdots\!46}{33\!\cdots\!99}a^{6}+\frac{16\!\cdots\!96}{16\!\cdots\!11}a^{5}-\frac{26\!\cdots\!20}{33\!\cdots\!99}a^{4}+\frac{43\!\cdots\!88}{33\!\cdots\!99}a^{3}+\frac{12\!\cdots\!50}{33\!\cdots\!99}a^{2}+\frac{20\!\cdots\!34}{30\!\cdots\!09}a+\frac{26\!\cdots\!72}{30\!\cdots\!09}$, $\frac{58603090470754}{33\!\cdots\!99}a^{26}-\frac{280645673040025}{33\!\cdots\!99}a^{25}+\frac{17\!\cdots\!74}{33\!\cdots\!99}a^{24}-\frac{34\!\cdots\!14}{33\!\cdots\!99}a^{23}+\frac{170475784401463}{17\!\cdots\!21}a^{22}+\frac{13\!\cdots\!81}{33\!\cdots\!99}a^{21}-\frac{24\!\cdots\!29}{33\!\cdots\!99}a^{20}-\frac{108220402115015}{37\!\cdots\!91}a^{19}+\frac{88\!\cdots\!68}{33\!\cdots\!99}a^{18}+\frac{66\!\cdots\!90}{33\!\cdots\!99}a^{17}-\frac{59\!\cdots\!02}{33\!\cdots\!99}a^{16}+\frac{44\!\cdots\!92}{33\!\cdots\!99}a^{15}+\frac{38\!\cdots\!52}{25\!\cdots\!23}a^{14}-\frac{34\!\cdots\!74}{33\!\cdots\!99}a^{13}-\frac{28\!\cdots\!30}{33\!\cdots\!99}a^{12}+\frac{23\!\cdots\!36}{33\!\cdots\!99}a^{11}-\frac{77\!\cdots\!19}{37\!\cdots\!91}a^{10}-\frac{87\!\cdots\!66}{33\!\cdots\!99}a^{9}+\frac{44\!\cdots\!48}{33\!\cdots\!99}a^{8}-\frac{62\!\cdots\!54}{33\!\cdots\!99}a^{7}-\frac{16\!\cdots\!09}{33\!\cdots\!99}a^{6}-\frac{32\!\cdots\!83}{33\!\cdots\!99}a^{5}-\frac{11\!\cdots\!52}{46\!\cdots\!63}a^{4}+\frac{13\!\cdots\!19}{17\!\cdots\!21}a^{3}-\frac{20\!\cdots\!87}{46\!\cdots\!63}a^{2}-\frac{52\!\cdots\!79}{33\!\cdots\!99}a-\frac{12\!\cdots\!76}{19\!\cdots\!47}$, $\frac{1694790120748}{46\!\cdots\!63}a^{26}-\frac{26115648147623}{19\!\cdots\!47}a^{25}+\frac{31\!\cdots\!88}{33\!\cdots\!99}a^{24}-\frac{36\!\cdots\!95}{33\!\cdots\!99}a^{23}+\frac{48\!\cdots\!04}{33\!\cdots\!99}a^{22}+\frac{33\!\cdots\!02}{30\!\cdots\!09}a^{21}-\frac{76\!\cdots\!26}{33\!\cdots\!99}a^{20}+\frac{18\!\cdots\!36}{33\!\cdots\!99}a^{19}+\frac{25\!\cdots\!11}{33\!\cdots\!99}a^{18}+\frac{19\!\cdots\!98}{17\!\cdots\!21}a^{17}-\frac{44\!\cdots\!89}{33\!\cdots\!99}a^{16}+\frac{13\!\cdots\!62}{33\!\cdots\!99}a^{15}+\frac{21\!\cdots\!50}{25\!\cdots\!23}a^{14}-\frac{24\!\cdots\!14}{33\!\cdots\!99}a^{13}+\frac{26\!\cdots\!59}{33\!\cdots\!99}a^{12}+\frac{10\!\cdots\!44}{33\!\cdots\!99}a^{11}-\frac{23\!\cdots\!75}{33\!\cdots\!99}a^{10}-\frac{29\!\cdots\!37}{33\!\cdots\!99}a^{9}+\frac{21\!\cdots\!54}{30\!\cdots\!09}a^{8}+\frac{26\!\cdots\!80}{33\!\cdots\!99}a^{7}-\frac{10\!\cdots\!32}{33\!\cdots\!99}a^{6}+\frac{27\!\cdots\!16}{33\!\cdots\!99}a^{5}+\frac{14\!\cdots\!93}{33\!\cdots\!99}a^{4}+\frac{34\!\cdots\!52}{33\!\cdots\!99}a^{3}+\frac{12\!\cdots\!50}{33\!\cdots\!99}a^{2}+\frac{39\!\cdots\!45}{33\!\cdots\!99}a+\frac{71\!\cdots\!52}{30\!\cdots\!09}$, $\frac{612640990063035}{33\!\cdots\!99}a^{26}-\frac{28157816286239}{37\!\cdots\!91}a^{25}+\frac{19\!\cdots\!11}{33\!\cdots\!99}a^{24}-\frac{33\!\cdots\!86}{33\!\cdots\!99}a^{23}+\frac{489697085455665}{16\!\cdots\!11}a^{22}+\frac{14\!\cdots\!67}{31\!\cdots\!57}a^{21}-\frac{40\!\cdots\!33}{19\!\cdots\!47}a^{20}+\frac{10\!\cdots\!84}{33\!\cdots\!99}a^{19}+\frac{20\!\cdots\!48}{33\!\cdots\!99}a^{18}+\frac{15\!\cdots\!25}{33\!\cdots\!99}a^{17}+\frac{19\!\cdots\!12}{33\!\cdots\!99}a^{16}+\frac{21\!\cdots\!51}{33\!\cdots\!99}a^{15}+\frac{46\!\cdots\!18}{15\!\cdots\!19}a^{14}-\frac{76\!\cdots\!09}{33\!\cdots\!99}a^{13}+\frac{90\!\cdots\!08}{33\!\cdots\!99}a^{12}+\frac{69\!\cdots\!92}{37\!\cdots\!91}a^{11}-\frac{42\!\cdots\!85}{30\!\cdots\!09}a^{10}+\frac{83\!\cdots\!53}{19\!\cdots\!47}a^{9}+\frac{25\!\cdots\!08}{33\!\cdots\!99}a^{8}-\frac{95\!\cdots\!53}{30\!\cdots\!09}a^{7}+\frac{64\!\cdots\!69}{33\!\cdots\!99}a^{6}+\frac{39\!\cdots\!68}{33\!\cdots\!99}a^{5}-\frac{65\!\cdots\!47}{33\!\cdots\!99}a^{4}+\frac{84\!\cdots\!95}{33\!\cdots\!99}a^{3}+\frac{36\!\cdots\!38}{33\!\cdots\!99}a^{2}+\frac{14\!\cdots\!89}{33\!\cdots\!99}a+\frac{15\!\cdots\!06}{33\!\cdots\!99}$, $\frac{16467946162112}{30\!\cdots\!09}a^{26}-\frac{52275691503778}{19\!\cdots\!47}a^{25}+\frac{54\!\cdots\!31}{33\!\cdots\!99}a^{24}-\frac{11\!\cdots\!90}{33\!\cdots\!99}a^{23}+\frac{13\!\cdots\!81}{33\!\cdots\!99}a^{22}+\frac{40\!\cdots\!20}{30\!\cdots\!09}a^{21}-\frac{43\!\cdots\!30}{17\!\cdots\!21}a^{20}+\frac{23\!\cdots\!47}{33\!\cdots\!99}a^{19}+\frac{35\!\cdots\!37}{33\!\cdots\!99}a^{18}-\frac{98\!\cdots\!67}{33\!\cdots\!99}a^{17}-\frac{15\!\cdots\!45}{33\!\cdots\!99}a^{16}+\frac{25\!\cdots\!04}{33\!\cdots\!99}a^{15}+\frac{13\!\cdots\!86}{25\!\cdots\!23}a^{14}-\frac{10\!\cdots\!01}{33\!\cdots\!99}a^{13}+\frac{77\!\cdots\!21}{46\!\cdots\!63}a^{12}+\frac{11\!\cdots\!54}{33\!\cdots\!99}a^{11}-\frac{26\!\cdots\!05}{33\!\cdots\!99}a^{10}-\frac{12\!\cdots\!50}{33\!\cdots\!99}a^{9}+\frac{19\!\cdots\!17}{17\!\cdots\!71}a^{8}-\frac{27\!\cdots\!01}{33\!\cdots\!99}a^{7}-\frac{47\!\cdots\!80}{33\!\cdots\!99}a^{6}+\frac{44\!\cdots\!79}{33\!\cdots\!99}a^{5}-\frac{15\!\cdots\!88}{33\!\cdots\!99}a^{4}-\frac{98\!\cdots\!36}{33\!\cdots\!99}a^{3}-\frac{92\!\cdots\!01}{33\!\cdots\!99}a^{2}+\frac{72\!\cdots\!47}{33\!\cdots\!99}a+\frac{30\!\cdots\!42}{30\!\cdots\!09}$, $\frac{325226539785657}{33\!\cdots\!99}a^{26}-\frac{125791928695864}{30\!\cdots\!09}a^{25}+\frac{95\!\cdots\!95}{33\!\cdots\!99}a^{24}-\frac{16\!\cdots\!66}{33\!\cdots\!99}a^{23}+\frac{15\!\cdots\!55}{17\!\cdots\!21}a^{22}+\frac{85\!\cdots\!57}{33\!\cdots\!99}a^{21}-\frac{65\!\cdots\!95}{33\!\cdots\!99}a^{20}+\frac{95\!\cdots\!16}{17\!\cdots\!21}a^{19}+\frac{78\!\cdots\!01}{33\!\cdots\!99}a^{18}+\frac{68\!\cdots\!76}{33\!\cdots\!99}a^{17}-\frac{57\!\cdots\!30}{19\!\cdots\!47}a^{16}+\frac{84\!\cdots\!02}{46\!\cdots\!63}a^{15}+\frac{46\!\cdots\!95}{25\!\cdots\!23}a^{14}-\frac{82\!\cdots\!45}{33\!\cdots\!99}a^{13}+\frac{20\!\cdots\!97}{33\!\cdots\!99}a^{12}+\frac{14\!\cdots\!21}{17\!\cdots\!21}a^{11}-\frac{10\!\cdots\!47}{19\!\cdots\!47}a^{10}+\frac{16\!\cdots\!44}{33\!\cdots\!99}a^{9}+\frac{82\!\cdots\!38}{33\!\cdots\!99}a^{8}-\frac{20\!\cdots\!62}{33\!\cdots\!99}a^{7}-\frac{11\!\cdots\!80}{33\!\cdots\!99}a^{6}+\frac{10\!\cdots\!11}{33\!\cdots\!99}a^{5}-\frac{99\!\cdots\!56}{19\!\cdots\!47}a^{4}+\frac{26\!\cdots\!40}{33\!\cdots\!99}a^{3}-\frac{26\!\cdots\!89}{30\!\cdots\!09}a^{2}+\frac{37\!\cdots\!78}{33\!\cdots\!99}a+\frac{54\!\cdots\!88}{33\!\cdots\!99}$, $\frac{1485658268895}{31\!\cdots\!57}a^{26}-\frac{225908377298039}{33\!\cdots\!99}a^{25}+\frac{26026374483894}{31\!\cdots\!57}a^{24}+\frac{46\!\cdots\!49}{33\!\cdots\!99}a^{23}-\frac{51\!\cdots\!69}{33\!\cdots\!99}a^{22}+\frac{38\!\cdots\!17}{19\!\cdots\!47}a^{21}+\frac{10\!\cdots\!47}{33\!\cdots\!99}a^{20}-\frac{98\!\cdots\!52}{33\!\cdots\!99}a^{19}+\frac{22\!\cdots\!95}{17\!\cdots\!21}a^{18}+\frac{14\!\cdots\!40}{33\!\cdots\!99}a^{17}+\frac{53\!\cdots\!43}{33\!\cdots\!99}a^{16}+\frac{24\!\cdots\!17}{33\!\cdots\!99}a^{15}+\frac{65\!\cdots\!77}{25\!\cdots\!23}a^{14}+\frac{76\!\cdots\!50}{37\!\cdots\!91}a^{13}-\frac{60\!\cdots\!85}{33\!\cdots\!99}a^{12}+\frac{19\!\cdots\!45}{33\!\cdots\!99}a^{11}+\frac{33\!\cdots\!52}{33\!\cdots\!99}a^{10}-\frac{14\!\cdots\!00}{33\!\cdots\!99}a^{9}+\frac{14\!\cdots\!71}{33\!\cdots\!99}a^{8}+\frac{70\!\cdots\!31}{33\!\cdots\!99}a^{7}+\frac{58\!\cdots\!70}{33\!\cdots\!99}a^{6}-\frac{30\!\cdots\!30}{33\!\cdots\!99}a^{5}+\frac{40\!\cdots\!08}{33\!\cdots\!99}a^{4}+\frac{12\!\cdots\!42}{33\!\cdots\!99}a^{3}+\frac{12\!\cdots\!86}{33\!\cdots\!99}a^{2}+\frac{64\!\cdots\!21}{19\!\cdots\!47}a+\frac{23\!\cdots\!75}{30\!\cdots\!09}$, $\frac{624284849990883}{33\!\cdots\!99}a^{26}-\frac{24\!\cdots\!58}{33\!\cdots\!99}a^{25}+\frac{17\!\cdots\!26}{33\!\cdots\!99}a^{24}-\frac{25\!\cdots\!08}{33\!\cdots\!99}a^{23}+\frac{43\!\cdots\!42}{33\!\cdots\!99}a^{22}+\frac{796138359651025}{16\!\cdots\!11}a^{21}-\frac{83\!\cdots\!82}{33\!\cdots\!99}a^{20}+\frac{23\!\cdots\!18}{33\!\cdots\!99}a^{19}+\frac{13\!\cdots\!50}{33\!\cdots\!99}a^{18}+\frac{14\!\cdots\!32}{33\!\cdots\!99}a^{17}-\frac{19\!\cdots\!00}{33\!\cdots\!99}a^{16}+\frac{93\!\cdots\!73}{33\!\cdots\!99}a^{15}+\frac{91\!\cdots\!09}{25\!\cdots\!23}a^{14}-\frac{76\!\cdots\!94}{18\!\cdots\!81}a^{13}+\frac{27\!\cdots\!98}{33\!\cdots\!99}a^{12}+\frac{47\!\cdots\!68}{33\!\cdots\!99}a^{11}-\frac{24\!\cdots\!29}{33\!\cdots\!99}a^{10}+\frac{13\!\cdots\!77}{33\!\cdots\!99}a^{9}+\frac{12\!\cdots\!79}{33\!\cdots\!99}a^{8}-\frac{24\!\cdots\!64}{33\!\cdots\!99}a^{7}-\frac{27\!\cdots\!85}{33\!\cdots\!99}a^{6}+\frac{15\!\cdots\!61}{33\!\cdots\!99}a^{5}-\frac{33\!\cdots\!91}{33\!\cdots\!99}a^{4}+\frac{32\!\cdots\!98}{33\!\cdots\!99}a^{3}+\frac{24\!\cdots\!97}{33\!\cdots\!99}a^{2}+\frac{25\!\cdots\!00}{19\!\cdots\!47}a+\frac{48\!\cdots\!42}{33\!\cdots\!99}$, $\frac{439873321556082}{33\!\cdots\!99}a^{26}-\frac{21\!\cdots\!75}{33\!\cdots\!99}a^{25}+\frac{12\!\cdots\!53}{33\!\cdots\!99}a^{24}-\frac{21\!\cdots\!19}{30\!\cdots\!09}a^{23}+\frac{407203986604102}{19\!\cdots\!47}a^{22}+\frac{11\!\cdots\!58}{33\!\cdots\!99}a^{21}-\frac{20\!\cdots\!38}{33\!\cdots\!99}a^{20}-\frac{15\!\cdots\!12}{18\!\cdots\!77}a^{19}+\frac{31\!\cdots\!28}{19\!\cdots\!47}a^{18}-\frac{73\!\cdots\!49}{33\!\cdots\!99}a^{17}-\frac{57\!\cdots\!57}{33\!\cdots\!99}a^{16}+\frac{37\!\cdots\!12}{33\!\cdots\!99}a^{15}+\frac{35\!\cdots\!79}{25\!\cdots\!23}a^{14}-\frac{29\!\cdots\!02}{33\!\cdots\!99}a^{13}-\frac{16\!\cdots\!17}{30\!\cdots\!09}a^{12}+\frac{20\!\cdots\!77}{33\!\cdots\!99}a^{11}-\frac{54\!\cdots\!22}{33\!\cdots\!99}a^{10}-\frac{10\!\cdots\!14}{33\!\cdots\!99}a^{9}+\frac{26\!\cdots\!17}{33\!\cdots\!99}a^{8}-\frac{25\!\cdots\!53}{33\!\cdots\!99}a^{7}-\frac{17\!\cdots\!20}{33\!\cdots\!99}a^{6}-\frac{36\!\cdots\!32}{33\!\cdots\!99}a^{5}-\frac{16\!\cdots\!13}{33\!\cdots\!99}a^{4}-\frac{19\!\cdots\!25}{19\!\cdots\!47}a^{3}-\frac{40\!\cdots\!98}{17\!\cdots\!21}a^{2}-\frac{40\!\cdots\!85}{33\!\cdots\!99}a-\frac{82\!\cdots\!58}{33\!\cdots\!99}$, $\frac{198977109348742}{48\!\cdots\!57}a^{26}-\frac{114862612846418}{68\!\cdots\!51}a^{25}+\frac{56\!\cdots\!73}{48\!\cdots\!57}a^{24}-\frac{86\!\cdots\!14}{48\!\cdots\!57}a^{23}+\frac{15\!\cdots\!73}{48\!\cdots\!57}a^{22}+\frac{53\!\cdots\!97}{48\!\cdots\!57}a^{21}-\frac{34\!\cdots\!52}{48\!\cdots\!57}a^{20}+\frac{88\!\cdots\!94}{43\!\cdots\!87}a^{19}+\frac{65\!\cdots\!93}{68\!\cdots\!51}a^{18}+\frac{40\!\cdots\!83}{48\!\cdots\!57}a^{17}-\frac{59\!\cdots\!37}{48\!\cdots\!57}a^{16}+\frac{33\!\cdots\!66}{48\!\cdots\!57}a^{15}+\frac{26\!\cdots\!34}{37\!\cdots\!89}a^{14}-\frac{75\!\cdots\!29}{68\!\cdots\!51}a^{13}+\frac{11\!\cdots\!32}{48\!\cdots\!57}a^{12}+\frac{15\!\cdots\!56}{48\!\cdots\!57}a^{11}-\frac{13\!\cdots\!91}{48\!\cdots\!57}a^{10}+\frac{56\!\cdots\!95}{28\!\cdots\!21}a^{9}+\frac{48\!\cdots\!38}{48\!\cdots\!57}a^{8}-\frac{20\!\cdots\!13}{48\!\cdots\!57}a^{7}-\frac{15\!\cdots\!45}{62\!\cdots\!41}a^{6}+\frac{73\!\cdots\!86}{48\!\cdots\!57}a^{5}-\frac{18\!\cdots\!49}{43\!\cdots\!87}a^{4}-\frac{17\!\cdots\!14}{43\!\cdots\!87}a^{3}+\frac{18\!\cdots\!71}{48\!\cdots\!57}a^{2}+\frac{51\!\cdots\!37}{25\!\cdots\!03}a+\frac{44\!\cdots\!67}{48\!\cdots\!57}$, $\frac{43146593710894}{33\!\cdots\!99}a^{26}-\frac{211181151101616}{33\!\cdots\!99}a^{25}+\frac{796352301268240}{33\!\cdots\!99}a^{24}-\frac{16\!\cdots\!09}{33\!\cdots\!99}a^{23}-\frac{425096987443467}{19\!\cdots\!47}a^{22}+\frac{64\!\cdots\!43}{33\!\cdots\!99}a^{21}-\frac{24\!\cdots\!72}{33\!\cdots\!99}a^{20}-\frac{14\!\cdots\!72}{33\!\cdots\!99}a^{19}-\frac{67\!\cdots\!19}{17\!\cdots\!21}a^{18}-\frac{78\!\cdots\!04}{30\!\cdots\!09}a^{17}-\frac{11\!\cdots\!37}{33\!\cdots\!99}a^{16}-\frac{21\!\cdots\!31}{33\!\cdots\!99}a^{15}-\frac{34\!\cdots\!79}{25\!\cdots\!23}a^{14}-\frac{36\!\cdots\!33}{33\!\cdots\!99}a^{13}-\frac{11\!\cdots\!80}{33\!\cdots\!99}a^{12}-\frac{60\!\cdots\!26}{33\!\cdots\!99}a^{11}-\frac{24\!\cdots\!32}{33\!\cdots\!99}a^{10}-\frac{13\!\cdots\!25}{18\!\cdots\!77}a^{9}-\frac{28\!\cdots\!41}{33\!\cdots\!99}a^{8}-\frac{10\!\cdots\!63}{33\!\cdots\!99}a^{7}-\frac{17\!\cdots\!95}{33\!\cdots\!99}a^{6}-\frac{43\!\cdots\!82}{33\!\cdots\!99}a^{5}-\frac{17\!\cdots\!84}{33\!\cdots\!99}a^{4}-\frac{16\!\cdots\!52}{19\!\cdots\!47}a^{3}-\frac{14\!\cdots\!62}{30\!\cdots\!09}a^{2}-\frac{16\!\cdots\!49}{33\!\cdots\!99}a-\frac{99\!\cdots\!03}{33\!\cdots\!99}$ 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:  \( 1184470448230.2556 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 1184470448230.2556 \cdot 1}{2\cdot\sqrt{1287062756823986424273208421285283554009333351}}\cr\approx \mathstrut & 0.785348930660551 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^27 - 4*x^26 + 28*x^25 - 42*x^24 + 71*x^23 + 267*x^22 - 142*x^21 + 385*x^20 + 2210*x^19 + 2329*x^18 - 3315*x^17 + 15117*x^16 + 19014*x^15 - 23157*x^14 + 44479*x^13 + 77004*x^12 - 41903*x^11 + 19829*x^10 + 208952*x^9 - 35302*x^8 - 53453*x^7 + 258838*x^6 - 35701*x^5 + 60990*x^4 - 5393*x^3 + 76227*x^2 + 19031*x + 2209)
 
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 - 4*x^26 + 28*x^25 - 42*x^24 + 71*x^23 + 267*x^22 - 142*x^21 + 385*x^20 + 2210*x^19 + 2329*x^18 - 3315*x^17 + 15117*x^16 + 19014*x^15 - 23157*x^14 + 44479*x^13 + 77004*x^12 - 41903*x^11 + 19829*x^10 + 208952*x^9 - 35302*x^8 - 53453*x^7 + 258838*x^6 - 35701*x^5 + 60990*x^4 - 5393*x^3 + 76227*x^2 + 19031*x + 2209, 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 - 4*x^26 + 28*x^25 - 42*x^24 + 71*x^23 + 267*x^22 - 142*x^21 + 385*x^20 + 2210*x^19 + 2329*x^18 - 3315*x^17 + 15117*x^16 + 19014*x^15 - 23157*x^14 + 44479*x^13 + 77004*x^12 - 41903*x^11 + 19829*x^10 + 208952*x^9 - 35302*x^8 - 53453*x^7 + 258838*x^6 - 35701*x^5 + 60990*x^4 - 5393*x^3 + 76227*x^2 + 19031*x + 2209);
 
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 - 4*x^26 + 28*x^25 - 42*x^24 + 71*x^23 + 267*x^22 - 142*x^21 + 385*x^20 + 2210*x^19 + 2329*x^18 - 3315*x^17 + 15117*x^16 + 19014*x^15 - 23157*x^14 + 44479*x^13 + 77004*x^12 - 41903*x^11 + 19829*x^10 + 208952*x^9 - 35302*x^8 - 53453*x^7 + 258838*x^6 - 35701*x^5 + 60990*x^4 - 5393*x^3 + 76227*x^2 + 19031*x + 2209);
 
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

3.1.2951.1, 9.1.75836247976801.1

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 $27$ $27$ ${\href{/padicField/5.9.0.1}{9} }^{3}$ ${\href{/padicField/7.2.0.1}{2} }^{13}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.2.0.1}{2} }^{13}{,}\,{\href{/padicField/11.1.0.1}{1} }$ R ${\href{/padicField/17.2.0.1}{2} }^{13}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{13}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.3.0.1}{3} }^{9}$ $27$ $27$ $27$ ${\href{/padicField/41.9.0.1}{9} }^{3}$ $27$ ${\href{/padicField/47.2.0.1}{2} }^{13}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $27$ ${\href{/padicField/59.2.0.1}{2} }^{13}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(13\) Copy content Toggle raw display $\Q_{13}$$x + 11$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
\(227\) Copy content Toggle raw display $\Q_{227}$$x$$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$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2951.2t1.a.a$1$ $ 13 \cdot 227 $ \(\Q(\sqrt{-2951}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2951.3t2.a.a$2$ $ 13 \cdot 227 $ 3.1.2951.1 $S_3$ (as 3T2) $1$ $0$
* 2.2951.9t3.a.b$2$ $ 13 \cdot 227 $ 9.1.75836247976801.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2951.9t3.a.c$2$ $ 13 \cdot 227 $ 9.1.75836247976801.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2951.9t3.a.a$2$ $ 13 \cdot 227 $ 9.1.75836247976801.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2951.27t8.a.c$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.b$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.e$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.i$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.f$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.a$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.h$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.g$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.2951.27t8.a.d$2$ $ 13 \cdot 227 $ 27.1.1287062756823986424273208421285283554009333351.1 $D_{27}$ (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 *.