Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 14*x^24 + 406*x^22 + 637*x^20 + 1736*x^18 + 25613*x^16 + 45672*x^14 + 146510*x^12 + 216314*x^10 + 175987*x^8 + 135485*x^6 + 138712*x^4 + 36281*x^2 + 6241)
gp: K = bnfinit(y^28 + 14*y^24 + 406*y^22 + 637*y^20 + 1736*y^18 + 25613*y^16 + 45672*y^14 + 146510*y^12 + 216314*y^10 + 175987*y^8 + 135485*y^6 + 138712*y^4 + 36281*y^2 + 6241, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 + 14*x^24 + 406*x^22 + 637*x^20 + 1736*x^18 + 25613*x^16 + 45672*x^14 + 146510*x^12 + 216314*x^10 + 175987*x^8 + 135485*x^6 + 138712*x^4 + 36281*x^2 + 6241);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 + 14*x^24 + 406*x^22 + 637*x^20 + 1736*x^18 + 25613*x^16 + 45672*x^14 + 146510*x^12 + 216314*x^10 + 175987*x^8 + 135485*x^6 + 138712*x^4 + 36281*x^2 + 6241)
\( x^{28} + 14 x^{24} + 406 x^{22} + 637 x^{20} + 1736 x^{18} + 25613 x^{16} + 45672 x^{14} + 146510 x^{12} + \cdots + 6241 \)
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: |
\(482771783823117526797953812068649122826362283884544\)
\(\medspace = 2^{28}\cdot 7^{50}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(64.59\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2\cdot 7^{25/14}\approx 64.58529365319154$ | ||
| Ramified primes: |
\(2\), \(7\)
| 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: | \(196=2^{2}\cdot 7^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{196}(1,·)$, $\chi_{196}(195,·)$, $\chi_{196}(69,·)$, $\chi_{196}(71,·)$, $\chi_{196}(139,·)$, $\chi_{196}(13,·)$, $\chi_{196}(15,·)$, $\chi_{196}(141,·)$, $\chi_{196}(83,·)$, $\chi_{196}(85,·)$, $\chi_{196}(153,·)$, $\chi_{196}(155,·)$, $\chi_{196}(29,·)$, $\chi_{196}(97,·)$, $\chi_{196}(27,·)$, $\chi_{196}(167,·)$, $\chi_{196}(41,·)$, $\chi_{196}(43,·)$, $\chi_{196}(125,·)$, $\chi_{196}(111,·)$, $\chi_{196}(99,·)$, $\chi_{196}(113,·)$, $\chi_{196}(181,·)$, $\chi_{196}(55,·)$, $\chi_{196}(57,·)$, $\chi_{196}(183,·)$, $\chi_{196}(169,·)$, $\chi_{196}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{8192}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{91991}a^{24}+\frac{3060}{91991}a^{22}-\frac{34560}{91991}a^{20}+\frac{34418}{91991}a^{18}-\frac{19428}{91991}a^{16}+\frac{14963}{91991}a^{14}+\frac{32934}{91991}a^{12}+\frac{17885}{91991}a^{10}+\frac{35419}{91991}a^{8}+\frac{30248}{91991}a^{6}+\frac{24092}{91991}a^{4}+\frac{36749}{91991}a^{2}+\frac{37627}{91991}$, $\frac{1}{7267289}a^{25}-\frac{1008841}{7267289}a^{23}+\frac{2909152}{7267289}a^{21}-\frac{241555}{7267289}a^{19}-\frac{387392}{7267289}a^{17}-\frac{353001}{7267289}a^{15}-\frac{2174850}{7267289}a^{13}+\frac{3329561}{7267289}a^{11}-\frac{700509}{7267289}a^{9}-\frac{3189437}{7267289}a^{7}+\frac{760020}{7267289}a^{5}-\frac{1987053}{7267289}a^{3}-\frac{2906085}{7267289}a$, $\frac{1}{26\!\cdots\!79}a^{26}+\frac{47\!\cdots\!51}{26\!\cdots\!79}a^{24}-\frac{74\!\cdots\!44}{26\!\cdots\!79}a^{22}+\frac{85\!\cdots\!96}{26\!\cdots\!79}a^{20}+\frac{11\!\cdots\!61}{26\!\cdots\!79}a^{18}+\frac{39\!\cdots\!11}{26\!\cdots\!79}a^{16}+\frac{41\!\cdots\!95}{26\!\cdots\!79}a^{14}-\frac{15\!\cdots\!97}{26\!\cdots\!79}a^{12}+\frac{64\!\cdots\!08}{26\!\cdots\!79}a^{10}-\frac{44\!\cdots\!15}{26\!\cdots\!79}a^{8}+\frac{91\!\cdots\!07}{26\!\cdots\!79}a^{6}-\frac{10\!\cdots\!88}{26\!\cdots\!79}a^{4}+\frac{42\!\cdots\!96}{26\!\cdots\!79}a^{2}+\frac{11\!\cdots\!05}{33\!\cdots\!01}$, $\frac{1}{26\!\cdots\!79}a^{27}-\frac{64\!\cdots\!92}{26\!\cdots\!79}a^{25}-\frac{12\!\cdots\!39}{26\!\cdots\!79}a^{23}+\frac{30\!\cdots\!55}{26\!\cdots\!79}a^{21}-\frac{39\!\cdots\!53}{26\!\cdots\!79}a^{19}-\frac{77\!\cdots\!12}{26\!\cdots\!79}a^{17}-\frac{56\!\cdots\!41}{26\!\cdots\!79}a^{15}-\frac{44\!\cdots\!63}{26\!\cdots\!79}a^{13}+\frac{76\!\cdots\!59}{26\!\cdots\!79}a^{11}+\frac{23\!\cdots\!93}{26\!\cdots\!79}a^{9}+\frac{12\!\cdots\!24}{26\!\cdots\!79}a^{7}+\frac{70\!\cdots\!10}{26\!\cdots\!79}a^{5}-\frac{11\!\cdots\!41}{26\!\cdots\!79}a^{3}-\frac{11\!\cdots\!24}{26\!\cdots\!79}a$
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
$C_{71}$, which has order $71$ (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: |
\( \frac{1694604370159091923560}{48166563384628033225653779} a^{27} + \frac{1303749658382307238053}{48166563384628033225653779} a^{25} + \frac{23091480306941228567945}{48166563384628033225653779} a^{23} + \frac{706348121813264898059271}{48166563384628033225653779} a^{21} + \frac{1599630340084744788309013}{48166563384628033225653779} a^{19} + \frac{3516446537302038602278212}{48166563384628033225653779} a^{17} + \frac{45295872275776396519772716}{48166563384628033225653779} a^{15} + \frac{1636185412934435323346617}{718903931113851242173937} a^{13} + \frac{291545865208818568790686328}{48166563384628033225653779} a^{11} + \frac{530706564397956963731430086}{48166563384628033225653779} a^{9} + \frac{484268213751188869019094535}{48166563384628033225653779} a^{7} + \frac{319918831668409030699320007}{48166563384628033225653779} a^{5} + \frac{300713830725918321539129708}{48166563384628033225653779} a^{3} + \frac{131750254101061389231655791}{48166563384628033225653779} a \)
(order $4$)
| 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: |
$\frac{63\!\cdots\!68}{27\!\cdots\!07}a^{26}-\frac{63\!\cdots\!87}{27\!\cdots\!07}a^{24}+\frac{96\!\cdots\!48}{27\!\cdots\!07}a^{22}+\frac{24\!\cdots\!50}{27\!\cdots\!07}a^{20}+\frac{15\!\cdots\!73}{27\!\cdots\!07}a^{18}+\frac{99\!\cdots\!43}{27\!\cdots\!07}a^{16}+\frac{15\!\cdots\!32}{27\!\cdots\!07}a^{14}+\frac{13\!\cdots\!15}{27\!\cdots\!07}a^{12}+\frac{82\!\cdots\!72}{27\!\cdots\!07}a^{10}+\frac{71\!\cdots\!18}{27\!\cdots\!07}a^{8}+\frac{63\!\cdots\!30}{27\!\cdots\!07}a^{6}+\frac{84\!\cdots\!16}{27\!\cdots\!07}a^{4}+\frac{22\!\cdots\!95}{27\!\cdots\!07}a^{2}-\frac{10\!\cdots\!72}{34\!\cdots\!33}$, $\frac{14\!\cdots\!22}{27\!\cdots\!07}a^{26}-\frac{24\!\cdots\!01}{27\!\cdots\!07}a^{24}+\frac{21\!\cdots\!50}{27\!\cdots\!07}a^{22}+\frac{56\!\cdots\!98}{27\!\cdots\!07}a^{20}-\frac{49\!\cdots\!71}{27\!\cdots\!07}a^{18}+\frac{12\!\cdots\!80}{27\!\cdots\!07}a^{16}+\frac{33\!\cdots\!71}{27\!\cdots\!07}a^{14}+\frac{53\!\cdots\!49}{27\!\cdots\!07}a^{12}+\frac{11\!\cdots\!28}{27\!\cdots\!07}a^{10}-\frac{15\!\cdots\!43}{27\!\cdots\!07}a^{8}-\frac{18\!\cdots\!22}{27\!\cdots\!07}a^{6}-\frac{11\!\cdots\!27}{27\!\cdots\!07}a^{4}-\frac{27\!\cdots\!77}{27\!\cdots\!07}a^{2}-\frac{28\!\cdots\!75}{34\!\cdots\!33}$, $\frac{16\!\cdots\!21}{27\!\cdots\!07}a^{26}-\frac{26\!\cdots\!19}{27\!\cdots\!07}a^{24}+\frac{24\!\cdots\!59}{27\!\cdots\!07}a^{22}+\frac{63\!\cdots\!26}{27\!\cdots\!07}a^{20}-\frac{28\!\cdots\!53}{27\!\cdots\!07}a^{18}+\frac{16\!\cdots\!38}{27\!\cdots\!07}a^{16}+\frac{38\!\cdots\!97}{27\!\cdots\!07}a^{14}+\frac{97\!\cdots\!21}{27\!\cdots\!07}a^{12}+\frac{15\!\cdots\!76}{27\!\cdots\!07}a^{10}+\frac{27\!\cdots\!36}{27\!\cdots\!07}a^{8}-\frac{12\!\cdots\!48}{27\!\cdots\!07}a^{6}-\frac{48\!\cdots\!47}{27\!\cdots\!07}a^{4}-\frac{10\!\cdots\!14}{27\!\cdots\!07}a^{2}-\frac{26\!\cdots\!18}{34\!\cdots\!33}$, $\frac{62\!\cdots\!70}{27\!\cdots\!07}a^{26}-\frac{96\!\cdots\!02}{27\!\cdots\!07}a^{24}+\frac{92\!\cdots\!94}{27\!\cdots\!07}a^{22}+\frac{23\!\cdots\!74}{27\!\cdots\!07}a^{20}+\frac{97\!\cdots\!34}{27\!\cdots\!07}a^{18}+\frac{67\!\cdots\!83}{27\!\cdots\!07}a^{16}+\frac{14\!\cdots\!31}{27\!\cdots\!07}a^{14}+\frac{43\!\cdots\!28}{27\!\cdots\!07}a^{12}+\frac{59\!\cdots\!90}{27\!\cdots\!07}a^{10}+\frac{20\!\cdots\!19}{27\!\cdots\!07}a^{8}-\frac{29\!\cdots\!58}{27\!\cdots\!07}a^{6}-\frac{63\!\cdots\!85}{27\!\cdots\!07}a^{4}+\frac{10\!\cdots\!78}{40\!\cdots\!21}a^{2}+\frac{29\!\cdots\!17}{34\!\cdots\!33}$, $\frac{68\!\cdots\!50}{27\!\cdots\!07}a^{26}-\frac{94\!\cdots\!13}{27\!\cdots\!07}a^{24}+\frac{10\!\cdots\!34}{27\!\cdots\!07}a^{22}+\frac{26\!\cdots\!06}{27\!\cdots\!07}a^{20}+\frac{57\!\cdots\!70}{27\!\cdots\!07}a^{18}+\frac{76\!\cdots\!51}{27\!\cdots\!07}a^{16}+\frac{16\!\cdots\!93}{27\!\cdots\!07}a^{14}+\frac{76\!\cdots\!05}{27\!\cdots\!07}a^{12}+\frac{68\!\cdots\!87}{27\!\cdots\!07}a^{10}+\frac{25\!\cdots\!47}{27\!\cdots\!07}a^{8}-\frac{27\!\cdots\!84}{27\!\cdots\!07}a^{6}+\frac{35\!\cdots\!62}{27\!\cdots\!07}a^{4}+\frac{18\!\cdots\!91}{27\!\cdots\!07}a^{2}-\frac{61\!\cdots\!25}{34\!\cdots\!33}$, $\frac{22\!\cdots\!45}{27\!\cdots\!07}a^{26}-\frac{33\!\cdots\!61}{27\!\cdots\!07}a^{24}+\frac{32\!\cdots\!77}{27\!\cdots\!07}a^{22}+\frac{85\!\cdots\!33}{27\!\cdots\!07}a^{20}+\frac{61\!\cdots\!13}{27\!\cdots\!07}a^{18}+\frac{21\!\cdots\!18}{27\!\cdots\!07}a^{16}+\frac{51\!\cdots\!52}{27\!\cdots\!07}a^{14}+\frac{16\!\cdots\!48}{27\!\cdots\!07}a^{12}+\frac{20\!\cdots\!10}{27\!\cdots\!07}a^{10}+\frac{28\!\cdots\!83}{27\!\cdots\!07}a^{8}-\frac{19\!\cdots\!88}{27\!\cdots\!07}a^{6}-\frac{83\!\cdots\!24}{27\!\cdots\!07}a^{4}-\frac{27\!\cdots\!23}{40\!\cdots\!21}a^{2}-\frac{33\!\cdots\!59}{34\!\cdots\!33}$, $\frac{40\!\cdots\!77}{26\!\cdots\!79}a^{27}+\frac{11\!\cdots\!95}{26\!\cdots\!79}a^{26}-\frac{69\!\cdots\!32}{26\!\cdots\!79}a^{25}-\frac{48\!\cdots\!86}{26\!\cdots\!79}a^{24}+\frac{65\!\cdots\!90}{26\!\cdots\!79}a^{23}+\frac{16\!\cdots\!87}{26\!\cdots\!79}a^{22}+\frac{15\!\cdots\!30}{26\!\cdots\!79}a^{21}+\frac{47\!\cdots\!86}{26\!\cdots\!79}a^{20}-\frac{10\!\cdots\!79}{26\!\cdots\!79}a^{19}+\frac{54\!\cdots\!41}{26\!\cdots\!79}a^{18}+\frac{59\!\cdots\!74}{26\!\cdots\!79}a^{17}+\frac{16\!\cdots\!19}{26\!\cdots\!79}a^{16}+\frac{96\!\cdots\!63}{26\!\cdots\!79}a^{15}+\frac{29\!\cdots\!17}{26\!\cdots\!79}a^{14}+\frac{22\!\cdots\!95}{26\!\cdots\!79}a^{13}+\frac{40\!\cdots\!82}{26\!\cdots\!79}a^{12}+\frac{48\!\cdots\!52}{26\!\cdots\!79}a^{11}+\frac{14\!\cdots\!21}{26\!\cdots\!79}a^{10}+\frac{21\!\cdots\!15}{26\!\cdots\!79}a^{9}+\frac{16\!\cdots\!24}{26\!\cdots\!79}a^{8}+\frac{40\!\cdots\!04}{26\!\cdots\!79}a^{7}+\frac{58\!\cdots\!39}{26\!\cdots\!79}a^{6}+\frac{91\!\cdots\!67}{26\!\cdots\!79}a^{5}+\frac{23\!\cdots\!58}{26\!\cdots\!79}a^{4}+\frac{90\!\cdots\!59}{26\!\cdots\!79}a^{3}+\frac{37\!\cdots\!82}{26\!\cdots\!79}a^{2}+\frac{99\!\cdots\!75}{26\!\cdots\!79}a+\frac{61\!\cdots\!13}{33\!\cdots\!01}$, $\frac{63\!\cdots\!19}{26\!\cdots\!79}a^{27}+\frac{62\!\cdots\!93}{26\!\cdots\!79}a^{26}-\frac{15\!\cdots\!04}{26\!\cdots\!79}a^{25}+\frac{32\!\cdots\!29}{26\!\cdots\!79}a^{24}+\frac{87\!\cdots\!72}{26\!\cdots\!79}a^{23}+\frac{83\!\cdots\!59}{26\!\cdots\!79}a^{22}+\frac{23\!\cdots\!22}{26\!\cdots\!79}a^{21}+\frac{26\!\cdots\!70}{26\!\cdots\!79}a^{20}-\frac{24\!\cdots\!99}{26\!\cdots\!79}a^{19}+\frac{52\!\cdots\!74}{26\!\cdots\!79}a^{18}+\frac{59\!\cdots\!42}{26\!\cdots\!79}a^{17}+\frac{11\!\cdots\!14}{26\!\cdots\!79}a^{16}+\frac{13\!\cdots\!96}{26\!\cdots\!79}a^{15}+\frac{16\!\cdots\!86}{26\!\cdots\!79}a^{14}-\frac{11\!\cdots\!19}{26\!\cdots\!79}a^{13}+\frac{36\!\cdots\!25}{26\!\cdots\!79}a^{12}+\frac{18\!\cdots\!31}{26\!\cdots\!79}a^{11}+\frac{97\!\cdots\!31}{26\!\cdots\!79}a^{10}-\frac{11\!\cdots\!22}{33\!\cdots\!01}a^{9}+\frac{17\!\cdots\!19}{26\!\cdots\!79}a^{8}-\frac{22\!\cdots\!54}{26\!\cdots\!79}a^{7}+\frac{14\!\cdots\!00}{26\!\cdots\!79}a^{6}-\frac{15\!\cdots\!24}{26\!\cdots\!79}a^{5}+\frac{11\!\cdots\!58}{26\!\cdots\!79}a^{4}-\frac{45\!\cdots\!59}{26\!\cdots\!79}a^{3}+\frac{16\!\cdots\!40}{26\!\cdots\!79}a^{2}-\frac{12\!\cdots\!52}{26\!\cdots\!79}a+\frac{10\!\cdots\!38}{33\!\cdots\!01}$, $\frac{13\!\cdots\!10}{26\!\cdots\!79}a^{27}-\frac{57\!\cdots\!98}{26\!\cdots\!79}a^{26}+\frac{18\!\cdots\!07}{26\!\cdots\!79}a^{25}+\frac{61\!\cdots\!66}{26\!\cdots\!79}a^{24}+\frac{19\!\cdots\!20}{26\!\cdots\!79}a^{23}-\frac{79\!\cdots\!62}{26\!\cdots\!79}a^{22}+\frac{71\!\cdots\!16}{33\!\cdots\!01}a^{21}-\frac{22\!\cdots\!60}{26\!\cdots\!79}a^{20}+\frac{95\!\cdots\!37}{26\!\cdots\!79}a^{19}-\frac{11\!\cdots\!60}{26\!\cdots\!79}a^{18}+\frac{24\!\cdots\!78}{26\!\cdots\!79}a^{17}-\frac{59\!\cdots\!32}{26\!\cdots\!79}a^{16}+\frac{35\!\cdots\!24}{26\!\cdots\!79}a^{15}-\frac{13\!\cdots\!56}{26\!\cdots\!79}a^{14}+\frac{67\!\cdots\!71}{26\!\cdots\!79}a^{13}-\frac{10\!\cdots\!30}{26\!\cdots\!79}a^{12}+\frac{20\!\cdots\!57}{26\!\cdots\!79}a^{11}-\frac{55\!\cdots\!42}{26\!\cdots\!79}a^{10}+\frac{32\!\cdots\!49}{26\!\cdots\!79}a^{9}-\frac{36\!\cdots\!91}{26\!\cdots\!79}a^{8}+\frac{28\!\cdots\!70}{26\!\cdots\!79}a^{7}+\frac{37\!\cdots\!00}{26\!\cdots\!79}a^{6}+\frac{21\!\cdots\!44}{26\!\cdots\!79}a^{5}+\frac{20\!\cdots\!76}{26\!\cdots\!79}a^{4}+\frac{23\!\cdots\!62}{26\!\cdots\!79}a^{3}-\frac{10\!\cdots\!96}{26\!\cdots\!79}a^{2}+\frac{93\!\cdots\!14}{26\!\cdots\!79}a+\frac{39\!\cdots\!97}{33\!\cdots\!01}$, $\frac{19\!\cdots\!55}{26\!\cdots\!79}a^{27}-\frac{15\!\cdots\!92}{26\!\cdots\!79}a^{26}-\frac{11\!\cdots\!12}{26\!\cdots\!79}a^{25}+\frac{30\!\cdots\!76}{26\!\cdots\!79}a^{24}+\frac{34\!\cdots\!40}{33\!\cdots\!01}a^{23}-\frac{21\!\cdots\!60}{26\!\cdots\!79}a^{22}+\frac{76\!\cdots\!51}{26\!\cdots\!79}a^{21}-\frac{62\!\cdots\!70}{26\!\cdots\!79}a^{20}+\frac{78\!\cdots\!02}{26\!\cdots\!79}a^{19}-\frac{85\!\cdots\!86}{26\!\cdots\!79}a^{18}+\frac{29\!\cdots\!04}{26\!\cdots\!79}a^{17}-\frac{24\!\cdots\!05}{26\!\cdots\!79}a^{16}+\frac{47\!\cdots\!20}{26\!\cdots\!79}a^{15}-\frac{39\!\cdots\!86}{26\!\cdots\!79}a^{14}+\frac{60\!\cdots\!96}{26\!\cdots\!79}a^{13}-\frac{62\!\cdots\!90}{26\!\cdots\!79}a^{12}+\frac{24\!\cdots\!06}{26\!\cdots\!79}a^{11}-\frac{21\!\cdots\!54}{26\!\cdots\!79}a^{10}+\frac{27\!\cdots\!36}{26\!\cdots\!79}a^{9}-\frac{29\!\cdots\!48}{26\!\cdots\!79}a^{8}+\frac{19\!\cdots\!19}{26\!\cdots\!79}a^{7}-\frac{20\!\cdots\!90}{26\!\cdots\!79}a^{6}+\frac{14\!\cdots\!30}{26\!\cdots\!79}a^{5}-\frac{18\!\cdots\!00}{26\!\cdots\!79}a^{4}+\frac{17\!\cdots\!89}{26\!\cdots\!79}a^{3}-\frac{15\!\cdots\!42}{26\!\cdots\!79}a^{2}-\frac{15\!\cdots\!28}{26\!\cdots\!79}a-\frac{45\!\cdots\!43}{33\!\cdots\!01}$, $\frac{16\!\cdots\!60}{48\!\cdots\!79}a^{27}+\frac{16\!\cdots\!70}{26\!\cdots\!79}a^{26}+\frac{13\!\cdots\!53}{48\!\cdots\!79}a^{25}+\frac{41\!\cdots\!33}{26\!\cdots\!79}a^{24}+\frac{23\!\cdots\!45}{48\!\cdots\!79}a^{23}+\frac{20\!\cdots\!90}{26\!\cdots\!79}a^{22}+\frac{70\!\cdots\!71}{48\!\cdots\!79}a^{21}+\frac{73\!\cdots\!26}{26\!\cdots\!79}a^{20}+\frac{15\!\cdots\!13}{48\!\cdots\!79}a^{19}+\frac{27\!\cdots\!63}{26\!\cdots\!79}a^{18}+\frac{35\!\cdots\!12}{48\!\cdots\!79}a^{17}+\frac{46\!\cdots\!82}{26\!\cdots\!79}a^{16}+\frac{45\!\cdots\!16}{48\!\cdots\!79}a^{15}+\frac{48\!\cdots\!36}{26\!\cdots\!79}a^{14}+\frac{16\!\cdots\!17}{71\!\cdots\!37}a^{13}+\frac{17\!\cdots\!69}{26\!\cdots\!79}a^{12}+\frac{29\!\cdots\!28}{48\!\cdots\!79}a^{11}+\frac{37\!\cdots\!33}{26\!\cdots\!79}a^{10}+\frac{53\!\cdots\!86}{48\!\cdots\!79}a^{9}+\frac{88\!\cdots\!41}{26\!\cdots\!79}a^{8}+\frac{48\!\cdots\!35}{48\!\cdots\!79}a^{7}+\frac{84\!\cdots\!00}{26\!\cdots\!79}a^{6}+\frac{31\!\cdots\!07}{48\!\cdots\!79}a^{5}+\frac{45\!\cdots\!90}{26\!\cdots\!79}a^{4}+\frac{30\!\cdots\!08}{48\!\cdots\!79}a^{3}+\frac{40\!\cdots\!11}{26\!\cdots\!79}a^{2}+\frac{17\!\cdots\!70}{48\!\cdots\!79}a+\frac{55\!\cdots\!65}{33\!\cdots\!01}$, $\frac{65\!\cdots\!11}{26\!\cdots\!79}a^{27}+\frac{28\!\cdots\!73}{26\!\cdots\!79}a^{26}-\frac{18\!\cdots\!66}{26\!\cdots\!79}a^{25}-\frac{78\!\cdots\!68}{26\!\cdots\!79}a^{24}+\frac{91\!\cdots\!17}{26\!\cdots\!79}a^{23}+\frac{39\!\cdots\!21}{26\!\cdots\!79}a^{22}+\frac{26\!\cdots\!79}{26\!\cdots\!79}a^{21}+\frac{11\!\cdots\!34}{26\!\cdots\!79}a^{20}+\frac{61\!\cdots\!89}{39\!\cdots\!37}a^{19}+\frac{14\!\cdots\!72}{26\!\cdots\!79}a^{18}+\frac{11\!\cdots\!02}{26\!\cdots\!79}a^{17}+\frac{42\!\cdots\!47}{26\!\cdots\!79}a^{16}+\frac{16\!\cdots\!70}{26\!\cdots\!79}a^{15}+\frac{70\!\cdots\!67}{26\!\cdots\!79}a^{14}+\frac{29\!\cdots\!97}{26\!\cdots\!79}a^{13}+\frac{10\!\cdots\!89}{26\!\cdots\!79}a^{12}+\frac{94\!\cdots\!54}{26\!\cdots\!79}a^{11}+\frac{36\!\cdots\!85}{26\!\cdots\!79}a^{10}+\frac{13\!\cdots\!01}{26\!\cdots\!79}a^{9}+\frac{48\!\cdots\!24}{26\!\cdots\!79}a^{8}+\frac{10\!\cdots\!39}{26\!\cdots\!79}a^{7}+\frac{29\!\cdots\!30}{26\!\cdots\!79}a^{6}+\frac{79\!\cdots\!24}{26\!\cdots\!79}a^{5}+\frac{21\!\cdots\!72}{26\!\cdots\!79}a^{4}+\frac{88\!\cdots\!67}{26\!\cdots\!79}a^{3}+\frac{29\!\cdots\!88}{26\!\cdots\!79}a^{2}+\frac{18\!\cdots\!39}{26\!\cdots\!79}a-\frac{89\!\cdots\!67}{33\!\cdots\!01}$, $\frac{71\!\cdots\!98}{36\!\cdots\!59}a^{27}-\frac{22\!\cdots\!98}{33\!\cdots\!09}a^{26}+\frac{26\!\cdots\!63}{36\!\cdots\!59}a^{25}+\frac{30\!\cdots\!50}{33\!\cdots\!09}a^{24}+\frac{98\!\cdots\!83}{36\!\cdots\!59}a^{23}-\frac{31\!\cdots\!09}{33\!\cdots\!09}a^{22}+\frac{28\!\cdots\!09}{36\!\cdots\!59}a^{21}-\frac{90\!\cdots\!40}{33\!\cdots\!09}a^{20}+\frac{46\!\cdots\!77}{36\!\cdots\!59}a^{19}-\frac{12\!\cdots\!91}{33\!\cdots\!09}a^{18}+\frac{12\!\cdots\!74}{36\!\cdots\!59}a^{17}-\frac{36\!\cdots\!50}{33\!\cdots\!09}a^{16}+\frac{18\!\cdots\!25}{36\!\cdots\!59}a^{15}-\frac{56\!\cdots\!94}{33\!\cdots\!09}a^{14}+\frac{49\!\cdots\!21}{54\!\cdots\!77}a^{13}-\frac{93\!\cdots\!38}{33\!\cdots\!09}a^{12}+\frac{10\!\cdots\!72}{36\!\cdots\!59}a^{11}-\frac{30\!\cdots\!87}{33\!\cdots\!09}a^{10}+\frac{19\!\cdots\!16}{45\!\cdots\!21}a^{9}-\frac{44\!\cdots\!49}{33\!\cdots\!09}a^{8}+\frac{11\!\cdots\!55}{36\!\cdots\!59}a^{7}-\frac{31\!\cdots\!87}{33\!\cdots\!09}a^{6}+\frac{81\!\cdots\!89}{36\!\cdots\!59}a^{5}-\frac{28\!\cdots\!71}{33\!\cdots\!09}a^{4}+\frac{89\!\cdots\!01}{36\!\cdots\!59}a^{3}-\frac{29\!\cdots\!72}{33\!\cdots\!09}a^{2}+\frac{18\!\cdots\!28}{36\!\cdots\!59}a+\frac{47\!\cdots\!48}{33\!\cdots\!09}$
| 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: | \( 983258026102.4221 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 983258026102.4221 \cdot 71}{4\cdot\sqrt{482771783823117526797953812068649122826362283884544}}\cr\approx \mathstrut & 0.118717044151947 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 + 14*x^24 + 406*x^22 + 637*x^20 + 1736*x^18 + 25613*x^16 + 45672*x^14 + 146510*x^12 + 216314*x^10 + 175987*x^8 + 135485*x^6 + 138712*x^4 + 36281*x^2 + 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 + 14*x^24 + 406*x^22 + 637*x^20 + 1736*x^18 + 25613*x^16 + 45672*x^14 + 146510*x^12 + 216314*x^10 + 175987*x^8 + 135485*x^6 + 138712*x^4 + 36281*x^2 + 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 + 14*x^24 + 406*x^22 + 637*x^20 + 1736*x^18 + 25613*x^16 + 45672*x^14 + 146510*x^12 + 216314*x^10 + 175987*x^8 + 135485*x^6 + 138712*x^4 + 36281*x^2 + 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 + 14*x^24 + 406*x^22 + 637*x^20 + 1736*x^18 + 25613*x^16 + 45672*x^14 + 146510*x^12 + 216314*x^10 + 175987*x^8 + 135485*x^6 + 138712*x^4 + 36281*x^2 + 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
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(i, \sqrt{7})\), 7.7.13841287201.1, 14.0.1341068619663964900807.1, 14.0.3138866894939200133545984.1, 14.14.21972068264574400934821888.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.14.0.1}{14} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }^{2}$ | R | ${\href{/padicField/11.14.0.1}{14} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{14}$ | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{14}$ | ${\href{/padicField/37.7.0.1}{7} }^{4}$ | ${\href{/padicField/41.14.0.1}{14} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }^{4}$ | ${\href{/padicField/59.14.0.1}{14} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.14.14.38 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ |
| 2.14.14.38 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ | |
|
\(7\)
| Deg $28$ | $14$ | $2$ | $50$ |