Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 2*x^27 + 4*x^26 - 10*x^25 + 5*x^24 + 7*x^23 - 18*x^22 - 225*x^21 - 120*x^20 + 1331*x^19 - 2152*x^18 + 4025*x^17 + 6967*x^16 - 15748*x^15 + 11604*x^14 - 4171*x^13 - 26180*x^12 + 37520*x^11 - 9040*x^10 + 16909*x^9 - 40976*x^8 - 11561*x^7 + 79270*x^6 - 49005*x^5 + 81626*x^4 - 70530*x^3 + 35205*x^2 - 15236*x + 3229)
gp: K = bnfinit(y^28 - 2*y^27 + 4*y^26 - 10*y^25 + 5*y^24 + 7*y^23 - 18*y^22 - 225*y^21 - 120*y^20 + 1331*y^19 - 2152*y^18 + 4025*y^17 + 6967*y^16 - 15748*y^15 + 11604*y^14 - 4171*y^13 - 26180*y^12 + 37520*y^11 - 9040*y^10 + 16909*y^9 - 40976*y^8 - 11561*y^7 + 79270*y^6 - 49005*y^5 + 81626*y^4 - 70530*y^3 + 35205*y^2 - 15236*y + 3229, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 2*x^27 + 4*x^26 - 10*x^25 + 5*x^24 + 7*x^23 - 18*x^22 - 225*x^21 - 120*x^20 + 1331*x^19 - 2152*x^18 + 4025*x^17 + 6967*x^16 - 15748*x^15 + 11604*x^14 - 4171*x^13 - 26180*x^12 + 37520*x^11 - 9040*x^10 + 16909*x^9 - 40976*x^8 - 11561*x^7 + 79270*x^6 - 49005*x^5 + 81626*x^4 - 70530*x^3 + 35205*x^2 - 15236*x + 3229);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 2*x^27 + 4*x^26 - 10*x^25 + 5*x^24 + 7*x^23 - 18*x^22 - 225*x^21 - 120*x^20 + 1331*x^19 - 2152*x^18 + 4025*x^17 + 6967*x^16 - 15748*x^15 + 11604*x^14 - 4171*x^13 - 26180*x^12 + 37520*x^11 - 9040*x^10 + 16909*x^9 - 40976*x^8 - 11561*x^7 + 79270*x^6 - 49005*x^5 + 81626*x^4 - 70530*x^3 + 35205*x^2 - 15236*x + 3229)
\( x^{28} - 2 x^{27} + 4 x^{26} - 10 x^{25} + 5 x^{24} + 7 x^{23} - 18 x^{22} - 225 x^{21} - 120 x^{20} + \cdots + 3229 \)
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: |
\(650754790224967946962108515803180182241777\)
\(\medspace = 7^{21}\cdot 71^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.14\) | 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: | $7^{3/4}71^{1/2}\approx 36.262079388792664$ | ||
| Ramified primes: |
\(7\), \(71\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{497}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{11}a^{24}-\frac{2}{11}a^{23}-\frac{4}{11}a^{21}-\frac{5}{11}a^{20}-\frac{4}{11}a^{19}-\frac{1}{11}a^{18}-\frac{5}{11}a^{16}+\frac{3}{11}a^{15}+\frac{5}{11}a^{14}-\frac{3}{11}a^{13}+\frac{4}{11}a^{12}-\frac{3}{11}a^{11}-\frac{4}{11}a^{10}-\frac{1}{11}a^{8}+\frac{5}{11}a^{7}-\frac{4}{11}a^{6}-\frac{5}{11}a^{5}-\frac{3}{11}a^{4}+\frac{2}{11}a^{3}+\frac{5}{11}a^{2}+\frac{2}{11}a+\frac{2}{11}$, $\frac{1}{451}a^{25}-\frac{17}{451}a^{24}+\frac{96}{451}a^{23}-\frac{224}{451}a^{22}+\frac{15}{41}a^{21}-\frac{50}{451}a^{20}+\frac{158}{451}a^{19}-\frac{172}{451}a^{18}-\frac{2}{11}a^{17}+\frac{56}{451}a^{16}-\frac{62}{451}a^{15}-\frac{155}{451}a^{14}+\frac{159}{451}a^{13}+\frac{135}{451}a^{12}-\frac{190}{451}a^{11}+\frac{225}{451}a^{10}+\frac{54}{451}a^{9}-\frac{57}{451}a^{8}+\frac{42}{451}a^{7}-\frac{12}{41}a^{6}+\frac{116}{451}a^{5}+\frac{69}{451}a^{4}+\frac{129}{451}a^{3}-\frac{40}{451}a^{2}+\frac{104}{451}a+\frac{190}{451}$, $\frac{1}{134849}a^{26}-\frac{9}{10373}a^{25}+\frac{5117}{134849}a^{24}+\frac{49380}{134849}a^{23}+\frac{57743}{134849}a^{22}+\frac{2154}{5863}a^{21}+\frac{54399}{134849}a^{20}+\frac{33884}{134849}a^{19}-\frac{8}{5863}a^{18}-\frac{313}{134849}a^{17}-\frac{65563}{134849}a^{16}-\frac{7895}{134849}a^{15}-\frac{10581}{134849}a^{14}+\frac{20274}{134849}a^{13}+\frac{3438}{10373}a^{12}-\frac{229}{533}a^{11}-\frac{49260}{134849}a^{10}+\frac{65350}{134849}a^{9}+\frac{60600}{134849}a^{8}+\frac{48353}{134849}a^{7}+\frac{16719}{134849}a^{6}+\frac{53946}{134849}a^{5}-\frac{64540}{134849}a^{4}+\frac{41508}{134849}a^{3}+\frac{49573}{134849}a^{2}-\frac{20255}{134849}a+\frac{59802}{134849}$, $\frac{1}{43\!\cdots\!19}a^{27}-\frac{11\!\cdots\!57}{39\!\cdots\!29}a^{26}+\frac{19\!\cdots\!48}{18\!\cdots\!53}a^{25}+\frac{13\!\cdots\!47}{43\!\cdots\!19}a^{24}-\frac{13\!\cdots\!98}{43\!\cdots\!19}a^{23}-\frac{16\!\cdots\!00}{43\!\cdots\!19}a^{22}-\frac{15\!\cdots\!58}{43\!\cdots\!19}a^{21}+\frac{20\!\cdots\!46}{43\!\cdots\!19}a^{20}+\frac{31\!\cdots\!67}{25\!\cdots\!07}a^{19}-\frac{19\!\cdots\!69}{43\!\cdots\!19}a^{18}+\frac{20\!\cdots\!84}{43\!\cdots\!19}a^{17}-\frac{10\!\cdots\!91}{33\!\cdots\!63}a^{16}+\frac{21\!\cdots\!21}{43\!\cdots\!19}a^{15}+\frac{41\!\cdots\!18}{43\!\cdots\!19}a^{14}+\frac{20\!\cdots\!11}{43\!\cdots\!19}a^{13}-\frac{13\!\cdots\!46}{43\!\cdots\!19}a^{12}-\frac{15\!\cdots\!90}{43\!\cdots\!19}a^{11}-\frac{92\!\cdots\!86}{43\!\cdots\!19}a^{10}-\frac{39\!\cdots\!44}{39\!\cdots\!29}a^{9}+\frac{11\!\cdots\!33}{39\!\cdots\!29}a^{8}+\frac{76\!\cdots\!15}{43\!\cdots\!19}a^{7}-\frac{96\!\cdots\!33}{18\!\cdots\!53}a^{6}+\frac{12\!\cdots\!47}{43\!\cdots\!19}a^{5}+\frac{14\!\cdots\!37}{43\!\cdots\!19}a^{4}-\frac{18\!\cdots\!50}{43\!\cdots\!19}a^{3}+\frac{10\!\cdots\!58}{43\!\cdots\!19}a^{2}-\frac{91\!\cdots\!10}{18\!\cdots\!53}a+\frac{18\!\cdots\!32}{43\!\cdots\!19}$
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
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{43\!\cdots\!99}{43\!\cdots\!19}a^{27}+\frac{28\!\cdots\!21}{39\!\cdots\!29}a^{26}-\frac{12\!\cdots\!54}{43\!\cdots\!19}a^{25}+\frac{33\!\cdots\!90}{43\!\cdots\!19}a^{24}-\frac{87\!\cdots\!12}{43\!\cdots\!19}a^{23}+\frac{16\!\cdots\!60}{43\!\cdots\!19}a^{22}-\frac{23\!\cdots\!86}{43\!\cdots\!19}a^{21}+\frac{30\!\cdots\!08}{43\!\cdots\!19}a^{20}-\frac{17\!\cdots\!58}{61\!\cdots\!27}a^{19}+\frac{18\!\cdots\!17}{43\!\cdots\!19}a^{18}+\frac{16\!\cdots\!91}{39\!\cdots\!29}a^{17}-\frac{11\!\cdots\!50}{43\!\cdots\!19}a^{16}+\frac{30\!\cdots\!70}{43\!\cdots\!19}a^{15}-\frac{30\!\cdots\!00}{43\!\cdots\!19}a^{14}-\frac{15\!\cdots\!22}{43\!\cdots\!19}a^{13}+\frac{67\!\cdots\!66}{43\!\cdots\!19}a^{12}-\frac{11\!\cdots\!63}{43\!\cdots\!19}a^{11}+\frac{96\!\cdots\!37}{43\!\cdots\!19}a^{10}-\frac{71\!\cdots\!70}{43\!\cdots\!19}a^{9}-\frac{13\!\cdots\!38}{43\!\cdots\!19}a^{8}+\frac{36\!\cdots\!27}{43\!\cdots\!19}a^{7}-\frac{24\!\cdots\!59}{43\!\cdots\!19}a^{6}+\frac{39\!\cdots\!11}{43\!\cdots\!19}a^{5}-\frac{13\!\cdots\!58}{18\!\cdots\!53}a^{4}+\frac{27\!\cdots\!48}{43\!\cdots\!19}a^{3}-\frac{26\!\cdots\!56}{43\!\cdots\!19}a^{2}+\frac{12\!\cdots\!76}{43\!\cdots\!19}a-\frac{57\!\cdots\!18}{43\!\cdots\!19}$, $\frac{31\!\cdots\!00}{18\!\cdots\!53}a^{27}-\frac{13\!\cdots\!44}{39\!\cdots\!29}a^{26}+\frac{29\!\cdots\!43}{43\!\cdots\!19}a^{25}-\frac{76\!\cdots\!85}{43\!\cdots\!19}a^{24}+\frac{45\!\cdots\!47}{43\!\cdots\!19}a^{23}+\frac{36\!\cdots\!38}{43\!\cdots\!19}a^{22}-\frac{40\!\cdots\!66}{18\!\cdots\!53}a^{21}-\frac{17\!\cdots\!56}{43\!\cdots\!19}a^{20}-\frac{38\!\cdots\!14}{25\!\cdots\!07}a^{19}+\frac{42\!\cdots\!16}{18\!\cdots\!53}a^{18}-\frac{14\!\cdots\!63}{39\!\cdots\!29}a^{17}+\frac{29\!\cdots\!07}{43\!\cdots\!19}a^{16}+\frac{49\!\cdots\!01}{43\!\cdots\!19}a^{15}-\frac{11\!\cdots\!49}{43\!\cdots\!19}a^{14}+\frac{74\!\cdots\!03}{43\!\cdots\!19}a^{13}-\frac{16\!\cdots\!85}{43\!\cdots\!19}a^{12}-\frac{81\!\cdots\!41}{18\!\cdots\!53}a^{11}+\frac{25\!\cdots\!56}{43\!\cdots\!19}a^{10}-\frac{12\!\cdots\!15}{43\!\cdots\!19}a^{9}+\frac{63\!\cdots\!84}{43\!\cdots\!19}a^{8}-\frac{29\!\cdots\!58}{43\!\cdots\!19}a^{7}-\frac{64\!\cdots\!07}{43\!\cdots\!19}a^{6}+\frac{55\!\cdots\!33}{43\!\cdots\!19}a^{5}-\frac{24\!\cdots\!67}{43\!\cdots\!19}a^{4}+\frac{41\!\cdots\!08}{43\!\cdots\!19}a^{3}-\frac{39\!\cdots\!80}{43\!\cdots\!19}a^{2}+\frac{99\!\cdots\!10}{43\!\cdots\!19}a-\frac{21\!\cdots\!84}{43\!\cdots\!19}$, $\frac{76\!\cdots\!25}{43\!\cdots\!19}a^{27}-\frac{15\!\cdots\!25}{39\!\cdots\!29}a^{26}+\frac{38\!\cdots\!97}{43\!\cdots\!19}a^{25}-\frac{10\!\cdots\!08}{43\!\cdots\!19}a^{24}+\frac{11\!\cdots\!75}{43\!\cdots\!19}a^{23}-\frac{10\!\cdots\!30}{43\!\cdots\!19}a^{22}+\frac{13\!\cdots\!50}{43\!\cdots\!19}a^{21}-\frac{20\!\cdots\!94}{43\!\cdots\!19}a^{20}-\frac{23\!\cdots\!12}{25\!\cdots\!07}a^{19}+\frac{83\!\cdots\!94}{43\!\cdots\!19}a^{18}-\frac{13\!\cdots\!47}{39\!\cdots\!29}a^{17}+\frac{36\!\cdots\!51}{43\!\cdots\!19}a^{16}+\frac{68\!\cdots\!87}{10\!\cdots\!59}a^{15}-\frac{77\!\cdots\!16}{43\!\cdots\!19}a^{14}+\frac{57\!\cdots\!03}{43\!\cdots\!19}a^{13}-\frac{73\!\cdots\!83}{43\!\cdots\!19}a^{12}-\frac{74\!\cdots\!49}{43\!\cdots\!19}a^{11}+\frac{11\!\cdots\!43}{43\!\cdots\!19}a^{10}+\frac{53\!\cdots\!83}{43\!\cdots\!19}a^{9}+\frac{13\!\cdots\!47}{43\!\cdots\!19}a^{8}-\frac{39\!\cdots\!09}{43\!\cdots\!19}a^{7}+\frac{81\!\cdots\!08}{43\!\cdots\!19}a^{6}+\frac{20\!\cdots\!07}{43\!\cdots\!19}a^{5}+\frac{16\!\cdots\!61}{43\!\cdots\!19}a^{4}+\frac{15\!\cdots\!25}{43\!\cdots\!19}a^{3}-\frac{19\!\cdots\!97}{43\!\cdots\!19}a^{2}-\frac{99\!\cdots\!85}{43\!\cdots\!19}a-\frac{27\!\cdots\!64}{43\!\cdots\!19}$, $\frac{20\!\cdots\!09}{43\!\cdots\!19}a^{27}-\frac{78\!\cdots\!36}{10\!\cdots\!59}a^{26}+\frac{66\!\cdots\!84}{43\!\cdots\!19}a^{25}-\frac{17\!\cdots\!19}{43\!\cdots\!19}a^{24}+\frac{14\!\cdots\!75}{43\!\cdots\!19}a^{23}+\frac{18\!\cdots\!67}{43\!\cdots\!19}a^{22}-\frac{33\!\cdots\!49}{43\!\cdots\!19}a^{21}-\frac{47\!\cdots\!40}{43\!\cdots\!19}a^{20}-\frac{27\!\cdots\!66}{25\!\cdots\!07}a^{19}+\frac{26\!\cdots\!02}{43\!\cdots\!19}a^{18}-\frac{30\!\cdots\!96}{39\!\cdots\!29}a^{17}+\frac{66\!\cdots\!32}{43\!\cdots\!19}a^{16}+\frac{17\!\cdots\!23}{43\!\cdots\!19}a^{15}-\frac{25\!\cdots\!88}{43\!\cdots\!19}a^{14}+\frac{11\!\cdots\!92}{43\!\cdots\!19}a^{13}-\frac{35\!\cdots\!95}{43\!\cdots\!19}a^{12}-\frac{57\!\cdots\!97}{43\!\cdots\!19}a^{11}+\frac{54\!\cdots\!53}{43\!\cdots\!19}a^{10}+\frac{84\!\cdots\!47}{43\!\cdots\!19}a^{9}+\frac{33\!\cdots\!24}{43\!\cdots\!19}a^{8}-\frac{69\!\cdots\!55}{43\!\cdots\!19}a^{7}-\frac{56\!\cdots\!17}{43\!\cdots\!19}a^{6}+\frac{14\!\cdots\!74}{43\!\cdots\!19}a^{5}-\frac{39\!\cdots\!89}{39\!\cdots\!29}a^{4}+\frac{14\!\cdots\!38}{43\!\cdots\!19}a^{3}-\frac{78\!\cdots\!49}{43\!\cdots\!19}a^{2}+\frac{36\!\cdots\!95}{43\!\cdots\!19}a-\frac{74\!\cdots\!21}{43\!\cdots\!19}$, $\frac{25\!\cdots\!15}{10\!\cdots\!59}a^{27}-\frac{82\!\cdots\!33}{43\!\cdots\!19}a^{26}+\frac{14\!\cdots\!45}{43\!\cdots\!19}a^{25}-\frac{37\!\cdots\!61}{43\!\cdots\!19}a^{24}-\frac{12\!\cdots\!36}{43\!\cdots\!19}a^{23}+\frac{26\!\cdots\!65}{43\!\cdots\!19}a^{22}-\frac{38\!\cdots\!26}{43\!\cdots\!19}a^{21}-\frac{21\!\cdots\!84}{43\!\cdots\!19}a^{20}-\frac{28\!\cdots\!88}{25\!\cdots\!07}a^{19}+\frac{13\!\cdots\!35}{43\!\cdots\!19}a^{18}-\frac{85\!\cdots\!60}{39\!\cdots\!29}a^{17}+\frac{16\!\cdots\!21}{43\!\cdots\!19}a^{16}+\frac{13\!\cdots\!90}{43\!\cdots\!19}a^{15}-\frac{12\!\cdots\!45}{43\!\cdots\!19}a^{14}+\frac{10\!\cdots\!72}{43\!\cdots\!19}a^{13}+\frac{71\!\cdots\!86}{43\!\cdots\!19}a^{12}-\frac{41\!\cdots\!38}{43\!\cdots\!19}a^{11}+\frac{26\!\cdots\!87}{43\!\cdots\!19}a^{10}+\frac{10\!\cdots\!79}{43\!\cdots\!19}a^{9}+\frac{23\!\cdots\!17}{43\!\cdots\!19}a^{8}-\frac{20\!\cdots\!54}{43\!\cdots\!19}a^{7}-\frac{69\!\cdots\!35}{43\!\cdots\!19}a^{6}+\frac{10\!\cdots\!15}{43\!\cdots\!19}a^{5}-\frac{24\!\cdots\!51}{39\!\cdots\!29}a^{4}+\frac{27\!\cdots\!25}{10\!\cdots\!59}a^{3}-\frac{43\!\cdots\!66}{43\!\cdots\!19}a^{2}+\frac{18\!\cdots\!39}{43\!\cdots\!19}a-\frac{11\!\cdots\!26}{43\!\cdots\!19}$, $\frac{95\!\cdots\!24}{43\!\cdots\!19}a^{27}-\frac{24\!\cdots\!63}{43\!\cdots\!19}a^{26}+\frac{60\!\cdots\!35}{43\!\cdots\!19}a^{25}-\frac{14\!\cdots\!92}{43\!\cdots\!19}a^{24}+\frac{14\!\cdots\!76}{33\!\cdots\!63}a^{23}-\frac{17\!\cdots\!59}{43\!\cdots\!19}a^{22}+\frac{85\!\cdots\!55}{33\!\cdots\!63}a^{21}-\frac{24\!\cdots\!30}{43\!\cdots\!19}a^{20}+\frac{49\!\cdots\!03}{25\!\cdots\!07}a^{19}+\frac{98\!\cdots\!91}{43\!\cdots\!19}a^{18}-\frac{26\!\cdots\!11}{43\!\cdots\!19}a^{17}+\frac{59\!\cdots\!96}{43\!\cdots\!19}a^{16}+\frac{17\!\cdots\!66}{43\!\cdots\!19}a^{15}-\frac{11\!\cdots\!57}{43\!\cdots\!19}a^{14}+\frac{16\!\cdots\!75}{39\!\cdots\!29}a^{13}-\frac{15\!\cdots\!34}{43\!\cdots\!19}a^{12}-\frac{12\!\cdots\!51}{43\!\cdots\!19}a^{11}+\frac{29\!\cdots\!26}{43\!\cdots\!19}a^{10}-\frac{14\!\cdots\!49}{33\!\cdots\!63}a^{9}+\frac{17\!\cdots\!20}{43\!\cdots\!19}a^{8}-\frac{38\!\cdots\!50}{43\!\cdots\!19}a^{7}+\frac{32\!\cdots\!35}{43\!\cdots\!19}a^{6}+\frac{15\!\cdots\!19}{43\!\cdots\!19}a^{5}-\frac{33\!\cdots\!14}{43\!\cdots\!19}a^{4}+\frac{63\!\cdots\!09}{43\!\cdots\!19}a^{3}-\frac{28\!\cdots\!23}{43\!\cdots\!19}a^{2}+\frac{29\!\cdots\!20}{43\!\cdots\!19}a-\frac{12\!\cdots\!55}{43\!\cdots\!19}$, $\frac{10\!\cdots\!30}{13\!\cdots\!71}a^{27}-\frac{57\!\cdots\!15}{43\!\cdots\!19}a^{26}+\frac{89\!\cdots\!98}{33\!\cdots\!63}a^{25}-\frac{29\!\cdots\!77}{43\!\cdots\!19}a^{24}+\frac{24\!\cdots\!35}{43\!\cdots\!19}a^{23}+\frac{35\!\cdots\!01}{43\!\cdots\!19}a^{22}-\frac{28\!\cdots\!77}{18\!\cdots\!53}a^{21}-\frac{19\!\cdots\!22}{10\!\cdots\!59}a^{20}-\frac{45\!\cdots\!68}{25\!\cdots\!07}a^{19}+\frac{20\!\cdots\!83}{18\!\cdots\!53}a^{18}-\frac{60\!\cdots\!82}{43\!\cdots\!19}a^{17}+\frac{11\!\cdots\!73}{43\!\cdots\!19}a^{16}+\frac{31\!\cdots\!74}{43\!\cdots\!19}a^{15}-\frac{47\!\cdots\!89}{43\!\cdots\!19}a^{14}+\frac{24\!\cdots\!01}{43\!\cdots\!19}a^{13}-\frac{10\!\cdots\!99}{33\!\cdots\!63}a^{12}-\frac{44\!\cdots\!26}{18\!\cdots\!53}a^{11}+\frac{10\!\cdots\!07}{43\!\cdots\!19}a^{10}+\frac{53\!\cdots\!62}{43\!\cdots\!19}a^{9}+\frac{53\!\cdots\!21}{39\!\cdots\!29}a^{8}-\frac{12\!\cdots\!86}{43\!\cdots\!19}a^{7}-\frac{91\!\cdots\!46}{43\!\cdots\!19}a^{6}+\frac{27\!\cdots\!68}{43\!\cdots\!19}a^{5}-\frac{10\!\cdots\!96}{43\!\cdots\!19}a^{4}+\frac{27\!\cdots\!70}{43\!\cdots\!19}a^{3}-\frac{17\!\cdots\!67}{43\!\cdots\!19}a^{2}+\frac{78\!\cdots\!30}{43\!\cdots\!19}a-\frac{29\!\cdots\!38}{43\!\cdots\!19}$, $\frac{30\!\cdots\!02}{30\!\cdots\!33}a^{27}-\frac{59\!\cdots\!40}{33\!\cdots\!63}a^{26}+\frac{11\!\cdots\!98}{33\!\cdots\!63}a^{25}-\frac{32\!\cdots\!45}{33\!\cdots\!63}a^{24}+\frac{15\!\cdots\!64}{33\!\cdots\!63}a^{23}+\frac{13\!\cdots\!64}{33\!\cdots\!63}a^{22}-\frac{20\!\cdots\!27}{33\!\cdots\!63}a^{21}-\frac{81\!\cdots\!46}{33\!\cdots\!63}a^{20}-\frac{29\!\cdots\!00}{19\!\cdots\!39}a^{19}+\frac{10\!\cdots\!28}{81\!\cdots\!43}a^{18}-\frac{56\!\cdots\!19}{33\!\cdots\!63}a^{17}+\frac{11\!\cdots\!23}{33\!\cdots\!63}a^{16}+\frac{24\!\cdots\!75}{33\!\cdots\!63}a^{15}-\frac{42\!\cdots\!56}{33\!\cdots\!63}a^{14}+\frac{15\!\cdots\!75}{33\!\cdots\!63}a^{13}-\frac{32\!\cdots\!40}{33\!\cdots\!63}a^{12}-\frac{74\!\cdots\!82}{33\!\cdots\!63}a^{11}+\frac{81\!\cdots\!56}{33\!\cdots\!63}a^{10}+\frac{30\!\cdots\!52}{33\!\cdots\!63}a^{9}+\frac{41\!\cdots\!37}{33\!\cdots\!63}a^{8}-\frac{14\!\cdots\!43}{33\!\cdots\!63}a^{7}-\frac{70\!\cdots\!45}{33\!\cdots\!63}a^{6}+\frac{23\!\cdots\!50}{33\!\cdots\!63}a^{5}+\frac{33\!\cdots\!00}{33\!\cdots\!63}a^{4}+\frac{11\!\cdots\!58}{33\!\cdots\!63}a^{3}-\frac{16\!\cdots\!52}{33\!\cdots\!63}a^{2}+\frac{10\!\cdots\!92}{30\!\cdots\!33}a+\frac{11\!\cdots\!24}{33\!\cdots\!63}$, $\frac{18\!\cdots\!53}{43\!\cdots\!19}a^{27}-\frac{19\!\cdots\!66}{43\!\cdots\!19}a^{26}+\frac{46\!\cdots\!47}{43\!\cdots\!19}a^{25}-\frac{13\!\cdots\!11}{43\!\cdots\!19}a^{24}-\frac{45\!\cdots\!32}{43\!\cdots\!19}a^{23}+\frac{12\!\cdots\!20}{43\!\cdots\!19}a^{22}-\frac{12\!\cdots\!36}{43\!\cdots\!19}a^{21}-\frac{45\!\cdots\!72}{43\!\cdots\!19}a^{20}-\frac{36\!\cdots\!29}{25\!\cdots\!07}a^{19}+\frac{20\!\cdots\!18}{43\!\cdots\!19}a^{18}-\frac{16\!\cdots\!22}{43\!\cdots\!19}a^{17}+\frac{46\!\cdots\!84}{43\!\cdots\!19}a^{16}+\frac{18\!\cdots\!16}{43\!\cdots\!19}a^{15}-\frac{13\!\cdots\!56}{43\!\cdots\!19}a^{14}-\frac{27\!\cdots\!00}{43\!\cdots\!19}a^{13}+\frac{12\!\cdots\!26}{43\!\cdots\!19}a^{12}-\frac{44\!\cdots\!95}{43\!\cdots\!19}a^{11}+\frac{77\!\cdots\!16}{18\!\cdots\!53}a^{10}+\frac{33\!\cdots\!19}{43\!\cdots\!19}a^{9}+\frac{38\!\cdots\!59}{39\!\cdots\!29}a^{8}-\frac{55\!\cdots\!77}{43\!\cdots\!19}a^{7}-\frac{85\!\cdots\!04}{43\!\cdots\!19}a^{6}+\frac{97\!\cdots\!98}{43\!\cdots\!19}a^{5}+\frac{46\!\cdots\!06}{33\!\cdots\!63}a^{4}+\frac{10\!\cdots\!89}{43\!\cdots\!19}a^{3}-\frac{24\!\cdots\!40}{43\!\cdots\!19}a^{2}-\frac{13\!\cdots\!47}{43\!\cdots\!19}a-\frac{17\!\cdots\!65}{43\!\cdots\!19}$, $\frac{16\!\cdots\!47}{43\!\cdots\!19}a^{27}-\frac{21\!\cdots\!85}{43\!\cdots\!19}a^{26}+\frac{42\!\cdots\!16}{43\!\cdots\!19}a^{25}-\frac{85\!\cdots\!11}{33\!\cdots\!63}a^{24}-\frac{53\!\cdots\!28}{43\!\cdots\!19}a^{23}+\frac{20\!\cdots\!38}{39\!\cdots\!29}a^{22}-\frac{36\!\cdots\!28}{43\!\cdots\!19}a^{21}-\frac{33\!\cdots\!67}{39\!\cdots\!29}a^{20}-\frac{22\!\cdots\!81}{19\!\cdots\!39}a^{19}+\frac{20\!\cdots\!02}{43\!\cdots\!19}a^{18}-\frac{17\!\cdots\!23}{33\!\cdots\!63}a^{17}+\frac{43\!\cdots\!33}{43\!\cdots\!19}a^{16}+\frac{16\!\cdots\!31}{43\!\cdots\!19}a^{15}-\frac{19\!\cdots\!38}{43\!\cdots\!19}a^{14}+\frac{63\!\cdots\!46}{43\!\cdots\!19}a^{13}+\frac{35\!\cdots\!53}{43\!\cdots\!19}a^{12}-\frac{51\!\cdots\!55}{43\!\cdots\!19}a^{11}+\frac{10\!\cdots\!39}{10\!\cdots\!59}a^{10}+\frac{96\!\cdots\!85}{43\!\cdots\!19}a^{9}+\frac{28\!\cdots\!71}{43\!\cdots\!19}a^{8}-\frac{43\!\cdots\!12}{39\!\cdots\!29}a^{7}-\frac{66\!\cdots\!26}{43\!\cdots\!19}a^{6}+\frac{31\!\cdots\!81}{10\!\cdots\!59}a^{5}-\frac{34\!\cdots\!13}{43\!\cdots\!19}a^{4}+\frac{13\!\cdots\!55}{43\!\cdots\!19}a^{3}-\frac{26\!\cdots\!06}{18\!\cdots\!53}a^{2}+\frac{25\!\cdots\!87}{43\!\cdots\!19}a-\frac{10\!\cdots\!37}{43\!\cdots\!19}$, $\frac{92\!\cdots\!02}{25\!\cdots\!07}a^{27}-\frac{54\!\cdots\!01}{25\!\cdots\!07}a^{26}+\frac{37\!\cdots\!60}{25\!\cdots\!07}a^{25}-\frac{66\!\cdots\!57}{25\!\cdots\!07}a^{24}-\frac{17\!\cdots\!75}{25\!\cdots\!07}a^{23}+\frac{28\!\cdots\!81}{19\!\cdots\!39}a^{22}-\frac{56\!\cdots\!07}{25\!\cdots\!07}a^{21}-\frac{15\!\cdots\!38}{25\!\cdots\!07}a^{20}-\frac{50\!\cdots\!42}{25\!\cdots\!07}a^{19}+\frac{13\!\cdots\!97}{25\!\cdots\!07}a^{18}-\frac{84\!\cdots\!28}{25\!\cdots\!07}a^{17}+\frac{68\!\cdots\!16}{25\!\cdots\!07}a^{16}+\frac{14\!\cdots\!99}{25\!\cdots\!07}a^{15}-\frac{13\!\cdots\!29}{25\!\cdots\!07}a^{14}+\frac{12\!\cdots\!54}{25\!\cdots\!07}a^{13}+\frac{11\!\cdots\!64}{25\!\cdots\!07}a^{12}-\frac{35\!\cdots\!78}{19\!\cdots\!39}a^{11}+\frac{27\!\cdots\!13}{19\!\cdots\!39}a^{10}+\frac{23\!\cdots\!16}{25\!\cdots\!07}a^{9}+\frac{14\!\cdots\!84}{23\!\cdots\!37}a^{8}-\frac{31\!\cdots\!52}{61\!\cdots\!27}a^{7}-\frac{14\!\cdots\!45}{47\!\cdots\!79}a^{6}+\frac{12\!\cdots\!25}{25\!\cdots\!07}a^{5}-\frac{60\!\cdots\!73}{25\!\cdots\!07}a^{4}+\frac{98\!\cdots\!37}{19\!\cdots\!39}a^{3}-\frac{59\!\cdots\!20}{25\!\cdots\!07}a^{2}+\frac{29\!\cdots\!21}{25\!\cdots\!07}a-\frac{85\!\cdots\!40}{25\!\cdots\!07}$, $\frac{17\!\cdots\!28}{25\!\cdots\!07}a^{27}-\frac{28\!\cdots\!04}{25\!\cdots\!07}a^{26}+\frac{60\!\cdots\!98}{25\!\cdots\!07}a^{25}-\frac{15\!\cdots\!43}{25\!\cdots\!07}a^{24}+\frac{28\!\cdots\!35}{19\!\cdots\!39}a^{23}+\frac{10\!\cdots\!83}{25\!\cdots\!07}a^{22}-\frac{17\!\cdots\!70}{19\!\cdots\!39}a^{21}-\frac{41\!\cdots\!72}{25\!\cdots\!07}a^{20}-\frac{36\!\cdots\!52}{25\!\cdots\!07}a^{19}+\frac{21\!\cdots\!25}{25\!\cdots\!07}a^{18}-\frac{28\!\cdots\!18}{25\!\cdots\!07}a^{17}+\frac{54\!\cdots\!95}{23\!\cdots\!37}a^{16}+\frac{14\!\cdots\!12}{25\!\cdots\!07}a^{15}-\frac{90\!\cdots\!51}{11\!\cdots\!09}a^{14}+\frac{11\!\cdots\!30}{25\!\cdots\!07}a^{13}-\frac{34\!\cdots\!82}{25\!\cdots\!07}a^{12}-\frac{45\!\cdots\!62}{25\!\cdots\!07}a^{11}+\frac{44\!\cdots\!54}{25\!\cdots\!07}a^{10}+\frac{29\!\cdots\!92}{19\!\cdots\!39}a^{9}+\frac{31\!\cdots\!13}{25\!\cdots\!07}a^{8}-\frac{59\!\cdots\!03}{25\!\cdots\!07}a^{7}-\frac{41\!\cdots\!64}{25\!\cdots\!07}a^{6}+\frac{11\!\cdots\!54}{25\!\cdots\!07}a^{5}-\frac{30\!\cdots\!22}{25\!\cdots\!07}a^{4}+\frac{54\!\cdots\!95}{11\!\cdots\!09}a^{3}-\frac{66\!\cdots\!96}{25\!\cdots\!07}a^{2}+\frac{30\!\cdots\!02}{25\!\cdots\!07}a-\frac{12\!\cdots\!08}{25\!\cdots\!07}$, $\frac{24\!\cdots\!66}{43\!\cdots\!19}a^{27}-\frac{41\!\cdots\!06}{43\!\cdots\!19}a^{26}+\frac{85\!\cdots\!20}{43\!\cdots\!19}a^{25}-\frac{21\!\cdots\!56}{43\!\cdots\!19}a^{24}+\frac{58\!\cdots\!50}{43\!\cdots\!19}a^{23}+\frac{18\!\cdots\!79}{43\!\cdots\!19}a^{22}-\frac{39\!\cdots\!73}{43\!\cdots\!19}a^{21}-\frac{55\!\cdots\!71}{43\!\cdots\!19}a^{20}-\frac{26\!\cdots\!88}{25\!\cdots\!07}a^{19}+\frac{30\!\cdots\!05}{43\!\cdots\!19}a^{18}-\frac{43\!\cdots\!78}{43\!\cdots\!19}a^{17}+\frac{77\!\cdots\!58}{39\!\cdots\!29}a^{16}+\frac{19\!\cdots\!23}{43\!\cdots\!19}a^{15}-\frac{32\!\cdots\!35}{43\!\cdots\!19}a^{14}+\frac{20\!\cdots\!28}{43\!\cdots\!19}a^{13}-\frac{45\!\cdots\!16}{43\!\cdots\!19}a^{12}-\frac{65\!\cdots\!92}{43\!\cdots\!19}a^{11}+\frac{72\!\cdots\!50}{43\!\cdots\!19}a^{10}-\frac{39\!\cdots\!26}{43\!\cdots\!19}a^{9}+\frac{40\!\cdots\!21}{43\!\cdots\!19}a^{8}-\frac{65\!\cdots\!69}{33\!\cdots\!63}a^{7}-\frac{49\!\cdots\!34}{43\!\cdots\!19}a^{6}+\frac{17\!\cdots\!56}{43\!\cdots\!19}a^{5}-\frac{76\!\cdots\!00}{43\!\cdots\!19}a^{4}+\frac{18\!\cdots\!03}{43\!\cdots\!19}a^{3}-\frac{11\!\cdots\!87}{43\!\cdots\!19}a^{2}+\frac{41\!\cdots\!06}{30\!\cdots\!33}a-\frac{18\!\cdots\!38}{43\!\cdots\!19}$
| 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: | \( 3185481783.8925004 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 3185481783.8925004 \cdot 1}{2\cdot\sqrt{650754790224967946962108515803180182241777}}\cr\approx \mathstrut & 0.295090073420219 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 - 2*x^27 + 4*x^26 - 10*x^25 + 5*x^24 + 7*x^23 - 18*x^22 - 225*x^21 - 120*x^20 + 1331*x^19 - 2152*x^18 + 4025*x^17 + 6967*x^16 - 15748*x^15 + 11604*x^14 - 4171*x^13 - 26180*x^12 + 37520*x^11 - 9040*x^10 + 16909*x^9 - 40976*x^8 - 11561*x^7 + 79270*x^6 - 49005*x^5 + 81626*x^4 - 70530*x^3 + 35205*x^2 - 15236*x + 3229)
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 - 2*x^27 + 4*x^26 - 10*x^25 + 5*x^24 + 7*x^23 - 18*x^22 - 225*x^21 - 120*x^20 + 1331*x^19 - 2152*x^18 + 4025*x^17 + 6967*x^16 - 15748*x^15 + 11604*x^14 - 4171*x^13 - 26180*x^12 + 37520*x^11 - 9040*x^10 + 16909*x^9 - 40976*x^8 - 11561*x^7 + 79270*x^6 - 49005*x^5 + 81626*x^4 - 70530*x^3 + 35205*x^2 - 15236*x + 3229, 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 - 2*x^27 + 4*x^26 - 10*x^25 + 5*x^24 + 7*x^23 - 18*x^22 - 225*x^21 - 120*x^20 + 1331*x^19 - 2152*x^18 + 4025*x^17 + 6967*x^16 - 15748*x^15 + 11604*x^14 - 4171*x^13 - 26180*x^12 + 37520*x^11 - 9040*x^10 + 16909*x^9 - 40976*x^8 - 11561*x^7 + 79270*x^6 - 49005*x^5 + 81626*x^4 - 70530*x^3 + 35205*x^2 - 15236*x + 3229);
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 - 2*x^27 + 4*x^26 - 10*x^25 + 5*x^24 + 7*x^23 - 18*x^22 - 225*x^21 - 120*x^20 + 1331*x^19 - 2152*x^18 + 4025*x^17 + 6967*x^16 - 15748*x^15 + 11604*x^14 - 4171*x^13 - 26180*x^12 + 37520*x^11 - 9040*x^10 + 16909*x^9 - 40976*x^8 - 11561*x^7 + 79270*x^6 - 49005*x^5 + 81626*x^4 - 70530*x^3 + 35205*x^2 - 15236*x + 3229);
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
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
| \(\Q(\sqrt{-7}) \), 4.0.24353.1, 7.1.357911.1, 14.0.105496092121152103.1 |
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: | 28.2.46203590105972724234309704622025792939166167.1 |
| 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 | ${\href{/padicField/2.7.0.1}{7} }^{4}$ | $28$ | $28$ | R | ${\href{/padicField/11.2.0.1}{2} }^{13}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{14}$ | ${\href{/padicField/17.2.0.1}{2} }^{14}$ | $28$ | ${\href{/padicField/23.2.0.1}{2} }^{13}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{14}$ | ${\href{/padicField/37.14.0.1}{14} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{14}$ | ${\href{/padicField/43.7.0.1}{7} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{14}$ | ${\href{/padicField/53.2.0.1}{2} }^{13}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{14}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.4.3.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
| 7.8.6.2 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2660 x^{4} + 3408 x^{3} + 3312 x^{2} + 5184 x + 6304$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
| 7.8.6.2 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2660 x^{4} + 3408 x^{3} + 3312 x^{2} + 5184 x + 6304$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
| 7.8.6.2 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2660 x^{4} + 3408 x^{3} + 3312 x^{2} + 5184 x + 6304$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
|
\(71\)
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.497.2t1.a.a | $1$ | $ 7 \cdot 71 $ | \(\Q(\sqrt{497}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.71.2t1.a.a | $1$ | $ 71 $ | \(\Q(\sqrt{-71}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 2.3479.4t3.c.a | $2$ | $ 7^{2} \cdot 71 $ | 4.2.1729063.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| * | 2.3479.14t3.a.c | $2$ | $ 7^{2} \cdot 71 $ | 14.2.7490222540601799313.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| * | 2.3479.14t3.a.a | $2$ | $ 7^{2} \cdot 71 $ | 14.2.7490222540601799313.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| * | 2.71.7t2.a.b | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.71.7t2.a.a | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.3479.14t3.a.b | $2$ | $ 7^{2} \cdot 71 $ | 14.2.7490222540601799313.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| * | 2.71.7t2.a.c | $2$ | $ 71 $ | 7.1.357911.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.3479.28t10.b.f | $2$ | $ 7^{2} \cdot 71 $ | 28.0.650754790224967946962108515803180182241777.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
| * | 2.3479.28t10.b.e | $2$ | $ 7^{2} \cdot 71 $ | 28.0.650754790224967946962108515803180182241777.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
| * | 2.3479.28t10.b.d | $2$ | $ 7^{2} \cdot 71 $ | 28.0.650754790224967946962108515803180182241777.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
| * | 2.3479.28t10.b.a | $2$ | $ 7^{2} \cdot 71 $ | 28.0.650754790224967946962108515803180182241777.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
| * | 2.3479.28t10.b.c | $2$ | $ 7^{2} \cdot 71 $ | 28.0.650754790224967946962108515803180182241777.1 | $D_{28}$ (as 28T10) | $1$ | $0$ |
| * | 2.3479.28t10.b.b | $2$ | $ 7^{2} \cdot 71 $ | 28.0.650754790224967946962108515803180182241777.1 | $D_{28}$ (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 *.