Properties

Label 28.0.141...481.1
Degree $28$
Signature $[0, 14]$
Discriminant $1.412\times 10^{49}$
Root discriminant \(56.93\)
Ramified primes $3,43$
Class number $203$ (GRH)
Class group [203] (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - x^26 - 50*x^25 + 45*x^24 + 41*x^23 + 923*x^22 - 740*x^21 - 566*x^20 - 7986*x^19 + 5482*x^18 + 3755*x^17 + 34135*x^16 - 15421*x^15 - 15266*x^14 - 71623*x^13 - 3964*x^12 + 45122*x^11 + 87041*x^10 + 49398*x^9 - 13057*x^8 - 105029*x^7 - 52559*x^6 - 58937*x^5 + 57132*x^4 + 53384*x^3 + 9659*x^2 + 13272*x + 6241)
 
gp: K = bnfinit(y^28 - y^27 - y^26 - 50*y^25 + 45*y^24 + 41*y^23 + 923*y^22 - 740*y^21 - 566*y^20 - 7986*y^19 + 5482*y^18 + 3755*y^17 + 34135*y^16 - 15421*y^15 - 15266*y^14 - 71623*y^13 - 3964*y^12 + 45122*y^11 + 87041*y^10 + 49398*y^9 - 13057*y^8 - 105029*y^7 - 52559*y^6 - 58937*y^5 + 57132*y^4 + 53384*y^3 + 9659*y^2 + 13272*y + 6241, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - x^27 - x^26 - 50*x^25 + 45*x^24 + 41*x^23 + 923*x^22 - 740*x^21 - 566*x^20 - 7986*x^19 + 5482*x^18 + 3755*x^17 + 34135*x^16 - 15421*x^15 - 15266*x^14 - 71623*x^13 - 3964*x^12 + 45122*x^11 + 87041*x^10 + 49398*x^9 - 13057*x^8 - 105029*x^7 - 52559*x^6 - 58937*x^5 + 57132*x^4 + 53384*x^3 + 9659*x^2 + 13272*x + 6241);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - x^27 - x^26 - 50*x^25 + 45*x^24 + 41*x^23 + 923*x^22 - 740*x^21 - 566*x^20 - 7986*x^19 + 5482*x^18 + 3755*x^17 + 34135*x^16 - 15421*x^15 - 15266*x^14 - 71623*x^13 - 3964*x^12 + 45122*x^11 + 87041*x^10 + 49398*x^9 - 13057*x^8 - 105029*x^7 - 52559*x^6 - 58937*x^5 + 57132*x^4 + 53384*x^3 + 9659*x^2 + 13272*x + 6241)
 

\( x^{28} - x^{27} - x^{26} - 50 x^{25} + 45 x^{24} + 41 x^{23} + 923 x^{22} - 740 x^{21} - 566 x^{20} - 7986 x^{19} + 5482 x^{18} + 3755 x^{17} + 34135 x^{16} - 15421 x^{15} - 15266 x^{14} + \cdots + 6241 \) Copy content Toggle raw display

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(14121388821225670988853483488774192350843817726481\) \(\medspace = 3^{14}\cdot 43^{26}\) 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:  \(56.93\)
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:  $3^{1/2}43^{13/14}\approx 56.93151811241331$
Ramified primes:   \(3\), \(43\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $28$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(129=3\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{129}(128,·)$, $\chi_{129}(1,·)$, $\chi_{129}(2,·)$, $\chi_{129}(4,·)$, $\chi_{129}(70,·)$, $\chi_{129}(65,·)$, $\chi_{129}(8,·)$, $\chi_{129}(11,·)$, $\chi_{129}(64,·)$, $\chi_{129}(16,·)$, $\chi_{129}(82,·)$, $\chi_{129}(85,·)$, $\chi_{129}(22,·)$, $\chi_{129}(88,·)$, $\chi_{129}(94,·)$, $\chi_{129}(32,·)$, $\chi_{129}(97,·)$, $\chi_{129}(35,·)$, $\chi_{129}(41,·)$, $\chi_{129}(107,·)$, $\chi_{129}(44,·)$, $\chi_{129}(47,·)$, $\chi_{129}(113,·)$, $\chi_{129}(118,·)$, $\chi_{129}(121,·)$, $\chi_{129}(59,·)$, $\chi_{129}(125,·)$, $\chi_{129}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{79}a^{23}+\frac{4}{79}a^{22}+\frac{30}{79}a^{21}-\frac{33}{79}a^{20}+\frac{15}{79}a^{19}-\frac{29}{79}a^{18}-\frac{25}{79}a^{17}+\frac{9}{79}a^{16}+\frac{24}{79}a^{15}+\frac{30}{79}a^{14}+\frac{10}{79}a^{13}+\frac{1}{79}a^{12}+\frac{14}{79}a^{11}+\frac{18}{79}a^{10}-\frac{36}{79}a^{9}-\frac{21}{79}a^{8}+\frac{5}{79}a^{7}+\frac{8}{79}a^{6}+\frac{21}{79}a^{5}+\frac{13}{79}a^{4}+\frac{1}{79}a^{3}+\frac{23}{79}a^{2}-\frac{37}{79}a$, $\frac{1}{553}a^{24}+\frac{3}{553}a^{23}+\frac{26}{553}a^{22}-\frac{142}{553}a^{21}-\frac{27}{79}a^{20}+\frac{114}{553}a^{19}-\frac{75}{553}a^{18}+\frac{113}{553}a^{17}+\frac{15}{553}a^{16}-\frac{73}{553}a^{15}+\frac{138}{553}a^{14}-\frac{9}{553}a^{13}-\frac{66}{553}a^{12}-\frac{22}{79}a^{11}-\frac{19}{79}a^{10}-\frac{64}{553}a^{9}+\frac{184}{553}a^{8}+\frac{23}{79}a^{7}+\frac{92}{553}a^{6}-\frac{166}{553}a^{5}+\frac{67}{553}a^{4}+\frac{37}{79}a^{3}+\frac{256}{553}a^{2}+\frac{116}{553}a-\frac{1}{7}$, $\frac{1}{169771}a^{25}-\frac{150}{169771}a^{24}+\frac{463}{169771}a^{23}-\frac{48094}{169771}a^{22}-\frac{72137}{169771}a^{21}+\frac{38726}{169771}a^{20}+\frac{66707}{169771}a^{19}+\frac{19337}{169771}a^{18}-\frac{12024}{169771}a^{17}+\frac{22839}{169771}a^{16}-\frac{6452}{169771}a^{15}+\frac{26771}{169771}a^{14}+\frac{83267}{169771}a^{13}+\frac{30748}{169771}a^{12}+\frac{1979}{24253}a^{11}-\frac{33265}{169771}a^{10}+\frac{31361}{169771}a^{9}+\frac{72641}{169771}a^{8}+\frac{75608}{169771}a^{7}-\frac{65139}{169771}a^{6}+\frac{6677}{169771}a^{5}-\frac{67469}{169771}a^{4}+\frac{79314}{169771}a^{3}+\frac{10312}{169771}a^{2}-\frac{37154}{169771}a+\frac{132}{2149}$, $\frac{1}{12\!\cdots\!43}a^{26}+\frac{25500050657}{12\!\cdots\!43}a^{25}+\frac{8172513473303}{12\!\cdots\!43}a^{24}-\frac{35401708760840}{12\!\cdots\!43}a^{23}-\frac{54\!\cdots\!08}{12\!\cdots\!43}a^{22}-\frac{45333011933414}{154539746166617}a^{21}-\frac{545176098296287}{12\!\cdots\!43}a^{20}-\frac{21\!\cdots\!61}{12\!\cdots\!43}a^{19}-\frac{50\!\cdots\!87}{12\!\cdots\!43}a^{18}+\frac{56\!\cdots\!82}{12\!\cdots\!43}a^{17}-\frac{585010836262925}{12\!\cdots\!43}a^{16}+\frac{426512719512321}{12\!\cdots\!43}a^{15}-\frac{49\!\cdots\!76}{12\!\cdots\!43}a^{14}+\frac{51\!\cdots\!38}{12\!\cdots\!43}a^{13}-\frac{36109037429781}{154539746166617}a^{12}-\frac{35\!\cdots\!48}{12\!\cdots\!43}a^{11}+\frac{49\!\cdots\!56}{12\!\cdots\!43}a^{10}-\frac{43\!\cdots\!68}{12\!\cdots\!43}a^{9}+\frac{440703115311966}{17\!\cdots\!49}a^{8}+\frac{26\!\cdots\!75}{12\!\cdots\!43}a^{7}-\frac{22\!\cdots\!43}{12\!\cdots\!43}a^{6}+\frac{39\!\cdots\!45}{12\!\cdots\!43}a^{5}-\frac{419010392445237}{12\!\cdots\!43}a^{4}-\frac{58\!\cdots\!33}{12\!\cdots\!43}a^{3}-\frac{477653372780291}{17\!\cdots\!49}a^{2}-\frac{25\!\cdots\!99}{12\!\cdots\!43}a+\frac{4722382228694}{154539746166617}$, $\frac{1}{97\!\cdots\!71}a^{27}-\frac{30\!\cdots\!27}{97\!\cdots\!71}a^{26}+\frac{21\!\cdots\!95}{97\!\cdots\!71}a^{25}-\frac{71\!\cdots\!09}{97\!\cdots\!71}a^{24}-\frac{43\!\cdots\!44}{13\!\cdots\!53}a^{23}+\frac{22\!\cdots\!20}{97\!\cdots\!71}a^{22}-\frac{48\!\cdots\!76}{97\!\cdots\!71}a^{21}+\frac{60\!\cdots\!73}{97\!\cdots\!71}a^{20}-\frac{44\!\cdots\!08}{97\!\cdots\!71}a^{19}-\frac{23\!\cdots\!20}{97\!\cdots\!71}a^{18}+\frac{28\!\cdots\!02}{97\!\cdots\!71}a^{17}-\frac{36\!\cdots\!31}{97\!\cdots\!71}a^{16}-\frac{60\!\cdots\!99}{13\!\cdots\!53}a^{15}+\frac{35\!\cdots\!30}{13\!\cdots\!53}a^{14}+\frac{43\!\cdots\!46}{97\!\cdots\!71}a^{13}-\frac{14\!\cdots\!98}{12\!\cdots\!49}a^{12}-\frac{76\!\cdots\!02}{97\!\cdots\!71}a^{11}-\frac{24\!\cdots\!71}{97\!\cdots\!71}a^{10}-\frac{20\!\cdots\!43}{97\!\cdots\!71}a^{9}+\frac{63\!\cdots\!66}{13\!\cdots\!53}a^{8}+\frac{51\!\cdots\!34}{97\!\cdots\!71}a^{7}+\frac{35\!\cdots\!39}{97\!\cdots\!71}a^{6}+\frac{21\!\cdots\!34}{97\!\cdots\!71}a^{5}-\frac{23\!\cdots\!46}{97\!\cdots\!71}a^{4}-\frac{70\!\cdots\!42}{97\!\cdots\!71}a^{3}+\frac{74\!\cdots\!46}{97\!\cdots\!71}a^{2}+\frac{23\!\cdots\!71}{97\!\cdots\!71}a+\frac{25\!\cdots\!18}{12\!\cdots\!49}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{203}$, which has order $203$ (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:   \( \frac{1469565974509958327034868919014095}{13589686546556770354122015967062022987} a^{27} - \frac{2185707542180747637832708906726247}{13589686546556770354122015967062022987} a^{26} - \frac{7423076916778607575505668606219}{172021348690592029799012860342557253} a^{25} - \frac{73439535746083352207161767503807445}{13589686546556770354122015967062022987} a^{24} + \frac{102056386983756908318367238772596210}{13589686546556770354122015967062022987} a^{23} + \frac{20112019365816358240629209106930190}{13589686546556770354122015967062022987} a^{22} + \frac{1360145432505719268638469568936869224}{13589686546556770354122015967062022987} a^{21} - \frac{1753206778039274686307086151665928270}{13589686546556770354122015967062022987} a^{20} - \frac{163205604854518903203388760690261033}{13589686546556770354122015967062022987} a^{19} - \frac{11924696640706567412378496798402125083}{13589686546556770354122015967062022987} a^{18} + \frac{13848196158548853725689842851966575139}{13589686546556770354122015967062022987} a^{17} + \frac{417485373565694113786508502403038620}{13589686546556770354122015967062022987} a^{16} + \frac{52384474480375976058742825612964545713}{13589686546556770354122015967062022987} a^{15} - \frac{47527746331131048620434744262972068572}{13589686546556770354122015967062022987} a^{14} - \frac{6234873556739011773170419315120310751}{13589686546556770354122015967062022987} a^{13} - \frac{113173553420117357450659172523095744476}{13589686546556770354122015967062022987} a^{12} + \frac{44128646302373508697263147156693373974}{13589686546556770354122015967062022987} a^{11} + \frac{57441928696070197496832368274180043024}{13589686546556770354122015967062022987} a^{10} + \frac{17870280564843121166570103024525552756}{1941383792365252907731716566723146141} a^{9} + \frac{29156360210991818865782673380619600489}{13589686546556770354122015967062022987} a^{8} - \frac{40840584820498036407055156919440751331}{13589686546556770354122015967062022987} a^{7} - \frac{164990259650080922193773411795086730354}{13589686546556770354122015967062022987} a^{6} - \frac{30186355712350454417212570511258960916}{13589686546556770354122015967062022987} a^{5} - \frac{91841198497284455494610331339211512711}{13589686546556770354122015967062022987} a^{4} + \frac{19657519881538952482738490037871420732}{1941383792365252907731716566723146141} a^{3} + \frac{40280007276170701259769319385155426104}{13589686546556770354122015967062022987} a^{2} + \frac{22718577442160810286994597934338804912}{13589686546556770354122015967062022987} a + \frac{50996908933488818958189033790849093}{24574478384370289971287551477508179} \)  (order $6$) 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:   $\frac{14\!\cdots\!63}{97\!\cdots\!71}a^{27}-\frac{16\!\cdots\!29}{97\!\cdots\!71}a^{26}-\frac{64\!\cdots\!81}{97\!\cdots\!71}a^{25}-\frac{72\!\cdots\!52}{97\!\cdots\!71}a^{24}+\frac{74\!\cdots\!70}{97\!\cdots\!71}a^{23}+\frac{19\!\cdots\!76}{97\!\cdots\!71}a^{22}+\frac{19\!\cdots\!62}{13\!\cdots\!53}a^{21}-\frac{17\!\cdots\!53}{13\!\cdots\!53}a^{20}-\frac{15\!\cdots\!71}{13\!\cdots\!53}a^{19}-\frac{11\!\cdots\!53}{97\!\cdots\!71}a^{18}+\frac{94\!\cdots\!60}{97\!\cdots\!71}a^{17}-\frac{78\!\cdots\!04}{13\!\cdots\!53}a^{16}+\frac{49\!\cdots\!74}{97\!\cdots\!71}a^{15}-\frac{29\!\cdots\!27}{97\!\cdots\!71}a^{14}+\frac{14\!\cdots\!47}{97\!\cdots\!71}a^{13}-\frac{98\!\cdots\!64}{97\!\cdots\!71}a^{12}+\frac{13\!\cdots\!72}{97\!\cdots\!71}a^{11}+\frac{27\!\cdots\!06}{97\!\cdots\!71}a^{10}+\frac{95\!\cdots\!57}{97\!\cdots\!71}a^{9}+\frac{44\!\cdots\!01}{97\!\cdots\!71}a^{8}+\frac{67\!\cdots\!20}{97\!\cdots\!71}a^{7}-\frac{10\!\cdots\!78}{97\!\cdots\!71}a^{6}-\frac{14\!\cdots\!31}{97\!\cdots\!71}a^{5}-\frac{94\!\cdots\!63}{97\!\cdots\!71}a^{4}+\frac{81\!\cdots\!62}{97\!\cdots\!71}a^{3}+\frac{93\!\cdots\!18}{97\!\cdots\!71}a^{2}+\frac{28\!\cdots\!93}{13\!\cdots\!53}a+\frac{15\!\cdots\!74}{12\!\cdots\!49}$, $\frac{42\!\cdots\!31}{97\!\cdots\!71}a^{27}-\frac{37\!\cdots\!25}{97\!\cdots\!71}a^{26}-\frac{59\!\cdots\!51}{97\!\cdots\!71}a^{25}-\frac{21\!\cdots\!42}{97\!\cdots\!71}a^{24}+\frac{21\!\cdots\!60}{12\!\cdots\!49}a^{23}+\frac{25\!\cdots\!50}{97\!\cdots\!71}a^{22}+\frac{55\!\cdots\!02}{13\!\cdots\!53}a^{21}-\frac{40\!\cdots\!47}{13\!\cdots\!53}a^{20}-\frac{53\!\cdots\!65}{13\!\cdots\!53}a^{19}-\frac{32\!\cdots\!19}{97\!\cdots\!71}a^{18}+\frac{21\!\cdots\!30}{97\!\cdots\!71}a^{17}+\frac{37\!\cdots\!40}{13\!\cdots\!53}a^{16}+\frac{13\!\cdots\!92}{97\!\cdots\!71}a^{15}-\frac{64\!\cdots\!73}{97\!\cdots\!71}a^{14}-\frac{99\!\cdots\!59}{97\!\cdots\!71}a^{13}-\frac{27\!\cdots\!00}{97\!\cdots\!71}a^{12}+\frac{52\!\cdots\!16}{97\!\cdots\!71}a^{11}+\frac{22\!\cdots\!08}{97\!\cdots\!71}a^{10}+\frac{33\!\cdots\!71}{97\!\cdots\!71}a^{9}+\frac{13\!\cdots\!45}{97\!\cdots\!71}a^{8}-\frac{85\!\cdots\!14}{97\!\cdots\!71}a^{7}-\frac{39\!\cdots\!04}{97\!\cdots\!71}a^{6}-\frac{21\!\cdots\!67}{97\!\cdots\!71}a^{5}-\frac{17\!\cdots\!19}{97\!\cdots\!71}a^{4}+\frac{26\!\cdots\!46}{97\!\cdots\!71}a^{3}+\frac{27\!\cdots\!20}{12\!\cdots\!49}a^{2}+\frac{67\!\cdots\!22}{13\!\cdots\!53}a-\frac{83\!\cdots\!56}{12\!\cdots\!49}$, $\frac{81\!\cdots\!63}{97\!\cdots\!71}a^{27}-\frac{49\!\cdots\!95}{97\!\cdots\!71}a^{26}-\frac{11\!\cdots\!71}{97\!\cdots\!71}a^{25}-\frac{58\!\cdots\!80}{13\!\cdots\!53}a^{24}+\frac{29\!\cdots\!56}{13\!\cdots\!53}a^{23}+\frac{47\!\cdots\!72}{97\!\cdots\!71}a^{22}+\frac{75\!\cdots\!76}{97\!\cdots\!71}a^{21}-\frac{31\!\cdots\!55}{97\!\cdots\!71}a^{20}-\frac{68\!\cdots\!50}{97\!\cdots\!71}a^{19}-\frac{65\!\cdots\!59}{97\!\cdots\!71}a^{18}+\frac{20\!\cdots\!73}{97\!\cdots\!71}a^{17}+\frac{45\!\cdots\!72}{97\!\cdots\!71}a^{16}+\frac{40\!\cdots\!08}{13\!\cdots\!53}a^{15}-\frac{29\!\cdots\!01}{97\!\cdots\!71}a^{14}-\frac{15\!\cdots\!77}{97\!\cdots\!71}a^{13}-\frac{59\!\cdots\!44}{97\!\cdots\!71}a^{12}-\frac{20\!\cdots\!21}{97\!\cdots\!71}a^{11}+\frac{43\!\cdots\!56}{13\!\cdots\!53}a^{10}+\frac{75\!\cdots\!57}{97\!\cdots\!71}a^{9}+\frac{53\!\cdots\!39}{97\!\cdots\!71}a^{8}+\frac{87\!\cdots\!72}{97\!\cdots\!71}a^{7}-\frac{70\!\cdots\!23}{97\!\cdots\!71}a^{6}-\frac{55\!\cdots\!88}{97\!\cdots\!71}a^{5}-\frac{57\!\cdots\!73}{97\!\cdots\!71}a^{4}+\frac{15\!\cdots\!96}{97\!\cdots\!71}a^{3}+\frac{59\!\cdots\!03}{13\!\cdots\!53}a^{2}+\frac{27\!\cdots\!67}{97\!\cdots\!71}a+\frac{98\!\cdots\!53}{12\!\cdots\!49}$, $\frac{93\!\cdots\!39}{97\!\cdots\!71}a^{27}-\frac{13\!\cdots\!48}{97\!\cdots\!71}a^{26}-\frac{14\!\cdots\!40}{13\!\cdots\!53}a^{25}-\frac{46\!\cdots\!95}{97\!\cdots\!71}a^{24}+\frac{60\!\cdots\!65}{97\!\cdots\!71}a^{23}-\frac{15\!\cdots\!56}{97\!\cdots\!71}a^{22}+\frac{86\!\cdots\!41}{97\!\cdots\!71}a^{21}-\frac{18\!\cdots\!08}{17\!\cdots\!07}a^{20}+\frac{17\!\cdots\!67}{97\!\cdots\!71}a^{19}-\frac{75\!\cdots\!68}{97\!\cdots\!71}a^{18}+\frac{81\!\cdots\!07}{97\!\cdots\!71}a^{17}-\frac{21\!\cdots\!65}{97\!\cdots\!71}a^{16}+\frac{32\!\cdots\!93}{97\!\cdots\!71}a^{15}-\frac{40\!\cdots\!20}{13\!\cdots\!53}a^{14}+\frac{67\!\cdots\!93}{97\!\cdots\!71}a^{13}-\frac{66\!\cdots\!78}{97\!\cdots\!71}a^{12}+\frac{27\!\cdots\!28}{97\!\cdots\!71}a^{11}+\frac{14\!\cdots\!01}{97\!\cdots\!71}a^{10}+\frac{57\!\cdots\!13}{97\!\cdots\!71}a^{9}+\frac{13\!\cdots\!70}{97\!\cdots\!71}a^{8}-\frac{86\!\cdots\!29}{97\!\cdots\!71}a^{7}-\frac{70\!\cdots\!01}{97\!\cdots\!71}a^{6}+\frac{68\!\cdots\!06}{97\!\cdots\!71}a^{5}-\frac{56\!\cdots\!19}{97\!\cdots\!71}a^{4}+\frac{84\!\cdots\!31}{97\!\cdots\!71}a^{3}-\frac{14\!\cdots\!18}{97\!\cdots\!71}a^{2}+\frac{12\!\cdots\!14}{97\!\cdots\!71}a+\frac{44\!\cdots\!79}{12\!\cdots\!49}$, $\frac{22\!\cdots\!04}{97\!\cdots\!71}a^{27}-\frac{11\!\cdots\!97}{45\!\cdots\!79}a^{26}-\frac{15\!\cdots\!55}{13\!\cdots\!53}a^{25}-\frac{15\!\cdots\!58}{13\!\cdots\!53}a^{24}+\frac{10\!\cdots\!30}{97\!\cdots\!71}a^{23}+\frac{34\!\cdots\!91}{97\!\cdots\!71}a^{22}+\frac{29\!\cdots\!88}{13\!\cdots\!53}a^{21}-\frac{17\!\cdots\!65}{97\!\cdots\!71}a^{20}-\frac{22\!\cdots\!03}{97\!\cdots\!71}a^{19}-\frac{17\!\cdots\!82}{97\!\cdots\!71}a^{18}+\frac{13\!\cdots\!85}{97\!\cdots\!71}a^{17}-\frac{33\!\cdots\!41}{97\!\cdots\!71}a^{16}+\frac{75\!\cdots\!57}{97\!\cdots\!71}a^{15}-\frac{38\!\cdots\!75}{97\!\cdots\!71}a^{14}+\frac{51\!\cdots\!35}{97\!\cdots\!71}a^{13}-\frac{15\!\cdots\!64}{97\!\cdots\!71}a^{12}+\frac{27\!\cdots\!79}{97\!\cdots\!71}a^{11}+\frac{42\!\cdots\!04}{97\!\cdots\!71}a^{10}+\frac{14\!\cdots\!40}{97\!\cdots\!71}a^{9}+\frac{94\!\cdots\!75}{97\!\cdots\!71}a^{8}+\frac{11\!\cdots\!51}{97\!\cdots\!71}a^{7}-\frac{15\!\cdots\!88}{97\!\cdots\!71}a^{6}-\frac{51\!\cdots\!32}{97\!\cdots\!71}a^{5}-\frac{14\!\cdots\!25}{97\!\cdots\!71}a^{4}+\frac{98\!\cdots\!63}{97\!\cdots\!71}a^{3}+\frac{38\!\cdots\!65}{13\!\cdots\!53}a^{2}-\frac{30\!\cdots\!25}{97\!\cdots\!71}a+\frac{34\!\cdots\!54}{12\!\cdots\!49}$, $\frac{12\!\cdots\!42}{97\!\cdots\!71}a^{27}-\frac{15\!\cdots\!85}{97\!\cdots\!71}a^{26}-\frac{66\!\cdots\!44}{97\!\cdots\!71}a^{25}-\frac{62\!\cdots\!77}{97\!\cdots\!71}a^{24}+\frac{68\!\cdots\!50}{97\!\cdots\!71}a^{23}+\frac{33\!\cdots\!39}{13\!\cdots\!53}a^{22}+\frac{11\!\cdots\!37}{97\!\cdots\!71}a^{21}-\frac{11\!\cdots\!82}{97\!\cdots\!71}a^{20}-\frac{21\!\cdots\!69}{97\!\cdots\!71}a^{19}-\frac{10\!\cdots\!77}{97\!\cdots\!71}a^{18}+\frac{85\!\cdots\!43}{97\!\cdots\!71}a^{17}+\frac{65\!\cdots\!13}{97\!\cdots\!71}a^{16}+\frac{43\!\cdots\!92}{97\!\cdots\!71}a^{15}-\frac{25\!\cdots\!23}{97\!\cdots\!71}a^{14}-\frac{40\!\cdots\!41}{97\!\cdots\!71}a^{13}-\frac{92\!\cdots\!89}{97\!\cdots\!71}a^{12}+\frac{44\!\cdots\!21}{97\!\cdots\!71}a^{11}+\frac{34\!\cdots\!14}{97\!\cdots\!71}a^{10}+\frac{10\!\cdots\!81}{97\!\cdots\!71}a^{9}+\frac{59\!\cdots\!02}{97\!\cdots\!71}a^{8}+\frac{74\!\cdots\!96}{97\!\cdots\!71}a^{7}-\frac{13\!\cdots\!23}{97\!\cdots\!71}a^{6}-\frac{87\!\cdots\!10}{13\!\cdots\!53}a^{5}-\frac{87\!\cdots\!51}{97\!\cdots\!71}a^{4}+\frac{85\!\cdots\!84}{97\!\cdots\!71}a^{3}+\frac{47\!\cdots\!38}{97\!\cdots\!71}a^{2}+\frac{29\!\cdots\!27}{97\!\cdots\!71}a+\frac{21\!\cdots\!35}{12\!\cdots\!49}$, $\frac{55\!\cdots\!68}{97\!\cdots\!71}a^{27}-\frac{74\!\cdots\!17}{97\!\cdots\!71}a^{26}-\frac{54\!\cdots\!51}{13\!\cdots\!53}a^{25}-\frac{27\!\cdots\!61}{97\!\cdots\!71}a^{24}+\frac{34\!\cdots\!64}{97\!\cdots\!71}a^{23}+\frac{14\!\cdots\!65}{97\!\cdots\!71}a^{22}+\frac{49\!\cdots\!00}{97\!\cdots\!71}a^{21}-\frac{57\!\cdots\!79}{97\!\cdots\!71}a^{20}-\frac{32\!\cdots\!58}{17\!\cdots\!07}a^{19}-\frac{41\!\cdots\!70}{97\!\cdots\!71}a^{18}+\frac{43\!\cdots\!46}{97\!\cdots\!71}a^{17}+\frac{10\!\cdots\!84}{97\!\cdots\!71}a^{16}+\frac{16\!\cdots\!09}{97\!\cdots\!71}a^{15}-\frac{13\!\cdots\!64}{97\!\cdots\!71}a^{14}-\frac{54\!\cdots\!31}{97\!\cdots\!71}a^{13}-\frac{32\!\cdots\!43}{97\!\cdots\!71}a^{12}+\frac{58\!\cdots\!15}{97\!\cdots\!71}a^{11}+\frac{24\!\cdots\!71}{97\!\cdots\!71}a^{10}+\frac{61\!\cdots\!62}{17\!\cdots\!07}a^{9}+\frac{15\!\cdots\!85}{97\!\cdots\!71}a^{8}-\frac{10\!\cdots\!71}{97\!\cdots\!71}a^{7}-\frac{52\!\cdots\!88}{97\!\cdots\!71}a^{6}-\frac{13\!\cdots\!65}{13\!\cdots\!53}a^{5}-\frac{25\!\cdots\!39}{97\!\cdots\!71}a^{4}+\frac{29\!\cdots\!74}{97\!\cdots\!71}a^{3}+\frac{30\!\cdots\!16}{97\!\cdots\!71}a^{2}-\frac{81\!\cdots\!59}{97\!\cdots\!71}a+\frac{13\!\cdots\!29}{12\!\cdots\!49}$, $\frac{24\!\cdots\!56}{97\!\cdots\!71}a^{27}-\frac{31\!\cdots\!83}{97\!\cdots\!71}a^{26}-\frac{11\!\cdots\!84}{97\!\cdots\!71}a^{25}-\frac{12\!\cdots\!42}{97\!\cdots\!71}a^{24}+\frac{14\!\cdots\!92}{97\!\cdots\!71}a^{23}+\frac{53\!\cdots\!20}{13\!\cdots\!53}a^{22}+\frac{22\!\cdots\!10}{97\!\cdots\!71}a^{21}-\frac{24\!\cdots\!78}{97\!\cdots\!71}a^{20}-\frac{29\!\cdots\!42}{97\!\cdots\!71}a^{19}-\frac{28\!\cdots\!05}{13\!\cdots\!53}a^{18}+\frac{19\!\cdots\!75}{97\!\cdots\!71}a^{17}+\frac{47\!\cdots\!13}{97\!\cdots\!71}a^{16}+\frac{88\!\cdots\!10}{97\!\cdots\!71}a^{15}-\frac{64\!\cdots\!96}{97\!\cdots\!71}a^{14}-\frac{59\!\cdots\!14}{97\!\cdots\!71}a^{13}-\frac{19\!\cdots\!38}{97\!\cdots\!71}a^{12}+\frac{49\!\cdots\!98}{97\!\cdots\!71}a^{11}+\frac{73\!\cdots\!53}{97\!\cdots\!71}a^{10}+\frac{31\!\cdots\!35}{13\!\cdots\!53}a^{9}+\frac{65\!\cdots\!97}{97\!\cdots\!71}a^{8}-\frac{41\!\cdots\!90}{97\!\cdots\!71}a^{7}-\frac{26\!\cdots\!37}{97\!\cdots\!71}a^{6}-\frac{60\!\cdots\!72}{97\!\cdots\!71}a^{5}-\frac{14\!\cdots\!07}{97\!\cdots\!71}a^{4}+\frac{21\!\cdots\!11}{97\!\cdots\!71}a^{3}+\frac{58\!\cdots\!18}{97\!\cdots\!71}a^{2}+\frac{35\!\cdots\!69}{97\!\cdots\!71}a+\frac{26\!\cdots\!31}{12\!\cdots\!49}$, $\frac{23\!\cdots\!06}{97\!\cdots\!71}a^{27}-\frac{31\!\cdots\!27}{97\!\cdots\!71}a^{26}-\frac{12\!\cdots\!46}{97\!\cdots\!71}a^{25}-\frac{11\!\cdots\!05}{97\!\cdots\!71}a^{24}+\frac{14\!\cdots\!99}{97\!\cdots\!71}a^{23}+\frac{43\!\cdots\!36}{97\!\cdots\!71}a^{22}+\frac{22\!\cdots\!22}{97\!\cdots\!71}a^{21}-\frac{24\!\cdots\!37}{97\!\cdots\!71}a^{20}-\frac{41\!\cdots\!36}{97\!\cdots\!71}a^{19}-\frac{19\!\cdots\!94}{97\!\cdots\!71}a^{18}+\frac{19\!\cdots\!79}{97\!\cdots\!71}a^{17}+\frac{21\!\cdots\!20}{13\!\cdots\!53}a^{16}+\frac{86\!\cdots\!02}{97\!\cdots\!71}a^{15}-\frac{67\!\cdots\!67}{97\!\cdots\!71}a^{14}-\frac{10\!\cdots\!30}{97\!\cdots\!71}a^{13}-\frac{18\!\cdots\!86}{97\!\cdots\!71}a^{12}+\frac{58\!\cdots\!90}{97\!\cdots\!71}a^{11}+\frac{79\!\cdots\!24}{97\!\cdots\!71}a^{10}+\frac{21\!\cdots\!42}{97\!\cdots\!71}a^{9}+\frac{53\!\cdots\!64}{97\!\cdots\!71}a^{8}-\frac{48\!\cdots\!70}{97\!\cdots\!71}a^{7}-\frac{26\!\cdots\!18}{97\!\cdots\!71}a^{6}-\frac{63\!\cdots\!97}{97\!\cdots\!71}a^{5}-\frac{13\!\cdots\!35}{97\!\cdots\!71}a^{4}+\frac{21\!\cdots\!44}{97\!\cdots\!71}a^{3}+\frac{91\!\cdots\!17}{13\!\cdots\!53}a^{2}+\frac{36\!\cdots\!28}{97\!\cdots\!71}a+\frac{24\!\cdots\!30}{12\!\cdots\!49}$, $\frac{75\!\cdots\!45}{13\!\cdots\!53}a^{27}-\frac{40\!\cdots\!10}{97\!\cdots\!71}a^{26}-\frac{47\!\cdots\!03}{97\!\cdots\!71}a^{25}-\frac{26\!\cdots\!82}{97\!\cdots\!71}a^{24}+\frac{25\!\cdots\!12}{13\!\cdots\!53}a^{23}+\frac{18\!\cdots\!86}{97\!\cdots\!71}a^{22}+\frac{48\!\cdots\!65}{97\!\cdots\!71}a^{21}-\frac{26\!\cdots\!71}{97\!\cdots\!71}a^{20}-\frac{22\!\cdots\!90}{97\!\cdots\!71}a^{19}-\frac{57\!\cdots\!05}{13\!\cdots\!53}a^{18}+\frac{17\!\cdots\!53}{97\!\cdots\!71}a^{17}+\frac{12\!\cdots\!07}{97\!\cdots\!71}a^{16}+\frac{16\!\cdots\!43}{97\!\cdots\!71}a^{15}-\frac{46\!\cdots\!69}{13\!\cdots\!53}a^{14}-\frac{41\!\cdots\!62}{97\!\cdots\!71}a^{13}-\frac{28\!\cdots\!32}{97\!\cdots\!71}a^{12}-\frac{11\!\cdots\!82}{97\!\cdots\!71}a^{11}+\frac{12\!\cdots\!91}{97\!\cdots\!71}a^{10}+\frac{20\!\cdots\!64}{97\!\cdots\!71}a^{9}+\frac{32\!\cdots\!83}{12\!\cdots\!49}a^{8}+\frac{16\!\cdots\!52}{13\!\cdots\!53}a^{7}-\frac{20\!\cdots\!22}{97\!\cdots\!71}a^{6}+\frac{13\!\cdots\!59}{13\!\cdots\!53}a^{5}-\frac{39\!\cdots\!39}{97\!\cdots\!71}a^{4}+\frac{14\!\cdots\!54}{97\!\cdots\!71}a^{3}-\frac{33\!\cdots\!48}{97\!\cdots\!71}a^{2}-\frac{60\!\cdots\!28}{97\!\cdots\!71}a+\frac{22\!\cdots\!86}{12\!\cdots\!49}$, $\frac{51\!\cdots\!33}{13\!\cdots\!53}a^{27}-\frac{34\!\cdots\!94}{97\!\cdots\!71}a^{26}-\frac{63\!\cdots\!60}{97\!\cdots\!71}a^{25}-\frac{17\!\cdots\!65}{97\!\cdots\!71}a^{24}+\frac{15\!\cdots\!57}{97\!\cdots\!71}a^{23}+\frac{28\!\cdots\!76}{97\!\cdots\!71}a^{22}+\frac{31\!\cdots\!22}{97\!\cdots\!71}a^{21}-\frac{26\!\cdots\!91}{97\!\cdots\!71}a^{20}-\frac{44\!\cdots\!40}{97\!\cdots\!71}a^{19}-\frac{25\!\cdots\!39}{97\!\cdots\!71}a^{18}+\frac{21\!\cdots\!73}{97\!\cdots\!71}a^{17}+\frac{32\!\cdots\!26}{97\!\cdots\!71}a^{16}+\frac{97\!\cdots\!24}{97\!\cdots\!71}a^{15}-\frac{74\!\cdots\!02}{97\!\cdots\!71}a^{14}-\frac{12\!\cdots\!80}{97\!\cdots\!71}a^{13}-\frac{15\!\cdots\!25}{97\!\cdots\!71}a^{12}+\frac{29\!\cdots\!08}{38\!\cdots\!21}a^{11}+\frac{26\!\cdots\!38}{97\!\cdots\!71}a^{10}+\frac{14\!\cdots\!02}{97\!\cdots\!71}a^{9}-\frac{51\!\cdots\!79}{13\!\cdots\!53}a^{8}-\frac{12\!\cdots\!97}{97\!\cdots\!71}a^{7}-\frac{21\!\cdots\!72}{97\!\cdots\!71}a^{6}+\frac{20\!\cdots\!82}{97\!\cdots\!71}a^{5}-\frac{10\!\cdots\!06}{97\!\cdots\!71}a^{4}+\frac{12\!\cdots\!60}{97\!\cdots\!71}a^{3}+\frac{13\!\cdots\!72}{13\!\cdots\!53}a^{2}+\frac{36\!\cdots\!91}{97\!\cdots\!71}a+\frac{13\!\cdots\!50}{12\!\cdots\!49}$, $\frac{73\!\cdots\!46}{97\!\cdots\!71}a^{27}-\frac{13\!\cdots\!64}{97\!\cdots\!71}a^{26}-\frac{46\!\cdots\!12}{97\!\cdots\!71}a^{25}-\frac{35\!\cdots\!47}{97\!\cdots\!71}a^{24}+\frac{64\!\cdots\!23}{97\!\cdots\!71}a^{23}+\frac{20\!\cdots\!96}{97\!\cdots\!71}a^{22}+\frac{64\!\cdots\!62}{97\!\cdots\!71}a^{21}-\frac{11\!\cdots\!18}{97\!\cdots\!71}a^{20}-\frac{29\!\cdots\!14}{97\!\cdots\!71}a^{19}-\frac{53\!\cdots\!55}{97\!\cdots\!71}a^{18}+\frac{91\!\cdots\!95}{97\!\cdots\!71}a^{17}+\frac{34\!\cdots\!16}{13\!\cdots\!53}a^{16}+\frac{21\!\cdots\!88}{97\!\cdots\!71}a^{15}-\frac{32\!\cdots\!65}{97\!\cdots\!71}a^{14}-\frac{15\!\cdots\!51}{97\!\cdots\!71}a^{13}-\frac{40\!\cdots\!65}{97\!\cdots\!71}a^{12}+\frac{42\!\cdots\!77}{97\!\cdots\!71}a^{11}+\frac{64\!\cdots\!56}{97\!\cdots\!71}a^{10}+\frac{58\!\cdots\!72}{13\!\cdots\!53}a^{9}-\frac{32\!\cdots\!41}{13\!\cdots\!53}a^{8}-\frac{71\!\cdots\!73}{97\!\cdots\!71}a^{7}-\frac{88\!\cdots\!66}{97\!\cdots\!71}a^{6}+\frac{19\!\cdots\!70}{97\!\cdots\!71}a^{5}+\frac{33\!\cdots\!00}{13\!\cdots\!53}a^{4}+\frac{88\!\cdots\!58}{97\!\cdots\!71}a^{3}+\frac{29\!\cdots\!60}{97\!\cdots\!71}a^{2}-\frac{41\!\cdots\!52}{97\!\cdots\!71}a-\frac{23\!\cdots\!30}{12\!\cdots\!49}$, $\frac{38\!\cdots\!34}{97\!\cdots\!71}a^{27}-\frac{58\!\cdots\!52}{97\!\cdots\!71}a^{26}-\frac{16\!\cdots\!59}{97\!\cdots\!71}a^{25}-\frac{19\!\cdots\!37}{97\!\cdots\!71}a^{24}+\frac{27\!\cdots\!11}{97\!\cdots\!71}a^{23}+\frac{59\!\cdots\!17}{97\!\cdots\!71}a^{22}+\frac{35\!\cdots\!29}{97\!\cdots\!71}a^{21}-\frac{46\!\cdots\!13}{97\!\cdots\!71}a^{20}-\frac{55\!\cdots\!93}{97\!\cdots\!71}a^{19}-\frac{30\!\cdots\!78}{97\!\cdots\!71}a^{18}+\frac{36\!\cdots\!19}{97\!\cdots\!71}a^{17}+\frac{22\!\cdots\!50}{97\!\cdots\!71}a^{16}+\frac{13\!\cdots\!28}{97\!\cdots\!71}a^{15}-\frac{12\!\cdots\!23}{97\!\cdots\!71}a^{14}-\frac{22\!\cdots\!61}{97\!\cdots\!71}a^{13}-\frac{27\!\cdots\!50}{97\!\cdots\!71}a^{12}+\frac{16\!\cdots\!78}{13\!\cdots\!53}a^{11}+\frac{16\!\cdots\!31}{97\!\cdots\!71}a^{10}+\frac{29\!\cdots\!15}{97\!\cdots\!71}a^{9}+\frac{56\!\cdots\!33}{97\!\cdots\!71}a^{8}-\frac{12\!\cdots\!95}{97\!\cdots\!71}a^{7}-\frac{52\!\cdots\!08}{12\!\cdots\!49}a^{6}-\frac{46\!\cdots\!12}{97\!\cdots\!71}a^{5}-\frac{18\!\cdots\!84}{97\!\cdots\!71}a^{4}+\frac{34\!\cdots\!20}{97\!\cdots\!71}a^{3}+\frac{10\!\cdots\!67}{97\!\cdots\!71}a^{2}-\frac{30\!\cdots\!03}{97\!\cdots\!71}a+\frac{79\!\cdots\!24}{12\!\cdots\!49}$ 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:  \( 158620092291.7724 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 158620092291.7724 \cdot 203}{6\cdot\sqrt{14121388821225670988853483488774192350843817726481}}\cr\approx \mathstrut & 0.213443255955932 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - x^26 - 50*x^25 + 45*x^24 + 41*x^23 + 923*x^22 - 740*x^21 - 566*x^20 - 7986*x^19 + 5482*x^18 + 3755*x^17 + 34135*x^16 - 15421*x^15 - 15266*x^14 - 71623*x^13 - 3964*x^12 + 45122*x^11 + 87041*x^10 + 49398*x^9 - 13057*x^8 - 105029*x^7 - 52559*x^6 - 58937*x^5 + 57132*x^4 + 53384*x^3 + 9659*x^2 + 13272*x + 6241)
 
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^28 - x^27 - x^26 - 50*x^25 + 45*x^24 + 41*x^23 + 923*x^22 - 740*x^21 - 566*x^20 - 7986*x^19 + 5482*x^18 + 3755*x^17 + 34135*x^16 - 15421*x^15 - 15266*x^14 - 71623*x^13 - 3964*x^12 + 45122*x^11 + 87041*x^10 + 49398*x^9 - 13057*x^8 - 105029*x^7 - 52559*x^6 - 58937*x^5 + 57132*x^4 + 53384*x^3 + 9659*x^2 + 13272*x + 6241, 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^28 - x^27 - x^26 - 50*x^25 + 45*x^24 + 41*x^23 + 923*x^22 - 740*x^21 - 566*x^20 - 7986*x^19 + 5482*x^18 + 3755*x^17 + 34135*x^16 - 15421*x^15 - 15266*x^14 - 71623*x^13 - 3964*x^12 + 45122*x^11 + 87041*x^10 + 49398*x^9 - 13057*x^8 - 105029*x^7 - 52559*x^6 - 58937*x^5 + 57132*x^4 + 53384*x^3 + 9659*x^2 + 13272*x + 6241);
 
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^28 - x^27 - x^26 - 50*x^25 + 45*x^24 + 41*x^23 + 923*x^22 - 740*x^21 - 566*x^20 - 7986*x^19 + 5482*x^18 + 3755*x^17 + 34135*x^16 - 15421*x^15 - 15266*x^14 - 71623*x^13 - 3964*x^12 + 45122*x^11 + 87041*x^10 + 49398*x^9 - 13057*x^8 - 105029*x^7 - 52559*x^6 - 58937*x^5 + 57132*x^4 + 53384*x^3 + 9659*x^2 + 13272*x + 6241);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2\times C_{14}$ (as 28T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{129}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-43})\), 7.7.6321363049.1, 14.14.3757843639805369947326441.1, 14.0.1718264124282290785243.1, 14.0.87391712553613254588987.1

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 ${\href{/padicField/2.14.0.1}{14} }^{2}$ R ${\href{/padicField/5.14.0.1}{14} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{14}$ ${\href{/padicField/11.14.0.1}{14} }^{2}$ ${\href{/padicField/13.7.0.1}{7} }^{4}$ ${\href{/padicField/17.14.0.1}{14} }^{2}$ ${\href{/padicField/19.14.0.1}{14} }^{2}$ ${\href{/padicField/23.14.0.1}{14} }^{2}$ ${\href{/padicField/29.14.0.1}{14} }^{2}$ ${\href{/padicField/31.7.0.1}{7} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{14}$ ${\href{/padicField/41.14.0.1}{14} }^{2}$ R ${\href{/padicField/47.14.0.1}{14} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }^{2}$ ${\href{/padicField/59.14.0.1}{14} }^{2}$

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
\(3\) Copy content Toggle raw display Deg $28$$2$$14$$14$
\(43\) Copy content Toggle raw display 43.14.13.11$x^{14} + 43$$14$$1$$13$$C_{14}$$[\ ]_{14}$
43.14.13.11$x^{14} + 43$$14$$1$$13$$C_{14}$$[\ ]_{14}$