Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 3*x^27 - 34*x^26 + 144*x^25 + 434*x^24 - 2347*x^23 - 1826*x^22 + 20510*x^21 - 1765*x^20 - 109438*x^19 + 155561*x^18 + 274375*x^17 - 755172*x^16 + 932048*x^15 - 239571*x^14 - 6022955*x^13 + 13616855*x^12 - 2683804*x^11 - 29709892*x^10 + 50763183*x^9 - 10828500*x^8 + 60669907*x^7 + 337407873*x^6 - 138342771*x^5 - 258957409*x^4 - 71603500*x^3 + 372018822*x^2 + 287060780*x + 118082201)
gp: K = bnfinit(y^28 - 3*y^27 - 34*y^26 + 144*y^25 + 434*y^24 - 2347*y^23 - 1826*y^22 + 20510*y^21 - 1765*y^20 - 109438*y^19 + 155561*y^18 + 274375*y^17 - 755172*y^16 + 932048*y^15 - 239571*y^14 - 6022955*y^13 + 13616855*y^12 - 2683804*y^11 - 29709892*y^10 + 50763183*y^9 - 10828500*y^8 + 60669907*y^7 + 337407873*y^6 - 138342771*y^5 - 258957409*y^4 - 71603500*y^3 + 372018822*y^2 + 287060780*y + 118082201, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 3*x^27 - 34*x^26 + 144*x^25 + 434*x^24 - 2347*x^23 - 1826*x^22 + 20510*x^21 - 1765*x^20 - 109438*x^19 + 155561*x^18 + 274375*x^17 - 755172*x^16 + 932048*x^15 - 239571*x^14 - 6022955*x^13 + 13616855*x^12 - 2683804*x^11 - 29709892*x^10 + 50763183*x^9 - 10828500*x^8 + 60669907*x^7 + 337407873*x^6 - 138342771*x^5 - 258957409*x^4 - 71603500*x^3 + 372018822*x^2 + 287060780*x + 118082201);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 3*x^27 - 34*x^26 + 144*x^25 + 434*x^24 - 2347*x^23 - 1826*x^22 + 20510*x^21 - 1765*x^20 - 109438*x^19 + 155561*x^18 + 274375*x^17 - 755172*x^16 + 932048*x^15 - 239571*x^14 - 6022955*x^13 + 13616855*x^12 - 2683804*x^11 - 29709892*x^10 + 50763183*x^9 - 10828500*x^8 + 60669907*x^7 + 337407873*x^6 - 138342771*x^5 - 258957409*x^4 - 71603500*x^3 + 372018822*x^2 + 287060780*x + 118082201)
\( x^{28} - 3 x^{27} - 34 x^{26} + 144 x^{25} + 434 x^{24} - 2347 x^{23} - 1826 x^{22} + 20510 x^{21} + \cdots + 118082201 \)
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: |
\(30928748566898619200779449873164521915364034557154874272253\)
\(\medspace = 13^{21}\cdot 29^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(122.73\) | 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: | $13^{3/4}29^{6/7}\approx 122.7273939845637$ | ||
| Ramified primes: |
\(13\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $28$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(377=13\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{377}(320,·)$, $\chi_{377}(1,·)$, $\chi_{377}(194,·)$, $\chi_{377}(326,·)$, $\chi_{377}(343,·)$, $\chi_{377}(268,·)$, $\chi_{377}(83,·)$, $\chi_{377}(25,·)$, $\chi_{377}(281,·)$, $\chi_{377}(285,·)$, $\chi_{377}(161,·)$, $\chi_{377}(226,·)$, $\chi_{377}(291,·)$, $\chi_{377}(103,·)$, $\chi_{377}(168,·)$, $\chi_{377}(233,·)$, $\chi_{377}(170,·)$, $\chi_{377}(239,·)$, $\chi_{377}(112,·)$, $\chi_{377}(339,·)$, $\chi_{377}(372,·)$, $\chi_{377}(53,·)$, $\chi_{377}(248,·)$, $\chi_{377}(313,·)$, $\chi_{377}(255,·)$, $\chi_{377}(252,·)$, $\chi_{377}(190,·)$, $\chi_{377}(181,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{8192}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3}a^{21}-\frac{1}{3}a^{19}-\frac{1}{3}a^{15}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{22}-\frac{1}{3}a^{20}-\frac{1}{3}a^{16}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{23}-\frac{1}{3}a^{19}-\frac{1}{3}a^{17}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{51}a^{24}+\frac{1}{17}a^{23}+\frac{5}{51}a^{22}+\frac{4}{51}a^{21}+\frac{5}{17}a^{20}+\frac{5}{51}a^{19}+\frac{8}{51}a^{18}+\frac{8}{17}a^{17}+\frac{7}{17}a^{16}+\frac{5}{51}a^{15}+\frac{25}{51}a^{14}+\frac{6}{17}a^{13}-\frac{11}{51}a^{12}+\frac{25}{51}a^{11}-\frac{14}{51}a^{10}-\frac{5}{17}a^{9}-\frac{25}{51}a^{8}-\frac{7}{17}a^{7}+\frac{4}{51}a^{6}+\frac{25}{51}a^{5}-\frac{1}{51}a^{4}+\frac{4}{51}a^{3}+\frac{1}{3}a^{2}-\frac{2}{17}a-\frac{10}{51}$, $\frac{1}{51}a^{25}-\frac{4}{51}a^{23}+\frac{2}{17}a^{22}+\frac{1}{17}a^{21}-\frac{2}{17}a^{20}-\frac{7}{51}a^{19}-\frac{8}{17}a^{16}+\frac{10}{51}a^{15}-\frac{2}{17}a^{14}-\frac{14}{51}a^{13}+\frac{8}{17}a^{12}+\frac{13}{51}a^{11}-\frac{7}{51}a^{10}+\frac{20}{51}a^{9}+\frac{20}{51}a^{8}-\frac{1}{51}a^{7}-\frac{4}{51}a^{6}+\frac{3}{17}a^{5}-\frac{10}{51}a^{4}-\frac{4}{17}a^{3}-\frac{2}{17}a^{2}-\frac{3}{17}a-\frac{7}{17}$, $\frac{1}{9741}a^{26}-\frac{88}{9741}a^{25}+\frac{4}{573}a^{24}+\frac{149}{9741}a^{23}-\frac{1100}{9741}a^{22}+\frac{40}{3247}a^{21}+\frac{1363}{9741}a^{20}+\frac{1300}{3247}a^{19}+\frac{498}{3247}a^{18}+\frac{4373}{9741}a^{17}+\frac{1319}{3247}a^{16}-\frac{3416}{9741}a^{15}-\frac{1460}{9741}a^{14}+\frac{1321}{3247}a^{13}+\frac{1784}{9741}a^{12}+\frac{919}{3247}a^{11}+\frac{3946}{9741}a^{10}-\frac{3925}{9741}a^{9}+\frac{598}{3247}a^{8}-\frac{29}{573}a^{7}-\frac{4859}{9741}a^{6}-\frac{838}{9741}a^{5}-\frac{3233}{9741}a^{4}-\frac{1994}{9741}a^{3}+\frac{1267}{9741}a^{2}+\frac{4487}{9741}a+\frac{2420}{9741}$, $\frac{1}{62\!\cdots\!91}a^{27}+\frac{28\!\cdots\!21}{20\!\cdots\!97}a^{26}+\frac{11\!\cdots\!52}{20\!\cdots\!97}a^{25}-\frac{39\!\cdots\!80}{62\!\cdots\!91}a^{24}+\frac{27\!\cdots\!77}{20\!\cdots\!97}a^{23}-\frac{10\!\cdots\!38}{62\!\cdots\!91}a^{22}-\frac{64\!\cdots\!39}{62\!\cdots\!91}a^{21}+\frac{92\!\cdots\!85}{62\!\cdots\!91}a^{20}-\frac{22\!\cdots\!19}{36\!\cdots\!23}a^{19}+\frac{17\!\cdots\!89}{62\!\cdots\!91}a^{18}-\frac{21\!\cdots\!17}{20\!\cdots\!97}a^{17}-\frac{16\!\cdots\!65}{62\!\cdots\!91}a^{16}+\frac{21\!\cdots\!46}{62\!\cdots\!91}a^{15}+\frac{14\!\cdots\!01}{62\!\cdots\!91}a^{14}+\frac{65\!\cdots\!29}{20\!\cdots\!97}a^{13}-\frac{14\!\cdots\!41}{62\!\cdots\!91}a^{12}+\frac{87\!\cdots\!46}{62\!\cdots\!91}a^{11}+\frac{11\!\cdots\!63}{62\!\cdots\!91}a^{10}-\frac{13\!\cdots\!69}{62\!\cdots\!91}a^{9}+\frac{52\!\cdots\!30}{62\!\cdots\!91}a^{8}+\frac{14\!\cdots\!30}{62\!\cdots\!91}a^{7}+\frac{50\!\cdots\!84}{20\!\cdots\!97}a^{6}-\frac{27\!\cdots\!14}{62\!\cdots\!91}a^{5}-\frac{26\!\cdots\!75}{62\!\cdots\!91}a^{4}+\frac{19\!\cdots\!42}{20\!\cdots\!97}a^{3}+\frac{17\!\cdots\!53}{62\!\cdots\!91}a^{2}-\frac{31\!\cdots\!04}{62\!\cdots\!91}a-\frac{11\!\cdots\!04}{20\!\cdots\!97}$
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
$C_{2}\times C_{14}\times C_{9422}$, which has order $263816$ (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: |
$\frac{63\!\cdots\!20}{20\!\cdots\!97}a^{27}-\frac{23\!\cdots\!97}{20\!\cdots\!97}a^{26}-\frac{20\!\cdots\!14}{20\!\cdots\!97}a^{25}+\frac{10\!\cdots\!56}{20\!\cdots\!97}a^{24}+\frac{20\!\cdots\!79}{20\!\cdots\!97}a^{23}-\frac{16\!\cdots\!19}{20\!\cdots\!97}a^{22}-\frac{38\!\cdots\!07}{20\!\cdots\!97}a^{21}+\frac{13\!\cdots\!43}{20\!\cdots\!97}a^{20}-\frac{10\!\cdots\!02}{20\!\cdots\!97}a^{19}-\frac{66\!\cdots\!04}{20\!\cdots\!97}a^{18}+\frac{14\!\cdots\!24}{20\!\cdots\!97}a^{17}+\frac{90\!\cdots\!83}{20\!\cdots\!97}a^{16}-\frac{59\!\cdots\!58}{20\!\cdots\!97}a^{15}+\frac{98\!\cdots\!49}{20\!\cdots\!97}a^{14}-\frac{64\!\cdots\!24}{20\!\cdots\!97}a^{13}-\frac{37\!\cdots\!87}{20\!\cdots\!97}a^{12}+\frac{11\!\cdots\!68}{20\!\cdots\!97}a^{11}-\frac{87\!\cdots\!77}{20\!\cdots\!97}a^{10}-\frac{16\!\cdots\!69}{20\!\cdots\!97}a^{9}+\frac{47\!\cdots\!73}{20\!\cdots\!97}a^{8}-\frac{35\!\cdots\!39}{20\!\cdots\!97}a^{7}+\frac{46\!\cdots\!20}{20\!\cdots\!97}a^{6}+\frac{19\!\cdots\!00}{20\!\cdots\!97}a^{5}-\frac{24\!\cdots\!96}{20\!\cdots\!97}a^{4}-\frac{53\!\cdots\!66}{20\!\cdots\!97}a^{3}+\frac{92\!\cdots\!15}{20\!\cdots\!97}a^{2}+\frac{17\!\cdots\!90}{20\!\cdots\!97}a+\frac{34\!\cdots\!45}{20\!\cdots\!97}$, $\frac{30\!\cdots\!88}{12\!\cdots\!41}a^{27}-\frac{10\!\cdots\!31}{12\!\cdots\!41}a^{26}-\frac{10\!\cdots\!23}{12\!\cdots\!41}a^{25}+\frac{46\!\cdots\!03}{12\!\cdots\!41}a^{24}+\frac{11\!\cdots\!10}{12\!\cdots\!41}a^{23}-\frac{73\!\cdots\!24}{12\!\cdots\!41}a^{22}-\frac{14\!\cdots\!59}{63\!\cdots\!51}a^{21}+\frac{61\!\cdots\!54}{12\!\cdots\!41}a^{20}-\frac{27\!\cdots\!60}{12\!\cdots\!41}a^{19}-\frac{30\!\cdots\!58}{12\!\cdots\!41}a^{18}+\frac{58\!\cdots\!50}{12\!\cdots\!41}a^{17}+\frac{54\!\cdots\!77}{12\!\cdots\!41}a^{16}-\frac{24\!\cdots\!67}{12\!\cdots\!41}a^{15}+\frac{39\!\cdots\!57}{12\!\cdots\!41}a^{14}-\frac{22\!\cdots\!51}{12\!\cdots\!41}a^{13}-\frac{17\!\cdots\!66}{12\!\cdots\!41}a^{12}+\frac{46\!\cdots\!60}{12\!\cdots\!41}a^{11}-\frac{26\!\cdots\!07}{12\!\cdots\!41}a^{10}-\frac{73\!\cdots\!61}{12\!\cdots\!41}a^{9}+\frac{18\!\cdots\!62}{12\!\cdots\!41}a^{8}-\frac{10\!\cdots\!56}{12\!\cdots\!41}a^{7}+\frac{21\!\cdots\!25}{12\!\cdots\!41}a^{6}+\frac{98\!\cdots\!17}{12\!\cdots\!41}a^{5}-\frac{54\!\cdots\!36}{12\!\cdots\!41}a^{4}-\frac{38\!\cdots\!02}{12\!\cdots\!41}a^{3}-\frac{22\!\cdots\!17}{12\!\cdots\!41}a^{2}+\frac{22\!\cdots\!95}{12\!\cdots\!41}a-\frac{26\!\cdots\!17}{12\!\cdots\!41}$, $\frac{20\!\cdots\!34}{20\!\cdots\!97}a^{27}-\frac{76\!\cdots\!92}{20\!\cdots\!97}a^{26}-\frac{66\!\cdots\!89}{20\!\cdots\!97}a^{25}+\frac{18\!\cdots\!21}{10\!\cdots\!67}a^{24}+\frac{69\!\cdots\!28}{20\!\cdots\!97}a^{23}-\frac{54\!\cdots\!68}{20\!\cdots\!97}a^{22}-\frac{30\!\cdots\!22}{20\!\cdots\!97}a^{21}+\frac{44\!\cdots\!03}{20\!\cdots\!97}a^{20}-\frac{34\!\cdots\!31}{20\!\cdots\!97}a^{19}-\frac{21\!\cdots\!67}{20\!\cdots\!97}a^{18}+\frac{48\!\cdots\!55}{20\!\cdots\!97}a^{17}+\frac{30\!\cdots\!99}{20\!\cdots\!97}a^{16}-\frac{19\!\cdots\!18}{20\!\cdots\!97}a^{15}+\frac{32\!\cdots\!87}{20\!\cdots\!97}a^{14}-\frac{20\!\cdots\!41}{20\!\cdots\!97}a^{13}-\frac{12\!\cdots\!55}{20\!\cdots\!97}a^{12}+\frac{37\!\cdots\!12}{20\!\cdots\!97}a^{11}-\frac{27\!\cdots\!09}{20\!\cdots\!97}a^{10}-\frac{55\!\cdots\!41}{20\!\cdots\!97}a^{9}+\frac{15\!\cdots\!26}{20\!\cdots\!97}a^{8}-\frac{11\!\cdots\!38}{20\!\cdots\!97}a^{7}+\frac{15\!\cdots\!36}{20\!\cdots\!97}a^{6}+\frac{65\!\cdots\!36}{20\!\cdots\!97}a^{5}-\frac{78\!\cdots\!56}{20\!\cdots\!97}a^{4}-\frac{18\!\cdots\!15}{20\!\cdots\!97}a^{3}+\frac{26\!\cdots\!00}{20\!\cdots\!97}a^{2}+\frac{54\!\cdots\!68}{20\!\cdots\!97}a+\frac{10\!\cdots\!84}{20\!\cdots\!97}$, $\frac{68\!\cdots\!86}{20\!\cdots\!97}a^{27}-\frac{25\!\cdots\!31}{20\!\cdots\!97}a^{26}-\frac{21\!\cdots\!28}{20\!\cdots\!97}a^{25}+\frac{11\!\cdots\!52}{20\!\cdots\!97}a^{24}+\frac{21\!\cdots\!46}{20\!\cdots\!97}a^{23}-\frac{18\!\cdots\!72}{20\!\cdots\!97}a^{22}+\frac{31\!\cdots\!89}{20\!\cdots\!97}a^{21}+\frac{14\!\cdots\!51}{20\!\cdots\!97}a^{20}-\frac{12\!\cdots\!39}{20\!\cdots\!97}a^{19}-\frac{71\!\cdots\!95}{20\!\cdots\!97}a^{18}+\frac{16\!\cdots\!95}{20\!\cdots\!97}a^{17}+\frac{91\!\cdots\!37}{20\!\cdots\!97}a^{16}-\frac{65\!\cdots\!33}{20\!\cdots\!97}a^{15}+\frac{11\!\cdots\!88}{20\!\cdots\!97}a^{14}-\frac{71\!\cdots\!20}{20\!\cdots\!97}a^{13}-\frac{40\!\cdots\!63}{20\!\cdots\!97}a^{12}+\frac{12\!\cdots\!72}{20\!\cdots\!97}a^{11}-\frac{97\!\cdots\!74}{20\!\cdots\!97}a^{10}-\frac{18\!\cdots\!42}{20\!\cdots\!97}a^{9}+\frac{53\!\cdots\!03}{20\!\cdots\!97}a^{8}-\frac{40\!\cdots\!94}{20\!\cdots\!97}a^{7}+\frac{49\!\cdots\!99}{20\!\cdots\!97}a^{6}+\frac{21\!\cdots\!45}{20\!\cdots\!97}a^{5}-\frac{27\!\cdots\!53}{20\!\cdots\!97}a^{4}-\frac{55\!\cdots\!01}{20\!\cdots\!97}a^{3}+\frac{11\!\cdots\!20}{20\!\cdots\!97}a^{2}+\frac{20\!\cdots\!37}{20\!\cdots\!97}a+\frac{13\!\cdots\!25}{20\!\cdots\!97}$, $\frac{12\!\cdots\!10}{20\!\cdots\!97}a^{27}-\frac{47\!\cdots\!01}{20\!\cdots\!97}a^{26}-\frac{40\!\cdots\!51}{20\!\cdots\!97}a^{25}+\frac{21\!\cdots\!57}{20\!\cdots\!97}a^{24}+\frac{40\!\cdots\!55}{20\!\cdots\!97}a^{23}-\frac{33\!\cdots\!83}{20\!\cdots\!97}a^{22}+\frac{41\!\cdots\!55}{20\!\cdots\!97}a^{21}+\frac{27\!\cdots\!76}{20\!\cdots\!97}a^{20}-\frac{22\!\cdots\!21}{20\!\cdots\!97}a^{19}-\frac{13\!\cdots\!13}{20\!\cdots\!97}a^{18}+\frac{30\!\cdots\!69}{20\!\cdots\!97}a^{17}+\frac{17\!\cdots\!11}{20\!\cdots\!97}a^{16}-\frac{12\!\cdots\!52}{20\!\cdots\!97}a^{15}+\frac{20\!\cdots\!68}{20\!\cdots\!97}a^{14}-\frac{13\!\cdots\!77}{20\!\cdots\!97}a^{13}-\frac{74\!\cdots\!28}{20\!\cdots\!97}a^{12}+\frac{23\!\cdots\!20}{20\!\cdots\!97}a^{11}-\frac{17\!\cdots\!32}{20\!\cdots\!97}a^{10}-\frac{34\!\cdots\!74}{20\!\cdots\!97}a^{9}+\frac{97\!\cdots\!22}{20\!\cdots\!97}a^{8}-\frac{74\!\cdots\!81}{20\!\cdots\!97}a^{7}+\frac{92\!\cdots\!94}{20\!\cdots\!97}a^{6}+\frac{39\!\cdots\!56}{20\!\cdots\!97}a^{5}-\frac{51\!\cdots\!37}{20\!\cdots\!97}a^{4}-\frac{10\!\cdots\!33}{20\!\cdots\!97}a^{3}+\frac{20\!\cdots\!24}{20\!\cdots\!97}a^{2}+\frac{37\!\cdots\!12}{20\!\cdots\!97}a+\frac{94\!\cdots\!53}{20\!\cdots\!97}$, $\frac{17\!\cdots\!76}{20\!\cdots\!97}a^{27}-\frac{63\!\cdots\!57}{20\!\cdots\!97}a^{26}-\frac{55\!\cdots\!62}{20\!\cdots\!97}a^{25}+\frac{28\!\cdots\!45}{20\!\cdots\!97}a^{24}+\frac{57\!\cdots\!59}{20\!\cdots\!97}a^{23}-\frac{45\!\cdots\!23}{20\!\cdots\!97}a^{22}-\frac{23\!\cdots\!73}{20\!\cdots\!97}a^{21}+\frac{37\!\cdots\!16}{20\!\cdots\!97}a^{20}-\frac{28\!\cdots\!32}{20\!\cdots\!97}a^{19}-\frac{18\!\cdots\!87}{20\!\cdots\!97}a^{18}+\frac{40\!\cdots\!09}{20\!\cdots\!97}a^{17}+\frac{25\!\cdots\!63}{20\!\cdots\!97}a^{16}-\frac{16\!\cdots\!58}{20\!\cdots\!97}a^{15}+\frac{26\!\cdots\!39}{20\!\cdots\!97}a^{14}-\frac{17\!\cdots\!36}{20\!\cdots\!97}a^{13}-\frac{10\!\cdots\!45}{20\!\cdots\!97}a^{12}+\frac{31\!\cdots\!54}{20\!\cdots\!97}a^{11}-\frac{23\!\cdots\!45}{20\!\cdots\!97}a^{10}-\frac{46\!\cdots\!96}{20\!\cdots\!97}a^{9}+\frac{12\!\cdots\!96}{20\!\cdots\!97}a^{8}-\frac{95\!\cdots\!33}{20\!\cdots\!97}a^{7}+\frac{12\!\cdots\!46}{20\!\cdots\!97}a^{6}+\frac{53\!\cdots\!33}{20\!\cdots\!97}a^{5}-\frac{66\!\cdots\!33}{20\!\cdots\!97}a^{4}-\frac{15\!\cdots\!77}{20\!\cdots\!97}a^{3}+\frac{23\!\cdots\!45}{20\!\cdots\!97}a^{2}+\frac{46\!\cdots\!77}{20\!\cdots\!97}a+\frac{10\!\cdots\!91}{20\!\cdots\!97}$, $\frac{25\!\cdots\!86}{43\!\cdots\!41}a^{27}-\frac{44\!\cdots\!54}{14\!\cdots\!47}a^{26}+\frac{38\!\cdots\!87}{43\!\cdots\!41}a^{25}+\frac{49\!\cdots\!68}{43\!\cdots\!41}a^{24}-\frac{22\!\cdots\!33}{43\!\cdots\!41}a^{23}-\frac{66\!\cdots\!33}{43\!\cdots\!41}a^{22}+\frac{13\!\cdots\!94}{14\!\cdots\!47}a^{21}+\frac{11\!\cdots\!84}{14\!\cdots\!47}a^{20}-\frac{14\!\cdots\!72}{14\!\cdots\!47}a^{19}+\frac{31\!\cdots\!91}{43\!\cdots\!41}a^{18}+\frac{28\!\cdots\!27}{43\!\cdots\!41}a^{17}-\frac{89\!\cdots\!88}{14\!\cdots\!47}a^{16}-\frac{11\!\cdots\!51}{43\!\cdots\!41}a^{15}+\frac{19\!\cdots\!24}{43\!\cdots\!41}a^{14}+\frac{44\!\cdots\!07}{14\!\cdots\!47}a^{13}-\frac{15\!\cdots\!77}{14\!\cdots\!47}a^{12}+\frac{66\!\cdots\!42}{43\!\cdots\!41}a^{11}-\frac{13\!\cdots\!32}{43\!\cdots\!41}a^{10}-\frac{70\!\cdots\!53}{14\!\cdots\!47}a^{9}+\frac{10\!\cdots\!10}{43\!\cdots\!41}a^{8}+\frac{32\!\cdots\!15}{43\!\cdots\!41}a^{7}-\frac{23\!\cdots\!87}{43\!\cdots\!41}a^{6}-\frac{87\!\cdots\!54}{43\!\cdots\!41}a^{5}+\frac{53\!\cdots\!50}{14\!\cdots\!47}a^{4}+\frac{12\!\cdots\!77}{43\!\cdots\!41}a^{3}-\frac{11\!\cdots\!02}{43\!\cdots\!41}a^{2}-\frac{38\!\cdots\!81}{14\!\cdots\!47}a-\frac{13\!\cdots\!93}{43\!\cdots\!41}$, $\frac{14\!\cdots\!40}{62\!\cdots\!91}a^{27}-\frac{17\!\cdots\!27}{20\!\cdots\!97}a^{26}-\frac{44\!\cdots\!84}{62\!\cdots\!91}a^{25}+\frac{78\!\cdots\!43}{20\!\cdots\!97}a^{24}+\frac{15\!\cdots\!94}{20\!\cdots\!97}a^{23}-\frac{12\!\cdots\!60}{20\!\cdots\!97}a^{22}-\frac{56\!\cdots\!19}{20\!\cdots\!97}a^{21}+\frac{10\!\cdots\!93}{20\!\cdots\!97}a^{20}-\frac{13\!\cdots\!43}{36\!\cdots\!23}a^{19}-\frac{49\!\cdots\!26}{20\!\cdots\!97}a^{18}+\frac{32\!\cdots\!29}{62\!\cdots\!91}a^{17}+\frac{67\!\cdots\!94}{20\!\cdots\!97}a^{16}-\frac{25\!\cdots\!55}{12\!\cdots\!41}a^{15}+\frac{72\!\cdots\!50}{20\!\cdots\!97}a^{14}-\frac{14\!\cdots\!64}{62\!\cdots\!91}a^{13}-\frac{83\!\cdots\!59}{62\!\cdots\!91}a^{12}+\frac{85\!\cdots\!44}{20\!\cdots\!97}a^{11}-\frac{18\!\cdots\!02}{62\!\cdots\!91}a^{10}-\frac{12\!\cdots\!61}{20\!\cdots\!97}a^{9}+\frac{10\!\cdots\!79}{62\!\cdots\!91}a^{8}-\frac{78\!\cdots\!78}{62\!\cdots\!91}a^{7}+\frac{10\!\cdots\!33}{62\!\cdots\!91}a^{6}+\frac{43\!\cdots\!87}{62\!\cdots\!91}a^{5}-\frac{53\!\cdots\!60}{62\!\cdots\!91}a^{4}-\frac{12\!\cdots\!42}{62\!\cdots\!91}a^{3}+\frac{18\!\cdots\!06}{62\!\cdots\!91}a^{2}+\frac{12\!\cdots\!87}{20\!\cdots\!97}a+\frac{88\!\cdots\!76}{62\!\cdots\!91}$, $\frac{30\!\cdots\!44}{20\!\cdots\!97}a^{27}-\frac{32\!\cdots\!52}{62\!\cdots\!91}a^{26}-\frac{29\!\cdots\!49}{62\!\cdots\!91}a^{25}+\frac{14\!\cdots\!86}{62\!\cdots\!91}a^{24}+\frac{31\!\cdots\!56}{62\!\cdots\!91}a^{23}-\frac{13\!\cdots\!23}{36\!\cdots\!23}a^{22}-\frac{33\!\cdots\!50}{62\!\cdots\!91}a^{21}+\frac{64\!\cdots\!77}{20\!\cdots\!97}a^{20}-\frac{13\!\cdots\!15}{62\!\cdots\!91}a^{19}-\frac{32\!\cdots\!32}{20\!\cdots\!97}a^{18}+\frac{69\!\cdots\!03}{20\!\cdots\!97}a^{17}+\frac{14\!\cdots\!14}{62\!\cdots\!91}a^{16}-\frac{29\!\cdots\!81}{20\!\cdots\!97}a^{15}+\frac{13\!\cdots\!41}{62\!\cdots\!91}a^{14}-\frac{70\!\cdots\!61}{62\!\cdots\!91}a^{13}-\frac{19\!\cdots\!43}{20\!\cdots\!97}a^{12}+\frac{16\!\cdots\!26}{62\!\cdots\!91}a^{11}-\frac{37\!\cdots\!36}{20\!\cdots\!97}a^{10}-\frac{26\!\cdots\!97}{62\!\cdots\!91}a^{9}+\frac{24\!\cdots\!12}{20\!\cdots\!97}a^{8}-\frac{16\!\cdots\!27}{20\!\cdots\!97}a^{7}+\frac{58\!\cdots\!66}{62\!\cdots\!91}a^{6}+\frac{29\!\cdots\!29}{62\!\cdots\!91}a^{5}-\frac{36\!\cdots\!54}{62\!\cdots\!91}a^{4}-\frac{24\!\cdots\!97}{20\!\cdots\!97}a^{3}+\frac{12\!\cdots\!75}{62\!\cdots\!91}a^{2}+\frac{25\!\cdots\!63}{62\!\cdots\!91}a+\frac{31\!\cdots\!68}{62\!\cdots\!91}$, $\frac{51\!\cdots\!47}{20\!\cdots\!97}a^{27}-\frac{10\!\cdots\!46}{62\!\cdots\!91}a^{26}-\frac{12\!\cdots\!17}{20\!\cdots\!97}a^{25}+\frac{13\!\cdots\!09}{20\!\cdots\!97}a^{24}-\frac{34\!\cdots\!09}{36\!\cdots\!23}a^{23}-\frac{62\!\cdots\!01}{62\!\cdots\!91}a^{22}+\frac{92\!\cdots\!39}{62\!\cdots\!91}a^{21}+\frac{15\!\cdots\!76}{20\!\cdots\!97}a^{20}-\frac{11\!\cdots\!22}{62\!\cdots\!91}a^{19}-\frac{19\!\cdots\!20}{62\!\cdots\!91}a^{18}+\frac{82\!\cdots\!91}{62\!\cdots\!91}a^{17}-\frac{22\!\cdots\!82}{62\!\cdots\!91}a^{16}-\frac{27\!\cdots\!99}{62\!\cdots\!91}a^{15}+\frac{52\!\cdots\!53}{62\!\cdots\!91}a^{14}-\frac{27\!\cdots\!45}{36\!\cdots\!23}a^{13}-\frac{54\!\cdots\!27}{36\!\cdots\!23}a^{12}+\frac{57\!\cdots\!36}{62\!\cdots\!91}a^{11}-\frac{68\!\cdots\!14}{62\!\cdots\!91}a^{10}-\frac{55\!\cdots\!54}{62\!\cdots\!91}a^{9}+\frac{23\!\cdots\!52}{62\!\cdots\!91}a^{8}-\frac{27\!\cdots\!71}{62\!\cdots\!91}a^{7}+\frac{44\!\cdots\!84}{20\!\cdots\!97}a^{6}+\frac{35\!\cdots\!98}{62\!\cdots\!91}a^{5}-\frac{20\!\cdots\!20}{62\!\cdots\!91}a^{4}+\frac{15\!\cdots\!92}{62\!\cdots\!91}a^{3}+\frac{59\!\cdots\!88}{20\!\cdots\!97}a^{2}+\frac{18\!\cdots\!95}{62\!\cdots\!91}a+\frac{61\!\cdots\!89}{62\!\cdots\!91}$, $\frac{51\!\cdots\!35}{20\!\cdots\!97}a^{27}-\frac{58\!\cdots\!15}{62\!\cdots\!91}a^{26}-\frac{49\!\cdots\!18}{62\!\cdots\!91}a^{25}+\frac{26\!\cdots\!06}{62\!\cdots\!91}a^{24}+\frac{50\!\cdots\!41}{62\!\cdots\!91}a^{23}-\frac{41\!\cdots\!70}{62\!\cdots\!91}a^{22}+\frac{13\!\cdots\!27}{62\!\cdots\!91}a^{21}+\frac{11\!\cdots\!51}{20\!\cdots\!97}a^{20}-\frac{27\!\cdots\!27}{62\!\cdots\!91}a^{19}-\frac{54\!\cdots\!24}{20\!\cdots\!97}a^{18}+\frac{37\!\cdots\!79}{62\!\cdots\!91}a^{17}+\frac{21\!\cdots\!47}{62\!\cdots\!91}a^{16}-\frac{49\!\cdots\!74}{20\!\cdots\!97}a^{15}+\frac{24\!\cdots\!94}{62\!\cdots\!91}a^{14}-\frac{53\!\cdots\!61}{20\!\cdots\!97}a^{13}-\frac{30\!\cdots\!60}{20\!\cdots\!97}a^{12}+\frac{95\!\cdots\!54}{20\!\cdots\!97}a^{11}-\frac{72\!\cdots\!11}{20\!\cdots\!97}a^{10}-\frac{24\!\cdots\!95}{36\!\cdots\!23}a^{9}+\frac{11\!\cdots\!87}{62\!\cdots\!91}a^{8}-\frac{89\!\cdots\!93}{62\!\cdots\!91}a^{7}+\frac{37\!\cdots\!13}{20\!\cdots\!97}a^{6}+\frac{15\!\cdots\!09}{20\!\cdots\!97}a^{5}-\frac{20\!\cdots\!75}{20\!\cdots\!97}a^{4}-\frac{43\!\cdots\!91}{20\!\cdots\!97}a^{3}+\frac{77\!\cdots\!83}{20\!\cdots\!97}a^{2}+\frac{43\!\cdots\!00}{62\!\cdots\!91}a+\frac{33\!\cdots\!51}{62\!\cdots\!91}$, $\frac{11\!\cdots\!75}{62\!\cdots\!91}a^{27}-\frac{42\!\cdots\!81}{62\!\cdots\!91}a^{26}-\frac{37\!\cdots\!73}{62\!\cdots\!91}a^{25}+\frac{19\!\cdots\!19}{62\!\cdots\!91}a^{24}+\frac{13\!\cdots\!44}{20\!\cdots\!97}a^{23}-\frac{30\!\cdots\!29}{62\!\cdots\!91}a^{22}-\frac{19\!\cdots\!07}{36\!\cdots\!23}a^{21}+\frac{24\!\cdots\!96}{62\!\cdots\!91}a^{20}-\frac{58\!\cdots\!32}{20\!\cdots\!97}a^{19}-\frac{40\!\cdots\!21}{20\!\cdots\!97}a^{18}+\frac{26\!\cdots\!06}{62\!\cdots\!91}a^{17}+\frac{59\!\cdots\!55}{20\!\cdots\!97}a^{16}-\frac{10\!\cdots\!23}{62\!\cdots\!91}a^{15}+\frac{17\!\cdots\!41}{62\!\cdots\!91}a^{14}-\frac{11\!\cdots\!76}{62\!\cdots\!91}a^{13}-\frac{68\!\cdots\!45}{62\!\cdots\!91}a^{12}+\frac{68\!\cdots\!77}{20\!\cdots\!97}a^{11}-\frac{48\!\cdots\!67}{20\!\cdots\!97}a^{10}-\frac{30\!\cdots\!62}{62\!\cdots\!91}a^{9}+\frac{83\!\cdots\!94}{62\!\cdots\!91}a^{8}-\frac{60\!\cdots\!99}{62\!\cdots\!91}a^{7}+\frac{50\!\cdots\!37}{36\!\cdots\!23}a^{6}+\frac{36\!\cdots\!23}{62\!\cdots\!91}a^{5}-\frac{13\!\cdots\!26}{20\!\cdots\!97}a^{4}-\frac{11\!\cdots\!73}{62\!\cdots\!91}a^{3}+\frac{38\!\cdots\!04}{20\!\cdots\!97}a^{2}+\frac{27\!\cdots\!17}{62\!\cdots\!91}a+\frac{64\!\cdots\!84}{62\!\cdots\!91}$, $\frac{17\!\cdots\!73}{62\!\cdots\!91}a^{27}-\frac{65\!\cdots\!37}{62\!\cdots\!91}a^{26}-\frac{55\!\cdots\!23}{62\!\cdots\!91}a^{25}+\frac{29\!\cdots\!57}{62\!\cdots\!91}a^{24}+\frac{57\!\cdots\!86}{62\!\cdots\!91}a^{23}-\frac{15\!\cdots\!60}{20\!\cdots\!97}a^{22}-\frac{26\!\cdots\!53}{20\!\cdots\!97}a^{21}+\frac{37\!\cdots\!93}{62\!\cdots\!91}a^{20}-\frac{30\!\cdots\!37}{62\!\cdots\!91}a^{19}-\frac{61\!\cdots\!58}{20\!\cdots\!97}a^{18}+\frac{13\!\cdots\!50}{20\!\cdots\!97}a^{17}+\frac{24\!\cdots\!67}{62\!\cdots\!91}a^{16}-\frac{16\!\cdots\!72}{62\!\cdots\!91}a^{15}+\frac{27\!\cdots\!80}{62\!\cdots\!91}a^{14}-\frac{17\!\cdots\!86}{62\!\cdots\!91}a^{13}-\frac{10\!\cdots\!94}{62\!\cdots\!91}a^{12}+\frac{32\!\cdots\!64}{62\!\cdots\!91}a^{11}-\frac{24\!\cdots\!24}{62\!\cdots\!91}a^{10}-\frac{27\!\cdots\!59}{36\!\cdots\!23}a^{9}+\frac{44\!\cdots\!91}{20\!\cdots\!97}a^{8}-\frac{33\!\cdots\!29}{20\!\cdots\!97}a^{7}+\frac{42\!\cdots\!03}{20\!\cdots\!97}a^{6}+\frac{54\!\cdots\!28}{62\!\cdots\!91}a^{5}-\frac{68\!\cdots\!75}{62\!\cdots\!91}a^{4}-\frac{14\!\cdots\!35}{62\!\cdots\!91}a^{3}+\frac{83\!\cdots\!20}{20\!\cdots\!97}a^{2}+\frac{16\!\cdots\!71}{20\!\cdots\!97}a+\frac{69\!\cdots\!17}{62\!\cdots\!91}$
| 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: | \( 1523489837639.8035 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 1523489837639.8035 \cdot 263816}{2\cdot\sqrt{30928748566898619200779449873164521915364034557154874272253}}\cr\approx \mathstrut & 0.170784196264912 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 - 3*x^27 - 34*x^26 + 144*x^25 + 434*x^24 - 2347*x^23 - 1826*x^22 + 20510*x^21 - 1765*x^20 - 109438*x^19 + 155561*x^18 + 274375*x^17 - 755172*x^16 + 932048*x^15 - 239571*x^14 - 6022955*x^13 + 13616855*x^12 - 2683804*x^11 - 29709892*x^10 + 50763183*x^9 - 10828500*x^8 + 60669907*x^7 + 337407873*x^6 - 138342771*x^5 - 258957409*x^4 - 71603500*x^3 + 372018822*x^2 + 287060780*x + 118082201)
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 - 3*x^27 - 34*x^26 + 144*x^25 + 434*x^24 - 2347*x^23 - 1826*x^22 + 20510*x^21 - 1765*x^20 - 109438*x^19 + 155561*x^18 + 274375*x^17 - 755172*x^16 + 932048*x^15 - 239571*x^14 - 6022955*x^13 + 13616855*x^12 - 2683804*x^11 - 29709892*x^10 + 50763183*x^9 - 10828500*x^8 + 60669907*x^7 + 337407873*x^6 - 138342771*x^5 - 258957409*x^4 - 71603500*x^3 + 372018822*x^2 + 287060780*x + 118082201, 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 - 3*x^27 - 34*x^26 + 144*x^25 + 434*x^24 - 2347*x^23 - 1826*x^22 + 20510*x^21 - 1765*x^20 - 109438*x^19 + 155561*x^18 + 274375*x^17 - 755172*x^16 + 932048*x^15 - 239571*x^14 - 6022955*x^13 + 13616855*x^12 - 2683804*x^11 - 29709892*x^10 + 50763183*x^9 - 10828500*x^8 + 60669907*x^7 + 337407873*x^6 - 138342771*x^5 - 258957409*x^4 - 71603500*x^3 + 372018822*x^2 + 287060780*x + 118082201);
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 - 3*x^27 - 34*x^26 + 144*x^25 + 434*x^24 - 2347*x^23 - 1826*x^22 + 20510*x^21 - 1765*x^20 - 109438*x^19 + 155561*x^18 + 274375*x^17 - 755172*x^16 + 932048*x^15 - 239571*x^14 - 6022955*x^13 + 13616855*x^12 - 2683804*x^11 - 29709892*x^10 + 50763183*x^9 - 10828500*x^8 + 60669907*x^7 + 337407873*x^6 - 138342771*x^5 - 258957409*x^4 - 71603500*x^3 + 372018822*x^2 + 287060780*x + 118082201);
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 cyclic group of order 28 |
| The 28 conjugacy class representatives for $C_{28}$ |
| Character table for $C_{28}$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.0.2197.1, 7.7.594823321.1, 14.14.22201352938819688612162197.1 |
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 | $28$ | ${\href{/padicField/3.7.0.1}{7} }^{4}$ | $28$ | $28$ | $28$ | R | ${\href{/padicField/17.2.0.1}{2} }^{14}$ | $28$ | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | R | $28$ | $28$ | ${\href{/padicField/41.4.0.1}{4} }^{7}$ | ${\href{/padicField/43.14.0.1}{14} }^{2}$ | $28$ | ${\href{/padicField/53.7.0.1}{7} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| Deg $28$ | $4$ | $7$ | $21$ | |||
|
\(29\)
| 29.7.6.2 | $x^{7} + 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 29.7.6.2 | $x^{7} + 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ | |
| 29.7.6.2 | $x^{7} + 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ | |
| 29.7.6.2 | $x^{7} + 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |