↑

Properties

Label 27.1.579...000.1
Degree $27$
Signature $[1, 13]$
Discriminant $-5.791\times 10^{61}$
Root discriminant \(193.87\)
Ramified primes $2,3,5$
Class number $27$ (GRH)
Class group [27] (GRH)
Galois group $D_{27}$ (as 27T8)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 27*x^25 - 162*x^24 + 567*x^23 + 7020*x^22 + 1845*x^21 - 137214*x^20 - 356859*x^19 + 1719144*x^18 + 11751723*x^17 + 15524730*x^16 - 75425499*x^15 - 340193844*x^14 - 302723001*x^13 + 1484785422*x^12 + 5038976790*x^11 + 2884521888*x^10 - 15795303852*x^9 - 34353915048*x^8 - 8492734008*x^7 + 44721504000*x^6 + 62398540512*x^5 + 101058043392*x^4 + 207332604288*x^3 - 84572356608*x^2 - 52216496640*x + 18041661440)
 
gp: K = bnfinit(y^27 - 27*y^25 - 162*y^24 + 567*y^23 + 7020*y^22 + 1845*y^21 - 137214*y^20 - 356859*y^19 + 1719144*y^18 + 11751723*y^17 + 15524730*y^16 - 75425499*y^15 - 340193844*y^14 - 302723001*y^13 + 1484785422*y^12 + 5038976790*y^11 + 2884521888*y^10 - 15795303852*y^9 - 34353915048*y^8 - 8492734008*y^7 + 44721504000*y^6 + 62398540512*y^5 + 101058043392*y^4 + 207332604288*y^3 - 84572356608*y^2 - 52216496640*y + 18041661440, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 27*x^25 - 162*x^24 + 567*x^23 + 7020*x^22 + 1845*x^21 - 137214*x^20 - 356859*x^19 + 1719144*x^18 + 11751723*x^17 + 15524730*x^16 - 75425499*x^15 - 340193844*x^14 - 302723001*x^13 + 1484785422*x^12 + 5038976790*x^11 + 2884521888*x^10 - 15795303852*x^9 - 34353915048*x^8 - 8492734008*x^7 + 44721504000*x^6 + 62398540512*x^5 + 101058043392*x^4 + 207332604288*x^3 - 84572356608*x^2 - 52216496640*x + 18041661440);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 27*x^25 - 162*x^24 + 567*x^23 + 7020*x^22 + 1845*x^21 - 137214*x^20 - 356859*x^19 + 1719144*x^18 + 11751723*x^17 + 15524730*x^16 - 75425499*x^15 - 340193844*x^14 - 302723001*x^13 + 1484785422*x^12 + 5038976790*x^11 + 2884521888*x^10 - 15795303852*x^9 - 34353915048*x^8 - 8492734008*x^7 + 44721504000*x^6 + 62398540512*x^5 + 101058043392*x^4 + 207332604288*x^3 - 84572356608*x^2 - 52216496640*x + 18041661440)
 

\( x^{27} - 27 x^{25} - 162 x^{24} + 567 x^{23} + 7020 x^{22} + 1845 x^{21} - 137214 x^{20} + \cdots + 18041661440 \) 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:   \(-57914576815317377556250885803017413799391835668480000000000000\) \(\medspace = -\,2^{26}\cdot 3^{94}\cdot 5^{13}\) 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:  \(193.87\)
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:  $2\cdot 3^{94/27}5^{1/2}\approx 204.92918390707015$
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:  \(\Q(\sqrt{-5}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{80}a^{10}-\frac{1}{16}a^{9}-\frac{1}{8}a^{7}+\frac{3}{80}a^{6}-\frac{3}{16}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}+\frac{1}{5}a^{2}-\frac{1}{2}a$, $\frac{1}{80}a^{11}-\frac{1}{16}a^{9}-\frac{7}{80}a^{7}-\frac{3}{16}a^{5}-\frac{1}{4}a^{4}+\frac{13}{40}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{80}a^{12}-\frac{1}{16}a^{9}+\frac{3}{80}a^{8}-\frac{1}{8}a^{7}-\frac{3}{16}a^{5}+\frac{1}{5}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{320}a^{13}-\frac{1}{160}a^{12}-\frac{1}{320}a^{11}-\frac{17}{320}a^{9}+\frac{7}{160}a^{8}+\frac{17}{320}a^{7}+\frac{1}{16}a^{6}+\frac{19}{80}a^{5}+\frac{1}{40}a^{4}+\frac{31}{80}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{3200}a^{14}+\frac{1}{1600}a^{13}+\frac{7}{3200}a^{12}+\frac{1}{200}a^{11}-\frac{13}{3200}a^{10}-\frac{87}{1600}a^{9}-\frac{159}{3200}a^{8}+\frac{97}{800}a^{7}-\frac{39}{400}a^{6}-\frac{19}{100}a^{5}-\frac{47}{800}a^{4}+\frac{13}{100}a^{3}-\frac{11}{200}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{3200}a^{15}+\frac{3}{3200}a^{13}+\frac{1}{1600}a^{12}-\frac{1}{640}a^{11}+\frac{3}{800}a^{10}-\frac{11}{3200}a^{9}-\frac{47}{1600}a^{8}+\frac{29}{400}a^{7}-\frac{19}{200}a^{6}-\frac{93}{800}a^{5}-\frac{1}{400}a^{4}-\frac{6}{25}a^{3}+\frac{13}{50}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{16000}a^{16}-\frac{1}{8000}a^{15}+\frac{1}{8000}a^{14}-\frac{3}{8000}a^{13}+\frac{3}{2000}a^{12}+\frac{3}{8000}a^{11}-\frac{31}{8000}a^{10}+\frac{51}{8000}a^{9}-\frac{101}{16000}a^{8}-\frac{89}{4000}a^{7}-\frac{93}{4000}a^{6}+\frac{17}{500}a^{5}+\frac{619}{4000}a^{4}-\frac{89}{500}a^{3}+\frac{357}{1000}a^{2}+\frac{1}{25}a-\frac{9}{25}$, $\frac{1}{32000}a^{17}-\frac{1}{16000}a^{15}-\frac{1}{16000}a^{14}+\frac{3}{8000}a^{13}-\frac{73}{16000}a^{12}-\frac{1}{640}a^{11}+\frac{89}{16000}a^{10}-\frac{1897}{32000}a^{9}+\frac{421}{16000}a^{8}-\frac{271}{8000}a^{7}-\frac{9}{80}a^{6}+\frac{1391}{8000}a^{5}-\frac{887}{4000}a^{4}+\frac{1}{2000}a^{3}+\frac{227}{1000}a^{2}+\frac{9}{25}a-\frac{9}{25}$, $\frac{1}{64000}a^{18}-\frac{1}{32000}a^{16}-\frac{1}{32000}a^{15}+\frac{1}{32000}a^{14}-\frac{33}{32000}a^{13}-\frac{1}{200}a^{12}-\frac{41}{32000}a^{11}-\frac{167}{64000}a^{10}-\frac{1559}{32000}a^{9}-\frac{1047}{32000}a^{8}-\frac{289}{3200}a^{7}-\frac{129}{16000}a^{6}+\frac{323}{8000}a^{5}-\frac{163}{8000}a^{4}-\frac{531}{4000}a^{3}+\frac{193}{400}a^{2}+\frac{79}{200}a+\frac{3}{10}$, $\frac{1}{128000}a^{19}-\frac{1}{64000}a^{17}-\frac{1}{64000}a^{16}+\frac{1}{64000}a^{15}-\frac{3}{64000}a^{14}-\frac{31}{64000}a^{12}-\frac{207}{128000}a^{11}+\frac{51}{64000}a^{10}-\frac{3967}{64000}a^{9}+\frac{87}{3200}a^{8}-\frac{2059}{32000}a^{7}-\frac{17}{16000}a^{6}-\frac{3823}{16000}a^{5}+\frac{233}{1000}a^{4}+\frac{81}{200}a^{3}-\frac{1}{100}a^{2}+\frac{17}{40}a+\frac{1}{10}$, $\frac{1}{1280000}a^{20}-\frac{1}{256000}a^{19}-\frac{1}{128000}a^{18}-\frac{1}{64000}a^{17}+\frac{7}{320000}a^{16}+\frac{17}{160000}a^{15}-\frac{13}{128000}a^{14}+\frac{117}{128000}a^{13}-\frac{3289}{1280000}a^{12}+\frac{3193}{1280000}a^{11}+\frac{1867}{320000}a^{10}+\frac{6273}{128000}a^{9}-\frac{529}{64000}a^{8}-\frac{4651}{320000}a^{7}-\frac{6431}{80000}a^{6}+\frac{6533}{32000}a^{5}-\frac{33}{160}a^{4}-\frac{13}{40}a^{3}+\frac{1501}{10000}a^{2}-\frac{19}{400}a+\frac{11}{500}$, $\frac{1}{2560000}a^{21}-\frac{1}{512000}a^{19}+\frac{1}{256000}a^{18}+\frac{7}{640000}a^{17}-\frac{3}{320000}a^{16}-\frac{11}{256000}a^{15}+\frac{1}{128000}a^{14}-\frac{1519}{2560000}a^{13}+\frac{1651}{320000}a^{12}-\frac{11817}{2560000}a^{11}-\frac{959}{256000}a^{10}-\frac{1167}{256000}a^{9}-\frac{2139}{160000}a^{8}-\frac{57789}{640000}a^{7}-\frac{3679}{64000}a^{6}-\frac{2817}{64000}a^{5}+\frac{59}{250}a^{4}+\frac{1791}{20000}a^{3}+\frac{147}{1000}a^{2}+\frac{679}{4000}a-\frac{67}{200}$, $\frac{1}{112640000}a^{22}-\frac{1}{5120000}a^{21}+\frac{7}{22528000}a^{20}-\frac{1}{704000}a^{19}+\frac{53}{7040000}a^{18}+\frac{1}{140800}a^{17}-\frac{991}{56320000}a^{16}+\frac{397}{2816000}a^{15}-\frac{8759}{112640000}a^{14}-\frac{84727}{56320000}a^{13}+\frac{36127}{112640000}a^{12}+\frac{73}{14080000}a^{11}+\frac{2203}{11264000}a^{10}+\frac{1540089}{28160000}a^{9}+\frac{779843}{28160000}a^{8}-\frac{78051}{880000}a^{7}+\frac{239561}{2816000}a^{6}-\frac{293801}{1408000}a^{5}-\frac{73669}{880000}a^{4}+\frac{12809}{440000}a^{3}-\frac{36681}{176000}a^{2}-\frac{11679}{88000}a+\frac{19}{80}$, $\frac{1}{563200000}a^{23}-\frac{53}{563200000}a^{21}+\frac{41}{281600000}a^{20}-\frac{503}{140800000}a^{19}+\frac{67}{70400000}a^{18}+\frac{1033}{281600000}a^{17}-\frac{4203}{140800000}a^{16}+\frac{42193}{563200000}a^{15}+\frac{9181}{70400000}a^{14}+\frac{118827}{112640000}a^{13}+\frac{1146129}{281600000}a^{12}+\frac{671521}{281600000}a^{11}+\frac{217077}{35200000}a^{10}-\frac{7455329}{140800000}a^{9}+\frac{4066769}{70400000}a^{8}+\frac{1070347}{14080000}a^{7}-\frac{130187}{2200000}a^{6}+\frac{2116957}{8800000}a^{5}+\frac{16803}{220000}a^{4}-\frac{267533}{4400000}a^{3}+\frac{426483}{1100000}a^{2}-\frac{2619}{10000}a-\frac{321}{2500}$, $\frac{1}{4505600000}a^{24}-\frac{1}{2252800000}a^{23}-\frac{1}{563200000}a^{22}+\frac{369}{2252800000}a^{21}+\frac{279}{4505600000}a^{20}+\frac{13}{20480000}a^{19}-\frac{16723}{2252800000}a^{18}+\frac{909}{70400000}a^{17}+\frac{27107}{4505600000}a^{16}+\frac{177371}{2252800000}a^{15}+\frac{40771}{1126400000}a^{14}+\frac{1778049}{2252800000}a^{13}+\frac{23996641}{4505600000}a^{12}+\frac{1247}{6400000}a^{11}-\frac{955639}{450560000}a^{10}+\frac{4548641}{102400000}a^{9}-\frac{19005311}{1126400000}a^{8}+\frac{2385191}{35200000}a^{7}+\frac{2372789}{563200000}a^{6}+\frac{44113149}{281600000}a^{5}-\frac{544151}{4400000}a^{4}+\frac{25573}{200000}a^{3}+\frac{14742831}{35200000}a^{2}-\frac{1544749}{3520000}a-\frac{15667}{80000}$, $\frac{1}{28\!\cdots\!00}a^{25}+\frac{8087}{109860044800000}a^{24}+\frac{39841}{357045145600000}a^{23}+\frac{3759753}{14\!\cdots\!00}a^{22}-\frac{377388441}{28\!\cdots\!00}a^{21}+\frac{23660331}{64917299200000}a^{20}-\frac{5183777987}{14\!\cdots\!00}a^{19}-\frac{569074989}{178522572800000}a^{18}-\frac{592804811}{51933839360000}a^{17}+\frac{43821262031}{14\!\cdots\!00}a^{16}-\frac{945134037}{19299737600000}a^{15}-\frac{138181912887}{14\!\cdots\!00}a^{14}+\frac{1601317453233}{28\!\cdots\!00}a^{13}-\frac{2002120782427}{357045145600000}a^{12}-\frac{6337562499}{2285088931840}a^{11}+\frac{1227461219823}{714090291200000}a^{10}+\frac{1299465412217}{714090291200000}a^{9}-\frac{586774146243}{17852257280000}a^{8}+\frac{22179088313333}{357045145600000}a^{7}-\frac{2396538726991}{178522572800000}a^{6}+\frac{3437802170833}{22315321600000}a^{5}+\frac{10624848313}{43584612500}a^{4}+\frac{7433439124431}{22315321600000}a^{3}+\frac{2853887145371}{11157660800000}a^{2}+\frac{242325074573}{557883040000}a-\frac{1555564183}{6339580000}$, $\frac{1}{15\!\cdots\!00}a^{26}+\frac{25\!\cdots\!03}{14\!\cdots\!00}a^{25}-\frac{23\!\cdots\!77}{38\!\cdots\!00}a^{24}+\frac{40\!\cdots\!67}{77\!\cdots\!00}a^{23}-\frac{30\!\cdots\!51}{15\!\cdots\!00}a^{22}-\frac{18\!\cdots\!83}{15\!\cdots\!00}a^{21}-\frac{13\!\cdots\!71}{38\!\cdots\!00}a^{20}+\frac{23\!\cdots\!87}{77\!\cdots\!00}a^{19}-\frac{48\!\cdots\!77}{15\!\cdots\!00}a^{18}-\frac{47\!\cdots\!97}{15\!\cdots\!00}a^{17}+\frac{26\!\cdots\!71}{29\!\cdots\!00}a^{16}-\frac{46\!\cdots\!07}{77\!\cdots\!00}a^{15}+\frac{71\!\cdots\!19}{15\!\cdots\!00}a^{14}+\frac{84\!\cdots\!69}{14\!\cdots\!00}a^{13}+\frac{20\!\cdots\!37}{38\!\cdots\!00}a^{12}+\frac{11\!\cdots\!93}{77\!\cdots\!00}a^{11}-\frac{23\!\cdots\!33}{38\!\cdots\!00}a^{10}-\frac{17\!\cdots\!67}{38\!\cdots\!00}a^{9}-\frac{31\!\cdots\!11}{97\!\cdots\!00}a^{8}+\frac{21\!\cdots\!73}{19\!\cdots\!00}a^{7}+\frac{44\!\cdots\!99}{49\!\cdots\!00}a^{6}+\frac{26\!\cdots\!61}{48\!\cdots\!00}a^{5}+\frac{56\!\cdots\!47}{93\!\cdots\!00}a^{4}+\frac{58\!\cdots\!07}{12\!\cdots\!00}a^{3}+\frac{25\!\cdots\!59}{15\!\cdots\!00}a^{2}+\frac{41\!\cdots\!21}{30\!\cdots\!00}a+\frac{12\!\cdots\!13}{13\!\cdots\!00}$ 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

$C_{27}$, which has order $27$ (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:   $\frac{22\!\cdots\!03}{38\!\cdots\!00}a^{26}+\frac{95\!\cdots\!09}{38\!\cdots\!00}a^{25}-\frac{14\!\cdots\!71}{96\!\cdots\!00}a^{24}-\frac{19\!\cdots\!39}{19\!\cdots\!00}a^{23}+\frac{10\!\cdots\!47}{38\!\cdots\!00}a^{22}+\frac{14\!\cdots\!51}{35\!\cdots\!00}a^{21}+\frac{13\!\cdots\!91}{48\!\cdots\!00}a^{20}-\frac{15\!\cdots\!89}{19\!\cdots\!00}a^{19}-\frac{83\!\cdots\!01}{35\!\cdots\!00}a^{18}+\frac{34\!\cdots\!59}{38\!\cdots\!00}a^{17}+\frac{53\!\cdots\!63}{74\!\cdots\!00}a^{16}+\frac{23\!\cdots\!39}{19\!\cdots\!00}a^{15}-\frac{14\!\cdots\!63}{38\!\cdots\!00}a^{14}-\frac{82\!\cdots\!41}{38\!\cdots\!00}a^{13}-\frac{12\!\cdots\!07}{48\!\cdots\!00}a^{12}+\frac{14\!\cdots\!89}{19\!\cdots\!00}a^{11}+\frac{31\!\cdots\!21}{96\!\cdots\!00}a^{10}+\frac{29\!\cdots\!69}{96\!\cdots\!00}a^{9}-\frac{94\!\cdots\!89}{12\!\cdots\!00}a^{8}-\frac{11\!\cdots\!31}{48\!\cdots\!00}a^{7}-\frac{35\!\cdots\!41}{24\!\cdots\!00}a^{6}+\frac{23\!\cdots\!13}{12\!\cdots\!00}a^{5}+\frac{10\!\cdots\!41}{23\!\cdots\!00}a^{4}+\frac{23\!\cdots\!51}{30\!\cdots\!00}a^{3}+\frac{28\!\cdots\!51}{18\!\cdots\!00}a^{2}+\frac{10\!\cdots\!23}{69\!\cdots\!00}a-\frac{83\!\cdots\!11}{34\!\cdots\!00}$, $\frac{33\!\cdots\!87}{23\!\cdots\!00}a^{26}-\frac{12\!\cdots\!23}{46\!\cdots\!00}a^{25}-\frac{44\!\cdots\!83}{11\!\cdots\!00}a^{24}-\frac{10\!\cdots\!81}{73\!\cdots\!00}a^{23}+\frac{13\!\cdots\!79}{11\!\cdots\!00}a^{22}+\frac{38\!\cdots\!43}{46\!\cdots\!00}a^{21}-\frac{34\!\cdots\!53}{21\!\cdots\!00}a^{20}-\frac{43\!\cdots\!07}{23\!\cdots\!00}a^{19}-\frac{22\!\cdots\!89}{23\!\cdots\!00}a^{18}+\frac{14\!\cdots\!87}{46\!\cdots\!00}a^{17}+\frac{78\!\cdots\!47}{73\!\cdots\!00}a^{16}-\frac{73\!\cdots\!79}{11\!\cdots\!00}a^{15}-\frac{33\!\cdots\!89}{29\!\cdots\!00}a^{14}-\frac{10\!\cdots\!59}{48\!\cdots\!00}a^{13}+\frac{62\!\cdots\!41}{23\!\cdots\!00}a^{12}+\frac{45\!\cdots\!17}{23\!\cdots\!00}a^{11}+\frac{72\!\cdots\!97}{29\!\cdots\!00}a^{10}-\frac{41\!\cdots\!21}{90\!\cdots\!00}a^{9}-\frac{72\!\cdots\!71}{45\!\cdots\!00}a^{8}-\frac{21\!\cdots\!03}{58\!\cdots\!00}a^{7}+\frac{40\!\cdots\!73}{19\!\cdots\!00}a^{6}+\frac{45\!\cdots\!39}{14\!\cdots\!00}a^{5}+\frac{96\!\cdots\!77}{18\!\cdots\!00}a^{4}+\frac{27\!\cdots\!23}{36\!\cdots\!00}a^{3}-\frac{87\!\cdots\!07}{22\!\cdots\!00}a^{2}-\frac{16\!\cdots\!81}{93\!\cdots\!00}a+\frac{29\!\cdots\!27}{41\!\cdots\!00}$, $\frac{34\!\cdots\!47}{70\!\cdots\!00}a^{26}+\frac{16\!\cdots\!17}{93\!\cdots\!00}a^{25}-\frac{47\!\cdots\!87}{36\!\cdots\!00}a^{24}-\frac{12\!\cdots\!51}{14\!\cdots\!00}a^{23}+\frac{11\!\cdots\!73}{46\!\cdots\!00}a^{22}+\frac{33\!\cdots\!03}{93\!\cdots\!00}a^{21}+\frac{25\!\cdots\!13}{11\!\cdots\!00}a^{20}-\frac{31\!\cdots\!67}{46\!\cdots\!00}a^{19}-\frac{23\!\cdots\!11}{11\!\cdots\!00}a^{18}+\frac{72\!\cdots\!07}{93\!\cdots\!00}a^{17}+\frac{28\!\cdots\!89}{46\!\cdots\!00}a^{16}+\frac{23\!\cdots\!01}{23\!\cdots\!00}a^{15}-\frac{15\!\cdots\!47}{46\!\cdots\!00}a^{14}-\frac{17\!\cdots\!19}{96\!\cdots\!00}a^{13}-\frac{49\!\cdots\!67}{23\!\cdots\!00}a^{12}+\frac{46\!\cdots\!43}{71\!\cdots\!00}a^{11}+\frac{63\!\cdots\!53}{23\!\cdots\!00}a^{10}+\frac{55\!\cdots\!37}{23\!\cdots\!00}a^{9}-\frac{40\!\cdots\!39}{58\!\cdots\!00}a^{8}-\frac{22\!\cdots\!23}{11\!\cdots\!00}a^{7}-\frac{34\!\cdots\!31}{30\!\cdots\!00}a^{6}+\frac{26\!\cdots\!97}{14\!\cdots\!00}a^{5}+\frac{17\!\cdots\!79}{45\!\cdots\!00}a^{4}+\frac{46\!\cdots\!23}{73\!\cdots\!00}a^{3}+\frac{45\!\cdots\!29}{36\!\cdots\!00}a^{2}+\frac{10\!\cdots\!51}{42\!\cdots\!00}a-\frac{10\!\cdots\!59}{41\!\cdots\!00}$, $\frac{17\!\cdots\!03}{93\!\cdots\!00}a^{26}+\frac{32\!\cdots\!39}{93\!\cdots\!00}a^{25}-\frac{43\!\cdots\!33}{10\!\cdots\!00}a^{24}-\frac{15\!\cdots\!49}{42\!\cdots\!00}a^{23}+\frac{32\!\cdots\!67}{93\!\cdots\!00}a^{22}+\frac{12\!\cdots\!91}{93\!\cdots\!00}a^{21}+\frac{39\!\cdots\!27}{14\!\cdots\!00}a^{20}-\frac{89\!\cdots\!99}{46\!\cdots\!00}a^{19}-\frac{68\!\cdots\!27}{72\!\cdots\!00}a^{18}+\frac{13\!\cdots\!29}{93\!\cdots\!00}a^{17}+\frac{13\!\cdots\!61}{58\!\cdots\!00}a^{16}+\frac{31\!\cdots\!19}{46\!\cdots\!00}a^{15}+\frac{97\!\cdots\!97}{93\!\cdots\!00}a^{14}-\frac{41\!\cdots\!03}{96\!\cdots\!00}a^{13}-\frac{10\!\cdots\!27}{11\!\cdots\!00}a^{12}+\frac{25\!\cdots\!19}{46\!\cdots\!00}a^{11}+\frac{13\!\cdots\!11}{23\!\cdots\!00}a^{10}+\frac{18\!\cdots\!99}{23\!\cdots\!00}a^{9}-\frac{15\!\cdots\!29}{29\!\cdots\!00}a^{8}-\frac{21\!\cdots\!21}{11\!\cdots\!00}a^{7}+\frac{19\!\cdots\!79}{15\!\cdots\!00}a^{6}+\frac{19\!\cdots\!13}{29\!\cdots\!00}a^{5}+\frac{48\!\cdots\!53}{73\!\cdots\!00}a^{4}+\frac{26\!\cdots\!87}{51\!\cdots\!00}a^{3}+\frac{30\!\cdots\!59}{16\!\cdots\!00}a^{2}-\frac{33\!\cdots\!27}{18\!\cdots\!00}a+\frac{31\!\cdots\!09}{83\!\cdots\!00}$, $\frac{42\!\cdots\!07}{19\!\cdots\!00}a^{26}+\frac{10\!\cdots\!57}{38\!\cdots\!00}a^{25}-\frac{59\!\cdots\!43}{97\!\cdots\!00}a^{24}-\frac{13\!\cdots\!43}{30\!\cdots\!00}a^{23}+\frac{84\!\cdots\!39}{97\!\cdots\!00}a^{22}+\frac{67\!\cdots\!63}{38\!\cdots\!00}a^{21}+\frac{39\!\cdots\!67}{17\!\cdots\!00}a^{20}-\frac{61\!\cdots\!07}{19\!\cdots\!00}a^{19}-\frac{23\!\cdots\!49}{19\!\cdots\!00}a^{18}+\frac{12\!\cdots\!27}{38\!\cdots\!00}a^{17}+\frac{77\!\cdots\!13}{24\!\cdots\!00}a^{16}+\frac{61\!\cdots\!41}{97\!\cdots\!00}a^{15}-\frac{49\!\cdots\!53}{30\!\cdots\!00}a^{14}-\frac{41\!\cdots\!19}{40\!\cdots\!00}a^{13}-\frac{25\!\cdots\!69}{17\!\cdots\!00}a^{12}+\frac{73\!\cdots\!37}{19\!\cdots\!00}a^{11}+\frac{42\!\cdots\!07}{24\!\cdots\!00}a^{10}+\frac{16\!\cdots\!87}{97\!\cdots\!00}a^{9}-\frac{24\!\cdots\!43}{48\!\cdots\!00}a^{8}-\frac{74\!\cdots\!43}{48\!\cdots\!00}a^{7}-\frac{51\!\cdots\!01}{80\!\cdots\!00}a^{6}+\frac{37\!\cdots\!59}{12\!\cdots\!00}a^{5}+\frac{71\!\cdots\!37}{15\!\cdots\!00}a^{4}+\frac{12\!\cdots\!63}{30\!\cdots\!00}a^{3}-\frac{13\!\cdots\!91}{86\!\cdots\!00}a^{2}-\frac{14\!\cdots\!61}{77\!\cdots\!00}a+\frac{56\!\cdots\!47}{34\!\cdots\!00}$, $\frac{15\!\cdots\!87}{38\!\cdots\!00}a^{26}+\frac{35\!\cdots\!01}{38\!\cdots\!00}a^{25}-\frac{17\!\cdots\!73}{19\!\cdots\!00}a^{24}-\frac{16\!\cdots\!81}{19\!\cdots\!00}a^{23}+\frac{14\!\cdots\!23}{38\!\cdots\!00}a^{22}+\frac{11\!\cdots\!09}{38\!\cdots\!00}a^{21}+\frac{14\!\cdots\!31}{19\!\cdots\!00}a^{20}-\frac{58\!\cdots\!97}{14\!\cdots\!00}a^{19}-\frac{91\!\cdots\!39}{38\!\cdots\!00}a^{18}+\frac{65\!\cdots\!71}{38\!\cdots\!00}a^{17}+\frac{10\!\cdots\!77}{19\!\cdots\!00}a^{16}+\frac{24\!\cdots\!47}{13\!\cdots\!00}a^{15}+\frac{37\!\cdots\!53}{38\!\cdots\!00}a^{14}-\frac{47\!\cdots\!37}{40\!\cdots\!00}a^{13}-\frac{77\!\cdots\!27}{19\!\cdots\!00}a^{12}-\frac{53\!\cdots\!89}{17\!\cdots\!00}a^{11}+\frac{67\!\cdots\!77}{48\!\cdots\!00}a^{10}+\frac{42\!\cdots\!91}{97\!\cdots\!00}a^{9}+\frac{17\!\cdots\!21}{48\!\cdots\!00}a^{8}-\frac{29\!\cdots\!19}{48\!\cdots\!00}a^{7}-\frac{22\!\cdots\!33}{12\!\cdots\!00}a^{6}-\frac{14\!\cdots\!49}{60\!\cdots\!00}a^{5}-\frac{91\!\cdots\!83}{30\!\cdots\!00}a^{4}-\frac{87\!\cdots\!01}{30\!\cdots\!00}a^{3}+\frac{25\!\cdots\!57}{13\!\cdots\!00}a^{2}+\frac{15\!\cdots\!93}{19\!\cdots\!00}a-\frac{54\!\cdots\!17}{17\!\cdots\!00}$, $\frac{39\!\cdots\!01}{77\!\cdots\!00}a^{26}+\frac{90\!\cdots\!63}{77\!\cdots\!00}a^{25}-\frac{42\!\cdots\!89}{38\!\cdots\!00}a^{24}-\frac{41\!\cdots\!83}{38\!\cdots\!00}a^{23}+\frac{29\!\cdots\!69}{77\!\cdots\!00}a^{22}+\frac{21\!\cdots\!19}{59\!\cdots\!00}a^{21}+\frac{36\!\cdots\!83}{38\!\cdots\!00}a^{20}-\frac{18\!\cdots\!23}{38\!\cdots\!00}a^{19}-\frac{22\!\cdots\!17}{77\!\cdots\!00}a^{18}+\frac{14\!\cdots\!73}{77\!\cdots\!00}a^{17}+\frac{24\!\cdots\!21}{38\!\cdots\!00}a^{16}+\frac{88\!\cdots\!23}{38\!\cdots\!00}a^{15}+\frac{10\!\cdots\!79}{77\!\cdots\!00}a^{14}-\frac{11\!\cdots\!91}{80\!\cdots\!00}a^{13}-\frac{19\!\cdots\!31}{38\!\cdots\!00}a^{12}-\frac{15\!\cdots\!77}{38\!\cdots\!00}a^{11}+\frac{62\!\cdots\!41}{37\!\cdots\!00}a^{10}+\frac{95\!\cdots\!83}{17\!\cdots\!00}a^{9}+\frac{45\!\cdots\!93}{97\!\cdots\!00}a^{8}-\frac{51\!\cdots\!09}{74\!\cdots\!00}a^{7}-\frac{54\!\cdots\!89}{25\!\cdots\!00}a^{6}-\frac{78\!\cdots\!03}{27\!\cdots\!00}a^{5}-\frac{22\!\cdots\!69}{60\!\cdots\!00}a^{4}-\frac{52\!\cdots\!29}{14\!\cdots\!00}a^{3}+\frac{13\!\cdots\!71}{63\!\cdots\!00}a^{2}+\frac{56\!\cdots\!91}{59\!\cdots\!00}a-\frac{33\!\cdots\!49}{86\!\cdots\!00}$, $\frac{16\!\cdots\!71}{38\!\cdots\!00}a^{26}-\frac{24\!\cdots\!51}{19\!\cdots\!00}a^{25}-\frac{30\!\cdots\!23}{35\!\cdots\!00}a^{24}-\frac{14\!\cdots\!69}{37\!\cdots\!00}a^{23}+\frac{39\!\cdots\!37}{10\!\cdots\!00}a^{22}+\frac{37\!\cdots\!01}{19\!\cdots\!00}a^{21}-\frac{15\!\cdots\!23}{27\!\cdots\!00}a^{20}-\frac{43\!\cdots\!19}{97\!\cdots\!00}a^{19}-\frac{96\!\cdots\!37}{38\!\cdots\!00}a^{18}+\frac{15\!\cdots\!49}{19\!\cdots\!00}a^{17}+\frac{96\!\cdots\!17}{38\!\cdots\!00}a^{16}-\frac{14\!\cdots\!91}{60\!\cdots\!00}a^{15}-\frac{10\!\cdots\!61}{38\!\cdots\!00}a^{14}-\frac{10\!\cdots\!73}{20\!\cdots\!00}a^{13}+\frac{26\!\cdots\!73}{38\!\cdots\!00}a^{12}+\frac{44\!\cdots\!89}{97\!\cdots\!00}a^{11}+\frac{10\!\cdots\!49}{19\!\cdots\!00}a^{10}-\frac{10\!\cdots\!87}{97\!\cdots\!00}a^{9}-\frac{92\!\cdots\!47}{26\!\cdots\!00}a^{8}-\frac{36\!\cdots\!61}{24\!\cdots\!00}a^{7}+\frac{10\!\cdots\!17}{25\!\cdots\!00}a^{6}+\frac{97\!\cdots\!67}{24\!\cdots\!00}a^{5}+\frac{25\!\cdots\!21}{30\!\cdots\!00}a^{4}+\frac{33\!\cdots\!01}{15\!\cdots\!00}a^{3}-\frac{24\!\cdots\!63}{30\!\cdots\!00}a^{2}-\frac{76\!\cdots\!33}{14\!\cdots\!00}a+\frac{12\!\cdots\!71}{69\!\cdots\!00}$, $\frac{25\!\cdots\!71}{77\!\cdots\!00}a^{26}+\frac{20\!\cdots\!09}{38\!\cdots\!00}a^{25}-\frac{47\!\cdots\!59}{38\!\cdots\!00}a^{24}-\frac{23\!\cdots\!43}{38\!\cdots\!00}a^{23}+\frac{39\!\cdots\!17}{21\!\cdots\!00}a^{22}+\frac{28\!\cdots\!79}{97\!\cdots\!00}a^{21}+\frac{25\!\cdots\!09}{19\!\cdots\!00}a^{20}-\frac{46\!\cdots\!49}{74\!\cdots\!00}a^{19}-\frac{11\!\cdots\!67}{77\!\cdots\!00}a^{18}+\frac{30\!\cdots\!69}{38\!\cdots\!00}a^{17}+\frac{18\!\cdots\!41}{38\!\cdots\!00}a^{16}+\frac{12\!\cdots\!81}{29\!\cdots\!00}a^{15}-\frac{30\!\cdots\!01}{77\!\cdots\!00}a^{14}-\frac{23\!\cdots\!69}{18\!\cdots\!00}a^{13}-\frac{18\!\cdots\!99}{97\!\cdots\!00}a^{12}+\frac{13\!\cdots\!99}{17\!\cdots\!00}a^{11}+\frac{74\!\cdots\!53}{44\!\cdots\!00}a^{10}-\frac{51\!\cdots\!41}{97\!\cdots\!00}a^{9}-\frac{52\!\cdots\!69}{65\!\cdots\!00}a^{8}-\frac{38\!\cdots\!21}{48\!\cdots\!00}a^{7}+\frac{27\!\cdots\!91}{25\!\cdots\!00}a^{6}+\frac{21\!\cdots\!81}{24\!\cdots\!00}a^{5}+\frac{22\!\cdots\!71}{55\!\cdots\!00}a^{4}+\frac{21\!\cdots\!61}{30\!\cdots\!00}a^{3}-\frac{56\!\cdots\!69}{30\!\cdots\!00}a^{2}-\frac{26\!\cdots\!09}{15\!\cdots\!00}a+\frac{36\!\cdots\!73}{69\!\cdots\!00}$, $\frac{10\!\cdots\!17}{15\!\cdots\!00}a^{26}-\frac{10\!\cdots\!41}{15\!\cdots\!00}a^{25}-\frac{15\!\cdots\!47}{77\!\cdots\!00}a^{24}-\frac{22\!\cdots\!51}{19\!\cdots\!00}a^{23}+\frac{15\!\cdots\!67}{35\!\cdots\!00}a^{22}+\frac{17\!\cdots\!93}{31\!\cdots\!00}a^{21}+\frac{23\!\cdots\!31}{77\!\cdots\!00}a^{20}-\frac{79\!\cdots\!37}{77\!\cdots\!00}a^{19}-\frac{26\!\cdots\!27}{77\!\cdots\!00}a^{18}+\frac{12\!\cdots\!81}{15\!\cdots\!00}a^{17}+\frac{14\!\cdots\!23}{19\!\cdots\!00}a^{16}+\frac{47\!\cdots\!53}{38\!\cdots\!00}a^{15}-\frac{37\!\cdots\!99}{93\!\cdots\!00}a^{14}-\frac{29\!\cdots\!31}{14\!\cdots\!00}a^{13}-\frac{62\!\cdots\!53}{31\!\cdots\!00}a^{12}+\frac{65\!\cdots\!47}{77\!\cdots\!00}a^{11}+\frac{28\!\cdots\!01}{97\!\cdots\!00}a^{10}+\frac{51\!\cdots\!39}{35\!\cdots\!00}a^{9}-\frac{37\!\cdots\!21}{38\!\cdots\!00}a^{8}-\frac{36\!\cdots\!01}{19\!\cdots\!00}a^{7}+\frac{75\!\cdots\!81}{21\!\cdots\!00}a^{6}+\frac{13\!\cdots\!53}{48\!\cdots\!00}a^{5}+\frac{16\!\cdots\!39}{60\!\cdots\!00}a^{4}+\frac{49\!\cdots\!93}{12\!\cdots\!00}a^{3}+\frac{19\!\cdots\!87}{19\!\cdots\!00}a^{2}-\frac{31\!\cdots\!31}{30\!\cdots\!00}a+\frac{29\!\cdots\!77}{13\!\cdots\!00}$, $\frac{21\!\cdots\!97}{77\!\cdots\!00}a^{26}-\frac{29\!\cdots\!57}{38\!\cdots\!00}a^{25}-\frac{39\!\cdots\!57}{60\!\cdots\!00}a^{24}-\frac{94\!\cdots\!31}{38\!\cdots\!00}a^{23}+\frac{18\!\cdots\!83}{77\!\cdots\!00}a^{22}+\frac{27\!\cdots\!01}{19\!\cdots\!00}a^{21}-\frac{14\!\cdots\!89}{35\!\cdots\!00}a^{20}-\frac{15\!\cdots\!17}{48\!\cdots\!00}a^{19}+\frac{54\!\cdots\!11}{77\!\cdots\!00}a^{18}+\frac{21\!\cdots\!03}{38\!\cdots\!00}a^{17}+\frac{33\!\cdots\!01}{19\!\cdots\!00}a^{16}-\frac{71\!\cdots\!79}{38\!\cdots\!00}a^{15}-\frac{10\!\cdots\!69}{54\!\cdots\!00}a^{14}-\frac{33\!\cdots\!59}{10\!\cdots\!00}a^{13}+\frac{17\!\cdots\!43}{35\!\cdots\!00}a^{12}+\frac{62\!\cdots\!83}{19\!\cdots\!00}a^{11}+\frac{66\!\cdots\!69}{19\!\cdots\!00}a^{10}-\frac{24\!\cdots\!01}{30\!\cdots\!00}a^{9}-\frac{22\!\cdots\!19}{97\!\cdots\!00}a^{8}-\frac{37\!\cdots\!27}{48\!\cdots\!00}a^{7}+\frac{19\!\cdots\!03}{64\!\cdots\!00}a^{6}+\frac{87\!\cdots\!33}{55\!\cdots\!00}a^{5}+\frac{45\!\cdots\!47}{60\!\cdots\!00}a^{4}+\frac{37\!\cdots\!27}{30\!\cdots\!00}a^{3}-\frac{20\!\cdots\!11}{38\!\cdots\!00}a^{2}-\frac{11\!\cdots\!47}{38\!\cdots\!00}a+\frac{18\!\cdots\!99}{17\!\cdots\!00}$, $\frac{21\!\cdots\!63}{29\!\cdots\!00}a^{26}-\frac{82\!\cdots\!53}{29\!\cdots\!00}a^{25}-\frac{22\!\cdots\!71}{14\!\cdots\!00}a^{24}-\frac{10\!\cdots\!53}{39\!\cdots\!00}a^{23}+\frac{21\!\cdots\!07}{29\!\cdots\!00}a^{22}+\frac{21\!\cdots\!67}{79\!\cdots\!00}a^{21}-\frac{27\!\cdots\!29}{14\!\cdots\!00}a^{20}-\frac{10\!\cdots\!99}{14\!\cdots\!00}a^{19}+\frac{11\!\cdots\!01}{58\!\cdots\!00}a^{18}+\frac{48\!\cdots\!79}{26\!\cdots\!00}a^{17}+\frac{24\!\cdots\!03}{14\!\cdots\!00}a^{16}-\frac{28\!\cdots\!03}{14\!\cdots\!00}a^{15}-\frac{14\!\cdots\!39}{29\!\cdots\!00}a^{14}+\frac{27\!\cdots\!41}{23\!\cdots\!00}a^{13}+\frac{13\!\cdots\!97}{14\!\cdots\!00}a^{12}+\frac{20\!\cdots\!17}{13\!\cdots\!00}a^{11}-\frac{62\!\cdots\!83}{28\!\cdots\!00}a^{10}-\frac{21\!\cdots\!39}{14\!\cdots\!00}a^{9}-\frac{83\!\cdots\!83}{36\!\cdots\!00}a^{8}+\frac{38\!\cdots\!79}{36\!\cdots\!00}a^{7}+\frac{65\!\cdots\!57}{96\!\cdots\!00}a^{6}+\frac{49\!\cdots\!41}{45\!\cdots\!00}a^{5}+\frac{19\!\cdots\!53}{22\!\cdots\!00}a^{4}+\frac{98\!\cdots\!37}{45\!\cdots\!00}a^{3}-\frac{98\!\cdots\!67}{11\!\cdots\!00}a^{2}-\frac{16\!\cdots\!77}{29\!\cdots\!00}a+\frac{25\!\cdots\!57}{13\!\cdots\!00}$, $\frac{22\!\cdots\!69}{77\!\cdots\!00}a^{26}+\frac{79\!\cdots\!67}{77\!\cdots\!00}a^{25}-\frac{30\!\cdots\!21}{38\!\cdots\!00}a^{24}-\frac{19\!\cdots\!07}{38\!\cdots\!00}a^{23}+\frac{11\!\cdots\!81}{77\!\cdots\!00}a^{22}+\frac{16\!\cdots\!83}{77\!\cdots\!00}a^{21}+\frac{49\!\cdots\!47}{38\!\cdots\!00}a^{20}-\frac{15\!\cdots\!07}{38\!\cdots\!00}a^{19}-\frac{91\!\cdots\!73}{77\!\cdots\!00}a^{18}+\frac{35\!\cdots\!17}{77\!\cdots\!00}a^{17}+\frac{10\!\cdots\!53}{29\!\cdots\!00}a^{16}+\frac{22\!\cdots\!47}{38\!\cdots\!00}a^{15}-\frac{15\!\cdots\!89}{77\!\cdots\!00}a^{14}-\frac{84\!\cdots\!99}{80\!\cdots\!00}a^{13}-\frac{48\!\cdots\!79}{38\!\cdots\!00}a^{12}+\frac{15\!\cdots\!67}{38\!\cdots\!00}a^{11}+\frac{19\!\cdots\!63}{12\!\cdots\!00}a^{10}+\frac{24\!\cdots\!47}{17\!\cdots\!00}a^{9}-\frac{39\!\cdots\!63}{97\!\cdots\!00}a^{8}-\frac{11\!\cdots\!13}{97\!\cdots\!00}a^{7}-\frac{12\!\cdots\!97}{19\!\cdots\!00}a^{6}+\frac{65\!\cdots\!11}{60\!\cdots\!00}a^{5}+\frac{10\!\cdots\!43}{46\!\cdots\!00}a^{4}+\frac{22\!\cdots\!73}{60\!\cdots\!00}a^{3}+\frac{22\!\cdots\!19}{30\!\cdots\!00}a^{2}+\frac{88\!\cdots\!97}{77\!\cdots\!00}a-\frac{25\!\cdots\!37}{17\!\cdots\!00}$ 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:  \( 1273307270352048000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 1273307270352048000000000 \cdot 27}{2\cdot\sqrt{57914576815317377556250885803017413799391835668480000000000000}}\cr\approx \mathstrut & 107458.637075087 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^27 - 27*x^25 - 162*x^24 + 567*x^23 + 7020*x^22 + 1845*x^21 - 137214*x^20 - 356859*x^19 + 1719144*x^18 + 11751723*x^17 + 15524730*x^16 - 75425499*x^15 - 340193844*x^14 - 302723001*x^13 + 1484785422*x^12 + 5038976790*x^11 + 2884521888*x^10 - 15795303852*x^9 - 34353915048*x^8 - 8492734008*x^7 + 44721504000*x^6 + 62398540512*x^5 + 101058043392*x^4 + 207332604288*x^3 - 84572356608*x^2 - 52216496640*x + 18041661440)
 
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 - 27*x^25 - 162*x^24 + 567*x^23 + 7020*x^22 + 1845*x^21 - 137214*x^20 - 356859*x^19 + 1719144*x^18 + 11751723*x^17 + 15524730*x^16 - 75425499*x^15 - 340193844*x^14 - 302723001*x^13 + 1484785422*x^12 + 5038976790*x^11 + 2884521888*x^10 - 15795303852*x^9 - 34353915048*x^8 - 8492734008*x^7 + 44721504000*x^6 + 62398540512*x^5 + 101058043392*x^4 + 207332604288*x^3 - 84572356608*x^2 - 52216496640*x + 18041661440, 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 - 27*x^25 - 162*x^24 + 567*x^23 + 7020*x^22 + 1845*x^21 - 137214*x^20 - 356859*x^19 + 1719144*x^18 + 11751723*x^17 + 15524730*x^16 - 75425499*x^15 - 340193844*x^14 - 302723001*x^13 + 1484785422*x^12 + 5038976790*x^11 + 2884521888*x^10 - 15795303852*x^9 - 34353915048*x^8 - 8492734008*x^7 + 44721504000*x^6 + 62398540512*x^5 + 101058043392*x^4 + 207332604288*x^3 - 84572356608*x^2 - 52216496640*x + 18041661440);
 
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 - 27*x^25 - 162*x^24 + 567*x^23 + 7020*x^22 + 1845*x^21 - 137214*x^20 - 356859*x^19 + 1719144*x^18 + 11751723*x^17 + 15524730*x^16 - 75425499*x^15 - 340193844*x^14 - 302723001*x^13 + 1484785422*x^12 + 5038976790*x^11 + 2884521888*x^10 - 15795303852*x^9 - 34353915048*x^8 - 8492734008*x^7 + 44721504000*x^6 + 62398540512*x^5 + 101058043392*x^4 + 207332604288*x^3 - 84572356608*x^2 - 52216496640*x + 18041661440);
 
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

3.1.1620.1, 9.1.5020969537440000.1

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 R $27$ ${\href{/padicField/11.2.0.1}{2} }^{13}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.2.0.1}{2} }^{13}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.2.0.1}{2} }^{13}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{13}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $27$ $27$ ${\href{/padicField/31.2.0.1}{2} }^{13}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{13}{,}\,{\href{/padicField/37.1.0.1}{1} }$ $27$ ${\href{/padicField/43.9.0.1}{9} }^{3}$ $27$ ${\href{/padicField/53.2.0.1}{2} }^{13}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{13}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x + 6$$2$$1$$2$$C_2$$[2]$
\(3\) Copy content Toggle raw display Deg $27$$27$$1$$94$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.20.2t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\sqrt{-5}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1620.3t2.b.a$2$ $ 2^{2} \cdot 3^{4} \cdot 5 $ 3.1.1620.1 $S_3$ (as 3T2) $1$ $0$
* 2.14580.9t3.a.a$2$ $ 2^{2} \cdot 3^{6} \cdot 5 $ 9.1.5020969537440000.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.14580.9t3.a.b$2$ $ 2^{2} \cdot 3^{6} \cdot 5 $ 9.1.5020969537440000.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.14580.9t3.a.c$2$ $ 2^{2} \cdot 3^{6} \cdot 5 $ 9.1.5020969537440000.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.131220.27t8.a.e$2$ $ 2^{2} \cdot 3^{8} \cdot 5 $ 27.1.57914576815317377556250885803017413799391835668480000000000000.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.131220.27t8.a.f$2$ $ 2^{2} \cdot 3^{8} \cdot 5 $ 27.1.57914576815317377556250885803017413799391835668480000000000000.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.131220.27t8.a.a$2$ $ 2^{2} \cdot 3^{8} \cdot 5 $ 27.1.57914576815317377556250885803017413799391835668480000000000000.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.131220.27t8.a.h$2$ $ 2^{2} \cdot 3^{8} \cdot 5 $ 27.1.57914576815317377556250885803017413799391835668480000000000000.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.131220.27t8.a.c$2$ $ 2^{2} \cdot 3^{8} \cdot 5 $ 27.1.57914576815317377556250885803017413799391835668480000000000000.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.131220.27t8.a.i$2$ $ 2^{2} \cdot 3^{8} \cdot 5 $ 27.1.57914576815317377556250885803017413799391835668480000000000000.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.131220.27t8.a.g$2$ $ 2^{2} \cdot 3^{8} \cdot 5 $ 27.1.57914576815317377556250885803017413799391835668480000000000000.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.131220.27t8.a.d$2$ $ 2^{2} \cdot 3^{8} \cdot 5 $ 27.1.57914576815317377556250885803017413799391835668480000000000000.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.131220.27t8.a.b$2$ $ 2^{2} \cdot 3^{8} \cdot 5 $ 27.1.57914576815317377556250885803017413799391835668480000000000000.1 $D_{27}$ (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 *.