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Properties

Label 28.0.192...877.1
Degree $28$
Signature $[0, 14]$
Discriminant $1.924\times 10^{45}$
Root discriminant \(41.43\)
Ramified primes $43,53$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $D_{28}$ (as 28T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 12*x^27 + 74*x^26 - 288*x^25 + 773*x^24 - 1407*x^23 + 1601*x^22 - 676*x^21 - 794*x^20 + 2173*x^19 - 6644*x^18 + 23336*x^17 - 53512*x^16 + 82551*x^15 - 80880*x^14 + 23133*x^13 + 117766*x^12 - 327568*x^11 + 536090*x^10 - 646700*x^9 + 621936*x^8 - 488735*x^7 + 317878*x^6 - 166984*x^5 + 68952*x^4 - 21502*x^3 + 5711*x^2 - 1133*x + 193)
 
gp: K = bnfinit(y^28 - 12*y^27 + 74*y^26 - 288*y^25 + 773*y^24 - 1407*y^23 + 1601*y^22 - 676*y^21 - 794*y^20 + 2173*y^19 - 6644*y^18 + 23336*y^17 - 53512*y^16 + 82551*y^15 - 80880*y^14 + 23133*y^13 + 117766*y^12 - 327568*y^11 + 536090*y^10 - 646700*y^9 + 621936*y^8 - 488735*y^7 + 317878*y^6 - 166984*y^5 + 68952*y^4 - 21502*y^3 + 5711*y^2 - 1133*y + 193, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 12*x^27 + 74*x^26 - 288*x^25 + 773*x^24 - 1407*x^23 + 1601*x^22 - 676*x^21 - 794*x^20 + 2173*x^19 - 6644*x^18 + 23336*x^17 - 53512*x^16 + 82551*x^15 - 80880*x^14 + 23133*x^13 + 117766*x^12 - 327568*x^11 + 536090*x^10 - 646700*x^9 + 621936*x^8 - 488735*x^7 + 317878*x^6 - 166984*x^5 + 68952*x^4 - 21502*x^3 + 5711*x^2 - 1133*x + 193);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 12*x^27 + 74*x^26 - 288*x^25 + 773*x^24 - 1407*x^23 + 1601*x^22 - 676*x^21 - 794*x^20 + 2173*x^19 - 6644*x^18 + 23336*x^17 - 53512*x^16 + 82551*x^15 - 80880*x^14 + 23133*x^13 + 117766*x^12 - 327568*x^11 + 536090*x^10 - 646700*x^9 + 621936*x^8 - 488735*x^7 + 317878*x^6 - 166984*x^5 + 68952*x^4 - 21502*x^3 + 5711*x^2 - 1133*x + 193)
 

\( x^{28} - 12 x^{27} + 74 x^{26} - 288 x^{25} + 773 x^{24} - 1407 x^{23} + 1601 x^{22} - 676 x^{21} + \cdots + 193 \) 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:   \(1923732503543401200540313355166872100640336877\) \(\medspace = 43^{14}\cdot 53^{13}\) 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:  \(41.43\)
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:  $43^{1/2}53^{1/2}\approx 47.738873049120045$
Ramified primes:   \(43\), \(53\) 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:  \(\Q(\sqrt{53}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{7}a^{20}+\frac{3}{7}a^{19}+\frac{3}{7}a^{18}+\frac{3}{7}a^{17}+\frac{3}{7}a^{16}-\frac{2}{7}a^{15}-\frac{3}{7}a^{13}-\frac{2}{7}a^{12}+\frac{1}{7}a^{11}-\frac{1}{7}a^{10}+\frac{3}{7}a^{9}+\frac{2}{7}a^{8}+\frac{1}{7}a^{7}-\frac{3}{7}a^{6}-\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{2}{7}a^{3}-\frac{3}{7}a^{2}-\frac{2}{7}a-\frac{2}{7}$, $\frac{1}{7}a^{21}+\frac{1}{7}a^{19}+\frac{1}{7}a^{18}+\frac{1}{7}a^{17}+\frac{3}{7}a^{16}-\frac{1}{7}a^{15}-\frac{3}{7}a^{14}+\frac{3}{7}a^{11}-\frac{1}{7}a^{10}+\frac{2}{7}a^{8}+\frac{1}{7}a^{7}-\frac{1}{7}a^{6}+\frac{3}{7}a^{5}-\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{22}-\frac{2}{7}a^{19}-\frac{2}{7}a^{18}+\frac{3}{7}a^{16}-\frac{1}{7}a^{15}+\frac{3}{7}a^{13}-\frac{2}{7}a^{12}-\frac{2}{7}a^{11}+\frac{1}{7}a^{10}-\frac{1}{7}a^{9}-\frac{1}{7}a^{8}-\frac{2}{7}a^{7}-\frac{1}{7}a^{6}+\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{2}{7}a^{3}+\frac{1}{7}a+\frac{2}{7}$, $\frac{1}{371}a^{23}-\frac{19}{371}a^{22}-\frac{3}{53}a^{21}+\frac{8}{371}a^{20}-\frac{172}{371}a^{19}-\frac{72}{371}a^{18}+\frac{40}{371}a^{17}+\frac{3}{53}a^{16}+\frac{111}{371}a^{15}+\frac{136}{371}a^{14}+\frac{44}{371}a^{13}-\frac{96}{371}a^{12}-\frac{19}{53}a^{11}+\frac{47}{371}a^{10}+\frac{146}{371}a^{9}+\frac{163}{371}a^{8}+\frac{103}{371}a^{7}+\frac{131}{371}a^{6}+\frac{13}{371}a^{5}+\frac{135}{371}a^{4}+\frac{163}{371}a^{3}-\frac{127}{371}a^{2}-\frac{65}{371}a-\frac{93}{371}$, $\frac{1}{2597}a^{24}-\frac{2}{2597}a^{23}+\frac{80}{2597}a^{22}+\frac{128}{2597}a^{21}+\frac{176}{2597}a^{20}-\frac{876}{2597}a^{19}-\frac{177}{2597}a^{18}-\frac{412}{2597}a^{17}+\frac{468}{2597}a^{16}-\frac{415}{2597}a^{15}+\frac{1296}{2597}a^{14}-\frac{81}{371}a^{13}-\frac{440}{2597}a^{12}+\frac{436}{2597}a^{11}-\frac{433}{2597}a^{10}-\frac{482}{2597}a^{9}+\frac{489}{2597}a^{8}-\frac{874}{2597}a^{7}+\frac{703}{2597}a^{6}-\frac{969}{2597}a^{5}-\frac{563}{2597}a^{4}+\frac{153}{2597}a^{3}-\frac{263}{2597}a^{2}+\frac{339}{2597}a-\frac{1263}{2597}$, $\frac{1}{2597}a^{25}-\frac{1}{2597}a^{23}-\frac{104}{2597}a^{22}-\frac{177}{2597}a^{21}-\frac{27}{2597}a^{20}+\frac{556}{2597}a^{19}-\frac{787}{2597}a^{18}+\frac{274}{2597}a^{17}+\frac{388}{2597}a^{16}-\frac{1032}{2597}a^{15}+\frac{828}{2597}a^{14}-\frac{881}{2597}a^{13}+\frac{641}{2597}a^{12}+\frac{1034}{2597}a^{11}-\frac{515}{2597}a^{10}+\frac{1268}{2597}a^{9}+\frac{167}{2597}a^{8}-\frac{1185}{2597}a^{7}-\frac{1117}{2597}a^{6}+\frac{950}{2597}a^{5}+\frac{19}{371}a^{4}+\frac{9}{49}a^{3}+\frac{1059}{2597}a^{2}-\frac{774}{2597}a+\frac{925}{2597}$, $\frac{1}{17979031}a^{26}-\frac{166}{2568433}a^{25}-\frac{709}{17979031}a^{24}+\frac{957}{781697}a^{23}-\frac{162853}{17979031}a^{22}-\frac{1160055}{17979031}a^{21}-\frac{1221526}{17979031}a^{20}+\frac{2436201}{17979031}a^{19}-\frac{2734302}{17979031}a^{18}-\frac{2754855}{17979031}a^{17}+\frac{4849983}{17979031}a^{16}-\frac{1811666}{17979031}a^{15}-\frac{8}{2568433}a^{14}+\frac{4179382}{17979031}a^{13}-\frac{8048323}{17979031}a^{12}+\frac{5315381}{17979031}a^{11}-\frac{395497}{2568433}a^{10}+\frac{312464}{17979031}a^{9}-\frac{2424409}{17979031}a^{8}-\frac{8946880}{17979031}a^{7}+\frac{6887253}{17979031}a^{6}+\frac{863936}{17979031}a^{5}+\frac{326678}{781697}a^{4}+\frac{3390761}{17979031}a^{3}+\frac{12603}{59731}a^{2}-\frac{8732278}{17979031}a-\frac{2330374}{17979031}$, $\frac{1}{61\!\cdots\!47}a^{27}+\frac{83\!\cdots\!76}{61\!\cdots\!47}a^{26}-\frac{44\!\cdots\!66}{61\!\cdots\!47}a^{25}-\frac{65\!\cdots\!45}{87\!\cdots\!21}a^{24}+\frac{11\!\cdots\!26}{11\!\cdots\!99}a^{23}-\frac{15\!\cdots\!34}{61\!\cdots\!47}a^{22}-\frac{15\!\cdots\!14}{61\!\cdots\!47}a^{21}+\frac{28\!\cdots\!56}{61\!\cdots\!47}a^{20}-\frac{60\!\cdots\!02}{12\!\cdots\!03}a^{19}+\frac{31\!\cdots\!00}{61\!\cdots\!47}a^{18}+\frac{19\!\cdots\!08}{19\!\cdots\!37}a^{17}-\frac{28\!\cdots\!41}{61\!\cdots\!47}a^{16}+\frac{95\!\cdots\!14}{61\!\cdots\!47}a^{15}-\frac{16\!\cdots\!13}{61\!\cdots\!47}a^{14}+\frac{17\!\cdots\!61}{61\!\cdots\!47}a^{13}+\frac{29\!\cdots\!77}{87\!\cdots\!21}a^{12}-\frac{73\!\cdots\!20}{61\!\cdots\!47}a^{11}-\frac{12\!\cdots\!79}{61\!\cdots\!47}a^{10}-\frac{15\!\cdots\!95}{61\!\cdots\!47}a^{9}+\frac{12\!\cdots\!58}{61\!\cdots\!47}a^{8}+\frac{24\!\cdots\!62}{61\!\cdots\!47}a^{7}+\frac{16\!\cdots\!59}{26\!\cdots\!89}a^{6}-\frac{11\!\cdots\!41}{61\!\cdots\!47}a^{5}-\frac{27\!\cdots\!95}{61\!\cdots\!47}a^{4}+\frac{19\!\cdots\!85}{61\!\cdots\!47}a^{3}-\frac{98\!\cdots\!29}{61\!\cdots\!47}a^{2}+\frac{73\!\cdots\!65}{17\!\cdots\!29}a-\frac{26\!\cdots\!09}{61\!\cdots\!47}$ Copy content Toggle raw display

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

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

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $13$
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:   $\frac{10\!\cdots\!74}{20\!\cdots\!93}a^{27}-\frac{12\!\cdots\!85}{20\!\cdots\!93}a^{26}+\frac{71\!\cdots\!33}{20\!\cdots\!93}a^{25}-\frac{38\!\cdots\!94}{29\!\cdots\!99}a^{24}+\frac{67\!\cdots\!10}{20\!\cdots\!93}a^{23}-\frac{11\!\cdots\!62}{20\!\cdots\!93}a^{22}+\frac{10\!\cdots\!58}{20\!\cdots\!93}a^{21}+\frac{30\!\cdots\!73}{20\!\cdots\!93}a^{20}-\frac{15\!\cdots\!10}{29\!\cdots\!99}a^{19}+\frac{18\!\cdots\!65}{20\!\cdots\!93}a^{18}-\frac{59\!\cdots\!91}{20\!\cdots\!93}a^{17}+\frac{21\!\cdots\!53}{20\!\cdots\!93}a^{16}-\frac{44\!\cdots\!17}{20\!\cdots\!93}a^{15}+\frac{61\!\cdots\!34}{20\!\cdots\!93}a^{14}-\frac{46\!\cdots\!95}{20\!\cdots\!93}a^{13}-\frac{15\!\cdots\!65}{29\!\cdots\!99}a^{12}+\frac{12\!\cdots\!02}{20\!\cdots\!93}a^{11}-\frac{28\!\cdots\!44}{20\!\cdots\!93}a^{10}+\frac{40\!\cdots\!98}{20\!\cdots\!93}a^{9}-\frac{43\!\cdots\!96}{20\!\cdots\!93}a^{8}+\frac{36\!\cdots\!69}{20\!\cdots\!93}a^{7}-\frac{24\!\cdots\!40}{20\!\cdots\!93}a^{6}+\frac{12\!\cdots\!22}{20\!\cdots\!93}a^{5}-\frac{47\!\cdots\!81}{20\!\cdots\!93}a^{4}+\frac{88\!\cdots\!87}{20\!\cdots\!93}a^{3}+\frac{14\!\cdots\!66}{20\!\cdots\!93}a^{2}-\frac{13\!\cdots\!71}{29\!\cdots\!99}a+\frac{21\!\cdots\!92}{20\!\cdots\!93}$, $\frac{18\!\cdots\!96}{61\!\cdots\!47}a^{27}-\frac{21\!\cdots\!27}{61\!\cdots\!47}a^{26}+\frac{11\!\cdots\!99}{61\!\cdots\!47}a^{25}-\frac{59\!\cdots\!85}{87\!\cdots\!21}a^{24}+\frac{96\!\cdots\!52}{61\!\cdots\!47}a^{23}-\frac{13\!\cdots\!20}{61\!\cdots\!47}a^{22}+\frac{28\!\cdots\!68}{26\!\cdots\!89}a^{21}+\frac{12\!\cdots\!80}{61\!\cdots\!47}a^{20}-\frac{31\!\cdots\!00}{87\!\cdots\!21}a^{19}+\frac{92\!\cdots\!95}{26\!\cdots\!89}a^{18}-\frac{28\!\cdots\!39}{19\!\cdots\!37}a^{17}+\frac{33\!\cdots\!51}{61\!\cdots\!47}a^{16}-\frac{61\!\cdots\!82}{61\!\cdots\!47}a^{15}+\frac{63\!\cdots\!73}{61\!\cdots\!47}a^{14}-\frac{17\!\cdots\!34}{61\!\cdots\!47}a^{13}-\frac{11\!\cdots\!41}{87\!\cdots\!21}a^{12}+\frac{23\!\cdots\!51}{61\!\cdots\!47}a^{11}-\frac{39\!\cdots\!18}{61\!\cdots\!47}a^{10}+\frac{43\!\cdots\!21}{61\!\cdots\!47}a^{9}-\frac{33\!\cdots\!95}{61\!\cdots\!47}a^{8}+\frac{73\!\cdots\!49}{26\!\cdots\!89}a^{7}-\frac{18\!\cdots\!36}{61\!\cdots\!47}a^{6}-\frac{61\!\cdots\!69}{61\!\cdots\!47}a^{5}+\frac{75\!\cdots\!71}{61\!\cdots\!47}a^{4}-\frac{51\!\cdots\!38}{61\!\cdots\!47}a^{3}+\frac{18\!\cdots\!64}{61\!\cdots\!47}a^{2}-\frac{46\!\cdots\!11}{87\!\cdots\!21}a+\frac{12\!\cdots\!05}{61\!\cdots\!47}$, $\frac{26\!\cdots\!54}{43\!\cdots\!89}a^{27}-\frac{75\!\cdots\!14}{43\!\cdots\!89}a^{26}+\frac{64\!\cdots\!30}{43\!\cdots\!89}a^{25}-\frac{45\!\cdots\!38}{61\!\cdots\!27}a^{24}+\frac{98\!\cdots\!80}{43\!\cdots\!89}a^{23}-\frac{20\!\cdots\!80}{43\!\cdots\!89}a^{22}+\frac{25\!\cdots\!12}{43\!\cdots\!89}a^{21}-\frac{11\!\cdots\!62}{43\!\cdots\!89}a^{20}-\frac{27\!\cdots\!92}{61\!\cdots\!27}a^{19}+\frac{26\!\cdots\!96}{43\!\cdots\!89}a^{18}-\frac{18\!\cdots\!57}{13\!\cdots\!19}a^{17}+\frac{25\!\cdots\!88}{43\!\cdots\!89}a^{16}-\frac{77\!\cdots\!01}{43\!\cdots\!89}a^{15}+\frac{12\!\cdots\!55}{43\!\cdots\!89}a^{14}-\frac{12\!\cdots\!27}{43\!\cdots\!89}a^{13}+\frac{81\!\cdots\!01}{88\!\cdots\!61}a^{12}+\frac{13\!\cdots\!61}{43\!\cdots\!89}a^{11}-\frac{47\!\cdots\!32}{43\!\cdots\!89}a^{10}+\frac{76\!\cdots\!10}{43\!\cdots\!89}a^{9}-\frac{92\!\cdots\!73}{43\!\cdots\!89}a^{8}+\frac{83\!\cdots\!86}{43\!\cdots\!89}a^{7}-\frac{63\!\cdots\!23}{43\!\cdots\!89}a^{6}+\frac{36\!\cdots\!74}{43\!\cdots\!89}a^{5}-\frac{18\!\cdots\!25}{43\!\cdots\!89}a^{4}+\frac{54\!\cdots\!42}{43\!\cdots\!89}a^{3}-\frac{20\!\cdots\!07}{43\!\cdots\!89}a^{2}+\frac{58\!\cdots\!78}{61\!\cdots\!27}a-\frac{17\!\cdots\!33}{43\!\cdots\!89}$, $\frac{98\!\cdots\!25}{61\!\cdots\!47}a^{27}-\frac{11\!\cdots\!42}{61\!\cdots\!47}a^{26}+\frac{29\!\cdots\!41}{26\!\cdots\!89}a^{25}-\frac{35\!\cdots\!90}{87\!\cdots\!21}a^{24}+\frac{61\!\cdots\!14}{61\!\cdots\!47}a^{23}-\frac{97\!\cdots\!24}{61\!\cdots\!47}a^{22}+\frac{83\!\cdots\!21}{61\!\cdots\!47}a^{21}+\frac{14\!\cdots\!58}{61\!\cdots\!47}a^{20}-\frac{14\!\cdots\!97}{87\!\cdots\!21}a^{19}+\frac{15\!\cdots\!14}{61\!\cdots\!47}a^{18}-\frac{17\!\cdots\!94}{19\!\cdots\!37}a^{17}+\frac{19\!\cdots\!22}{61\!\cdots\!47}a^{16}-\frac{40\!\cdots\!31}{61\!\cdots\!47}a^{15}+\frac{52\!\cdots\!97}{61\!\cdots\!47}a^{14}-\frac{16\!\cdots\!56}{26\!\cdots\!89}a^{13}-\frac{24\!\cdots\!79}{87\!\cdots\!21}a^{12}+\frac{12\!\cdots\!63}{61\!\cdots\!47}a^{11}-\frac{25\!\cdots\!61}{61\!\cdots\!47}a^{10}+\frac{35\!\cdots\!46}{61\!\cdots\!47}a^{9}-\frac{36\!\cdots\!16}{61\!\cdots\!47}a^{8}+\frac{28\!\cdots\!71}{61\!\cdots\!47}a^{7}-\frac{18\!\cdots\!64}{61\!\cdots\!47}a^{6}+\frac{85\!\cdots\!46}{61\!\cdots\!47}a^{5}-\frac{23\!\cdots\!07}{61\!\cdots\!47}a^{4}-\frac{21\!\cdots\!02}{61\!\cdots\!47}a^{3}+\frac{37\!\cdots\!97}{61\!\cdots\!47}a^{2}-\frac{20\!\cdots\!91}{87\!\cdots\!21}a+\frac{12\!\cdots\!35}{61\!\cdots\!47}$, $\frac{27\!\cdots\!59}{61\!\cdots\!47}a^{27}-\frac{30\!\cdots\!99}{61\!\cdots\!47}a^{26}+\frac{17\!\cdots\!17}{61\!\cdots\!47}a^{25}-\frac{20\!\cdots\!37}{20\!\cdots\!47}a^{24}+\frac{14\!\cdots\!93}{61\!\cdots\!47}a^{23}-\frac{23\!\cdots\!91}{61\!\cdots\!47}a^{22}+\frac{20\!\cdots\!01}{61\!\cdots\!47}a^{21}+\frac{33\!\cdots\!79}{61\!\cdots\!47}a^{20}-\frac{24\!\cdots\!37}{87\!\cdots\!21}a^{19}+\frac{39\!\cdots\!03}{61\!\cdots\!47}a^{18}-\frac{46\!\cdots\!63}{19\!\cdots\!37}a^{17}+\frac{49\!\cdots\!50}{61\!\cdots\!47}a^{16}-\frac{94\!\cdots\!46}{61\!\cdots\!47}a^{15}+\frac{12\!\cdots\!32}{61\!\cdots\!47}a^{14}-\frac{22\!\cdots\!37}{14\!\cdots\!29}a^{13}-\frac{31\!\cdots\!78}{87\!\cdots\!21}a^{12}+\frac{28\!\cdots\!85}{61\!\cdots\!47}a^{11}-\frac{58\!\cdots\!00}{61\!\cdots\!47}a^{10}+\frac{85\!\cdots\!40}{61\!\cdots\!47}a^{9}-\frac{92\!\cdots\!20}{61\!\cdots\!47}a^{8}+\frac{82\!\cdots\!70}{61\!\cdots\!47}a^{7}-\frac{58\!\cdots\!56}{61\!\cdots\!47}a^{6}+\frac{35\!\cdots\!57}{61\!\cdots\!47}a^{5}-\frac{16\!\cdots\!78}{61\!\cdots\!47}a^{4}+\frac{63\!\cdots\!17}{61\!\cdots\!47}a^{3}-\frac{18\!\cdots\!70}{61\!\cdots\!47}a^{2}+\frac{86\!\cdots\!55}{87\!\cdots\!21}a-\frac{18\!\cdots\!49}{61\!\cdots\!47}$, $\frac{68\!\cdots\!10}{61\!\cdots\!47}a^{27}-\frac{35\!\cdots\!11}{26\!\cdots\!89}a^{26}+\frac{48\!\cdots\!67}{61\!\cdots\!47}a^{25}-\frac{26\!\cdots\!99}{87\!\cdots\!21}a^{24}+\frac{48\!\cdots\!95}{61\!\cdots\!47}a^{23}-\frac{85\!\cdots\!37}{61\!\cdots\!47}a^{22}+\frac{89\!\cdots\!68}{61\!\cdots\!47}a^{21}-\frac{23\!\cdots\!48}{61\!\cdots\!47}a^{20}-\frac{13\!\cdots\!84}{12\!\cdots\!03}a^{19}+\frac{13\!\cdots\!26}{61\!\cdots\!47}a^{18}-\frac{13\!\cdots\!78}{19\!\cdots\!37}a^{17}+\frac{15\!\cdots\!16}{61\!\cdots\!47}a^{16}-\frac{33\!\cdots\!92}{61\!\cdots\!47}a^{15}+\frac{49\!\cdots\!91}{61\!\cdots\!47}a^{14}-\frac{43\!\cdots\!94}{61\!\cdots\!47}a^{13}+\frac{62\!\cdots\!98}{87\!\cdots\!21}a^{12}+\frac{83\!\cdots\!15}{61\!\cdots\!47}a^{11}-\frac{20\!\cdots\!81}{61\!\cdots\!47}a^{10}+\frac{32\!\cdots\!27}{61\!\cdots\!47}a^{9}-\frac{69\!\cdots\!76}{11\!\cdots\!99}a^{8}+\frac{33\!\cdots\!51}{61\!\cdots\!47}a^{7}-\frac{24\!\cdots\!54}{61\!\cdots\!47}a^{6}+\frac{15\!\cdots\!84}{61\!\cdots\!47}a^{5}-\frac{72\!\cdots\!27}{61\!\cdots\!47}a^{4}+\frac{25\!\cdots\!46}{61\!\cdots\!47}a^{3}-\frac{65\!\cdots\!00}{61\!\cdots\!47}a^{2}+\frac{33\!\cdots\!64}{12\!\cdots\!03}a-\frac{22\!\cdots\!03}{61\!\cdots\!47}$, $\frac{38\!\cdots\!45}{26\!\cdots\!89}a^{27}-\frac{10\!\cdots\!30}{61\!\cdots\!47}a^{26}+\frac{60\!\cdots\!54}{61\!\cdots\!47}a^{25}-\frac{31\!\cdots\!02}{87\!\cdots\!21}a^{24}+\frac{24\!\cdots\!49}{26\!\cdots\!89}a^{23}-\frac{91\!\cdots\!98}{61\!\cdots\!47}a^{22}+\frac{83\!\cdots\!64}{61\!\cdots\!47}a^{21}-\frac{80\!\cdots\!99}{61\!\cdots\!47}a^{20}-\frac{16\!\cdots\!05}{12\!\cdots\!03}a^{19}+\frac{14\!\cdots\!41}{61\!\cdots\!47}a^{18}-\frac{16\!\cdots\!95}{19\!\cdots\!37}a^{17}+\frac{17\!\cdots\!07}{61\!\cdots\!47}a^{16}-\frac{37\!\cdots\!26}{61\!\cdots\!47}a^{15}+\frac{50\!\cdots\!00}{61\!\cdots\!47}a^{14}-\frac{39\!\cdots\!73}{61\!\cdots\!47}a^{13}-\frac{10\!\cdots\!20}{87\!\cdots\!21}a^{12}+\frac{10\!\cdots\!73}{61\!\cdots\!47}a^{11}-\frac{23\!\cdots\!53}{61\!\cdots\!47}a^{10}+\frac{33\!\cdots\!81}{61\!\cdots\!47}a^{9}-\frac{36\!\cdots\!24}{61\!\cdots\!47}a^{8}+\frac{31\!\cdots\!41}{61\!\cdots\!47}a^{7}-\frac{22\!\cdots\!70}{61\!\cdots\!47}a^{6}+\frac{13\!\cdots\!57}{61\!\cdots\!47}a^{5}-\frac{26\!\cdots\!42}{26\!\cdots\!89}a^{4}+\frac{20\!\cdots\!97}{61\!\cdots\!47}a^{3}-\frac{63\!\cdots\!74}{61\!\cdots\!47}a^{2}+\frac{43\!\cdots\!93}{17\!\cdots\!29}a-\frac{21\!\cdots\!28}{61\!\cdots\!47}$, $\frac{14\!\cdots\!49}{26\!\cdots\!89}a^{27}-\frac{38\!\cdots\!19}{61\!\cdots\!47}a^{26}+\frac{23\!\cdots\!07}{61\!\cdots\!47}a^{25}-\frac{12\!\cdots\!42}{87\!\cdots\!21}a^{24}+\frac{96\!\cdots\!76}{26\!\cdots\!89}a^{23}-\frac{37\!\cdots\!66}{61\!\cdots\!47}a^{22}+\frac{35\!\cdots\!77}{61\!\cdots\!47}a^{21}-\frac{18\!\cdots\!92}{61\!\cdots\!47}a^{20}-\frac{49\!\cdots\!76}{87\!\cdots\!21}a^{19}+\frac{60\!\cdots\!51}{61\!\cdots\!47}a^{18}-\frac{61\!\cdots\!56}{19\!\cdots\!37}a^{17}+\frac{69\!\cdots\!09}{61\!\cdots\!47}a^{16}-\frac{14\!\cdots\!16}{61\!\cdots\!47}a^{15}+\frac{20\!\cdots\!84}{61\!\cdots\!47}a^{14}-\frac{16\!\cdots\!50}{61\!\cdots\!47}a^{13}-\frac{89\!\cdots\!81}{20\!\cdots\!47}a^{12}+\frac{41\!\cdots\!01}{61\!\cdots\!47}a^{11}-\frac{93\!\cdots\!10}{61\!\cdots\!47}a^{10}+\frac{13\!\cdots\!84}{61\!\cdots\!47}a^{9}-\frac{34\!\cdots\!63}{14\!\cdots\!29}a^{8}+\frac{12\!\cdots\!85}{61\!\cdots\!47}a^{7}-\frac{86\!\cdots\!93}{61\!\cdots\!47}a^{6}+\frac{48\!\cdots\!42}{61\!\cdots\!47}a^{5}-\frac{93\!\cdots\!11}{26\!\cdots\!89}a^{4}+\frac{68\!\cdots\!74}{61\!\cdots\!47}a^{3}-\frac{13\!\cdots\!24}{61\!\cdots\!47}a^{2}+\frac{35\!\cdots\!98}{87\!\cdots\!21}a+\frac{97\!\cdots\!69}{61\!\cdots\!47}$, $\frac{10\!\cdots\!27}{61\!\cdots\!47}a^{27}-\frac{94\!\cdots\!43}{61\!\cdots\!47}a^{26}+\frac{44\!\cdots\!77}{61\!\cdots\!47}a^{25}-\frac{17\!\cdots\!99}{87\!\cdots\!21}a^{24}+\frac{18\!\cdots\!89}{61\!\cdots\!47}a^{23}-\frac{38\!\cdots\!96}{61\!\cdots\!47}a^{22}-\frac{39\!\cdots\!83}{61\!\cdots\!47}a^{21}+\frac{74\!\cdots\!51}{61\!\cdots\!47}a^{20}-\frac{15\!\cdots\!42}{87\!\cdots\!21}a^{19}+\frac{39\!\cdots\!41}{61\!\cdots\!47}a^{18}-\frac{11\!\cdots\!23}{19\!\cdots\!37}a^{17}+\frac{96\!\cdots\!80}{61\!\cdots\!47}a^{16}-\frac{62\!\cdots\!07}{61\!\cdots\!47}a^{15}-\frac{28\!\cdots\!00}{61\!\cdots\!47}a^{14}+\frac{23\!\cdots\!52}{61\!\cdots\!47}a^{13}-\frac{56\!\cdots\!81}{87\!\cdots\!21}a^{12}+\frac{66\!\cdots\!13}{61\!\cdots\!47}a^{11}-\frac{44\!\cdots\!18}{61\!\cdots\!47}a^{10}+\frac{43\!\cdots\!24}{61\!\cdots\!47}a^{9}+\frac{58\!\cdots\!50}{61\!\cdots\!47}a^{8}-\frac{92\!\cdots\!09}{61\!\cdots\!47}a^{7}+\frac{98\!\cdots\!48}{61\!\cdots\!47}a^{6}-\frac{70\!\cdots\!22}{61\!\cdots\!47}a^{5}+\frac{45\!\cdots\!38}{61\!\cdots\!47}a^{4}-\frac{20\!\cdots\!45}{61\!\cdots\!47}a^{3}+\frac{63\!\cdots\!98}{61\!\cdots\!47}a^{2}-\frac{19\!\cdots\!60}{87\!\cdots\!21}a+\frac{50\!\cdots\!07}{61\!\cdots\!47}$, $\frac{17\!\cdots\!39}{61\!\cdots\!47}a^{27}-\frac{18\!\cdots\!64}{61\!\cdots\!47}a^{26}+\frac{97\!\cdots\!59}{61\!\cdots\!47}a^{25}-\frac{84\!\cdots\!91}{16\!\cdots\!57}a^{24}+\frac{65\!\cdots\!35}{61\!\cdots\!47}a^{23}-\frac{31\!\cdots\!57}{26\!\cdots\!89}a^{22}+\frac{57\!\cdots\!51}{61\!\cdots\!47}a^{21}+\frac{11\!\cdots\!26}{61\!\cdots\!47}a^{20}-\frac{14\!\cdots\!85}{87\!\cdots\!21}a^{19}+\frac{12\!\cdots\!99}{61\!\cdots\!47}a^{18}-\frac{24\!\cdots\!29}{19\!\cdots\!37}a^{17}+\frac{25\!\cdots\!80}{61\!\cdots\!47}a^{16}-\frac{36\!\cdots\!99}{61\!\cdots\!47}a^{15}+\frac{30\!\cdots\!73}{61\!\cdots\!47}a^{14}+\frac{58\!\cdots\!34}{61\!\cdots\!47}a^{13}-\frac{91\!\cdots\!60}{87\!\cdots\!21}a^{12}+\frac{16\!\cdots\!09}{61\!\cdots\!47}a^{11}-\frac{23\!\cdots\!09}{61\!\cdots\!47}a^{10}+\frac{22\!\cdots\!85}{61\!\cdots\!47}a^{9}-\frac{15\!\cdots\!33}{61\!\cdots\!47}a^{8}+\frac{64\!\cdots\!29}{61\!\cdots\!47}a^{7}+\frac{81\!\cdots\!69}{61\!\cdots\!47}a^{6}-\frac{87\!\cdots\!18}{14\!\cdots\!29}a^{5}+\frac{37\!\cdots\!81}{61\!\cdots\!47}a^{4}-\frac{20\!\cdots\!66}{61\!\cdots\!47}a^{3}+\frac{71\!\cdots\!52}{61\!\cdots\!47}a^{2}-\frac{21\!\cdots\!03}{87\!\cdots\!21}a+\frac{94\!\cdots\!79}{61\!\cdots\!47}$, $\frac{50\!\cdots\!08}{61\!\cdots\!47}a^{27}-\frac{62\!\cdots\!19}{61\!\cdots\!47}a^{26}+\frac{17\!\cdots\!11}{26\!\cdots\!89}a^{25}-\frac{21\!\cdots\!65}{87\!\cdots\!21}a^{24}+\frac{40\!\cdots\!82}{61\!\cdots\!47}a^{23}-\frac{71\!\cdots\!66}{61\!\cdots\!47}a^{22}+\frac{74\!\cdots\!70}{61\!\cdots\!47}a^{21}-\frac{16\!\cdots\!63}{61\!\cdots\!47}a^{20}-\frac{87\!\cdots\!06}{87\!\cdots\!21}a^{19}+\frac{10\!\cdots\!91}{61\!\cdots\!47}a^{18}-\frac{10\!\cdots\!00}{19\!\cdots\!37}a^{17}+\frac{12\!\cdots\!15}{61\!\cdots\!47}a^{16}-\frac{28\!\cdots\!08}{61\!\cdots\!47}a^{15}+\frac{40\!\cdots\!69}{61\!\cdots\!47}a^{14}-\frac{15\!\cdots\!52}{26\!\cdots\!89}a^{13}+\frac{35\!\cdots\!41}{12\!\cdots\!03}a^{12}+\frac{69\!\cdots\!01}{61\!\cdots\!47}a^{11}-\frac{17\!\cdots\!02}{61\!\cdots\!47}a^{10}+\frac{26\!\cdots\!19}{61\!\cdots\!47}a^{9}-\frac{29\!\cdots\!06}{61\!\cdots\!47}a^{8}+\frac{26\!\cdots\!27}{61\!\cdots\!47}a^{7}-\frac{19\!\cdots\!15}{61\!\cdots\!47}a^{6}+\frac{11\!\cdots\!44}{61\!\cdots\!47}a^{5}-\frac{54\!\cdots\!80}{61\!\cdots\!47}a^{4}+\frac{17\!\cdots\!08}{61\!\cdots\!47}a^{3}-\frac{50\!\cdots\!82}{61\!\cdots\!47}a^{2}+\frac{13\!\cdots\!29}{87\!\cdots\!21}a-\frac{24\!\cdots\!01}{61\!\cdots\!47}$, $\frac{19\!\cdots\!13}{61\!\cdots\!47}a^{27}-\frac{22\!\cdots\!89}{61\!\cdots\!47}a^{26}+\frac{13\!\cdots\!84}{61\!\cdots\!47}a^{25}-\frac{10\!\cdots\!88}{12\!\cdots\!03}a^{24}+\frac{12\!\cdots\!38}{61\!\cdots\!47}a^{23}-\frac{21\!\cdots\!70}{61\!\cdots\!47}a^{22}+\frac{21\!\cdots\!82}{61\!\cdots\!47}a^{21}-\frac{25\!\cdots\!31}{61\!\cdots\!47}a^{20}-\frac{26\!\cdots\!45}{87\!\cdots\!21}a^{19}+\frac{34\!\cdots\!15}{61\!\cdots\!47}a^{18}-\frac{83\!\cdots\!46}{46\!\cdots\!59}a^{17}+\frac{39\!\cdots\!59}{61\!\cdots\!47}a^{16}-\frac{85\!\cdots\!39}{61\!\cdots\!47}a^{15}+\frac{12\!\cdots\!65}{61\!\cdots\!47}a^{14}-\frac{99\!\cdots\!10}{61\!\cdots\!47}a^{13}-\frac{92\!\cdots\!01}{87\!\cdots\!21}a^{12}+\frac{23\!\cdots\!64}{61\!\cdots\!47}a^{11}-\frac{53\!\cdots\!53}{61\!\cdots\!47}a^{10}+\frac{79\!\cdots\!02}{61\!\cdots\!47}a^{9}-\frac{87\!\cdots\!88}{61\!\cdots\!47}a^{8}+\frac{76\!\cdots\!91}{61\!\cdots\!47}a^{7}-\frac{54\!\cdots\!64}{61\!\cdots\!47}a^{6}+\frac{31\!\cdots\!33}{61\!\cdots\!47}a^{5}-\frac{14\!\cdots\!42}{61\!\cdots\!47}a^{4}+\frac{46\!\cdots\!97}{61\!\cdots\!47}a^{3}-\frac{10\!\cdots\!87}{61\!\cdots\!47}a^{2}+\frac{46\!\cdots\!44}{87\!\cdots\!21}a-\frac{38\!\cdots\!35}{61\!\cdots\!47}$, $\frac{13\!\cdots\!86}{61\!\cdots\!47}a^{27}-\frac{14\!\cdots\!58}{61\!\cdots\!47}a^{26}+\frac{87\!\cdots\!26}{61\!\cdots\!47}a^{25}-\frac{45\!\cdots\!23}{87\!\cdots\!21}a^{24}+\frac{79\!\cdots\!54}{61\!\cdots\!47}a^{23}-\frac{12\!\cdots\!05}{61\!\cdots\!47}a^{22}+\frac{11\!\cdots\!90}{61\!\cdots\!47}a^{21}+\frac{10\!\cdots\!36}{26\!\cdots\!89}a^{20}-\frac{16\!\cdots\!39}{87\!\cdots\!21}a^{19}+\frac{20\!\cdots\!43}{61\!\cdots\!47}a^{18}-\frac{23\!\cdots\!44}{19\!\cdots\!37}a^{17}+\frac{25\!\cdots\!22}{61\!\cdots\!47}a^{16}-\frac{52\!\cdots\!46}{61\!\cdots\!47}a^{15}+\frac{70\!\cdots\!40}{61\!\cdots\!47}a^{14}-\frac{53\!\cdots\!12}{61\!\cdots\!47}a^{13}-\frac{18\!\cdots\!98}{87\!\cdots\!21}a^{12}+\frac{65\!\cdots\!96}{26\!\cdots\!89}a^{11}-\frac{32\!\cdots\!33}{61\!\cdots\!47}a^{10}+\frac{46\!\cdots\!02}{61\!\cdots\!47}a^{9}-\frac{50\!\cdots\!44}{61\!\cdots\!47}a^{8}+\frac{43\!\cdots\!75}{61\!\cdots\!47}a^{7}-\frac{30\!\cdots\!51}{61\!\cdots\!47}a^{6}+\frac{17\!\cdots\!05}{61\!\cdots\!47}a^{5}-\frac{77\!\cdots\!50}{61\!\cdots\!47}a^{4}+\frac{24\!\cdots\!49}{61\!\cdots\!47}a^{3}-\frac{65\!\cdots\!36}{61\!\cdots\!47}a^{2}+\frac{19\!\cdots\!48}{87\!\cdots\!21}a-\frac{22\!\cdots\!66}{61\!\cdots\!47}$ 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:  \( 71584737217.10019 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 71584737217.10019 \cdot 3}{2\cdot\sqrt{1923732503543401200540313355166872100640336877}}\cr\approx \mathstrut & 0.365895871823473 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - 12*x^27 + 74*x^26 - 288*x^25 + 773*x^24 - 1407*x^23 + 1601*x^22 - 676*x^21 - 794*x^20 + 2173*x^19 - 6644*x^18 + 23336*x^17 - 53512*x^16 + 82551*x^15 - 80880*x^14 + 23133*x^13 + 117766*x^12 - 327568*x^11 + 536090*x^10 - 646700*x^9 + 621936*x^8 - 488735*x^7 + 317878*x^6 - 166984*x^5 + 68952*x^4 - 21502*x^3 + 5711*x^2 - 1133*x + 193)
 
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 - 12*x^27 + 74*x^26 - 288*x^25 + 773*x^24 - 1407*x^23 + 1601*x^22 - 676*x^21 - 794*x^20 + 2173*x^19 - 6644*x^18 + 23336*x^17 - 53512*x^16 + 82551*x^15 - 80880*x^14 + 23133*x^13 + 117766*x^12 - 327568*x^11 + 536090*x^10 - 646700*x^9 + 621936*x^8 - 488735*x^7 + 317878*x^6 - 166984*x^5 + 68952*x^4 - 21502*x^3 + 5711*x^2 - 1133*x + 193, 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 - 12*x^27 + 74*x^26 - 288*x^25 + 773*x^24 - 1407*x^23 + 1601*x^22 - 676*x^21 - 794*x^20 + 2173*x^19 - 6644*x^18 + 23336*x^17 - 53512*x^16 + 82551*x^15 - 80880*x^14 + 23133*x^13 + 117766*x^12 - 327568*x^11 + 536090*x^10 - 646700*x^9 + 621936*x^8 - 488735*x^7 + 317878*x^6 - 166984*x^5 + 68952*x^4 - 21502*x^3 + 5711*x^2 - 1133*x + 193);
 
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 - 12*x^27 + 74*x^26 - 288*x^25 + 773*x^24 - 1407*x^23 + 1601*x^22 - 676*x^21 - 794*x^20 + 2173*x^19 - 6644*x^18 + 23336*x^17 - 53512*x^16 + 82551*x^15 - 80880*x^14 + 23133*x^13 + 117766*x^12 - 327568*x^11 + 536090*x^10 - 646700*x^9 + 621936*x^8 - 488735*x^7 + 317878*x^6 - 166984*x^5 + 68952*x^4 - 21502*x^3 + 5711*x^2 - 1133*x + 193);
 
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

$D_{28}$ (as 28T10):

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 56
The 17 conjugacy class representatives for $D_{28}$
Character table for $D_{28}$

Intermediate fields

\(\Q(\sqrt{-43}) \), 4.0.97997.1, 7.1.11836763639.1, 14.0.6024685858158758459803.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 28 sibling: deg 28
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $28$ $28$ $28$ ${\href{/padicField/7.2.0.1}{2} }^{14}$ ${\href{/padicField/11.7.0.1}{7} }^{4}$ ${\href{/padicField/13.14.0.1}{14} }^{2}$ ${\href{/padicField/17.7.0.1}{7} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{7}$ ${\href{/padicField/23.2.0.1}{2} }^{13}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{14}$ ${\href{/padicField/31.2.0.1}{2} }^{13}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{14}$ ${\href{/padicField/41.2.0.1}{2} }^{13}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ R ${\href{/padicField/47.7.0.1}{7} }^{4}$ R ${\href{/padicField/59.7.0.1}{7} }^{4}$

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
\(43\) Copy content Toggle raw display 43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 43$$2$$1$$1$$C_2$$[\ ]_{2}$
\(53\) Copy content Toggle raw display $\Q_{53}$$x + 51$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 51$$1$$1$$0$Trivial$[\ ]$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.43.2t1.a.a$1$ $ 43 $ \(\Q(\sqrt{-43}) \) $C_2$ (as 2T1) $1$ $-1$
1.2279.2t1.a.a$1$ $ 43 \cdot 53 $ \(\Q(\sqrt{-2279}) \) $C_2$ (as 2T1) $1$ $-1$
1.53.2t1.a.a$1$ $ 53 $ \(\Q(\sqrt{53}) \) $C_2$ (as 2T1) $1$ $1$
* 2.2279.4t3.a.a$2$ $ 43 \cdot 53 $ 4.0.97997.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.2279.14t3.a.a$2$ $ 43 \cdot 53 $ 14.0.6024685858158758459803.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.2279.7t2.a.b$2$ $ 43 \cdot 53 $ 7.1.11836763639.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.2279.14t3.a.c$2$ $ 43 \cdot 53 $ 14.0.6024685858158758459803.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.2279.7t2.a.c$2$ $ 43 \cdot 53 $ 7.1.11836763639.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.2279.7t2.a.a$2$ $ 43 \cdot 53 $ 7.1.11836763639.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.2279.14t3.a.b$2$ $ 43 \cdot 53 $ 14.0.6024685858158758459803.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.2279.28t10.a.c$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2279.28t10.a.f$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2279.28t10.a.d$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2279.28t10.a.e$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2279.28t10.a.b$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2279.28t10.a.a$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (as 28T10) $1$ $0$

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 *.