Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 197*x^24 + 9834*x^20 + 111451*x^16 + 395946*x^12 + 345566*x^8 + 1437*x^4 + 1)
gp: K = bnfinit(y^28 + 197*y^24 + 9834*y^20 + 111451*y^16 + 395946*y^12 + 345566*y^8 + 1437*y^4 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 + 197*x^24 + 9834*x^20 + 111451*x^16 + 395946*x^12 + 345566*x^8 + 1437*x^4 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 + 197*x^24 + 9834*x^20 + 111451*x^16 + 395946*x^12 + 345566*x^8 + 1437*x^4 + 1)
\( x^{28} + 197x^{24} + 9834x^{20} + 111451x^{16} + 395946x^{12} + 345566x^{8} + 1437x^{4} + 1 \)
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: |
\(9020522762586308771279473372699929575905800975548416\)
\(\medspace = 2^{56}\cdot 29^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(71.70\) | 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: | $2^{2}29^{6/7}\approx 71.7041000719408$ | ||
| Ramified primes: |
\(2\), \(29\)
| 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: | \(232=2^{3}\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{232}(1,·)$, $\chi_{232}(107,·)$, $\chi_{232}(197,·)$, $\chi_{232}(7,·)$, $\chi_{232}(139,·)$, $\chi_{232}(141,·)$, $\chi_{232}(81,·)$, $\chi_{232}(83,·)$, $\chi_{232}(23,·)$, $\chi_{232}(25,·)$, $\chi_{232}(219,·)$, $\chi_{232}(223,·)$, $\chi_{232}(161,·)$, $\chi_{232}(227,·)$, $\chi_{232}(165,·)$, $\chi_{232}(65,·)$, $\chi_{232}(103,·)$, $\chi_{232}(169,·)$, $\chi_{232}(199,·)$, $\chi_{232}(45,·)$, $\chi_{232}(175,·)$, $\chi_{232}(49,·)$, $\chi_{232}(123,·)$, $\chi_{232}(53,·)$, $\chi_{232}(111,·)$, $\chi_{232}(59,·)$, $\chi_{232}(117,·)$, $\chi_{232}(181,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{8192}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17}a^{16}-\frac{1}{17}$, $\frac{1}{17}a^{17}-\frac{1}{17}a$, $\frac{1}{17}a^{18}-\frac{1}{17}a^{2}$, $\frac{1}{17}a^{19}-\frac{1}{17}a^{3}$, $\frac{1}{17}a^{20}-\frac{1}{17}a^{4}$, $\frac{1}{17}a^{21}-\frac{1}{17}a^{5}$, $\frac{1}{17}a^{22}-\frac{1}{17}a^{6}$, $\frac{1}{17}a^{23}-\frac{1}{17}a^{7}$, $\frac{1}{61\!\cdots\!77}a^{24}-\frac{555760637084211}{61\!\cdots\!77}a^{20}-\frac{13\!\cdots\!47}{61\!\cdots\!77}a^{16}+\frac{11\!\cdots\!63}{36\!\cdots\!81}a^{12}-\frac{26\!\cdots\!22}{61\!\cdots\!77}a^{8}+\frac{14\!\cdots\!97}{61\!\cdots\!77}a^{4}-\frac{21\!\cdots\!64}{61\!\cdots\!77}$, $\frac{1}{61\!\cdots\!77}a^{25}-\frac{555760637084211}{61\!\cdots\!77}a^{21}-\frac{13\!\cdots\!47}{61\!\cdots\!77}a^{17}+\frac{11\!\cdots\!63}{36\!\cdots\!81}a^{13}-\frac{26\!\cdots\!22}{61\!\cdots\!77}a^{9}+\frac{14\!\cdots\!97}{61\!\cdots\!77}a^{5}-\frac{21\!\cdots\!64}{61\!\cdots\!77}a$, $\frac{1}{61\!\cdots\!77}a^{26}-\frac{555760637084211}{61\!\cdots\!77}a^{22}-\frac{13\!\cdots\!47}{61\!\cdots\!77}a^{18}+\frac{11\!\cdots\!63}{36\!\cdots\!81}a^{14}-\frac{26\!\cdots\!22}{61\!\cdots\!77}a^{10}+\frac{14\!\cdots\!97}{61\!\cdots\!77}a^{6}-\frac{21\!\cdots\!64}{61\!\cdots\!77}a^{2}$, $\frac{1}{61\!\cdots\!77}a^{27}-\frac{555760637084211}{61\!\cdots\!77}a^{23}-\frac{13\!\cdots\!47}{61\!\cdots\!77}a^{19}+\frac{11\!\cdots\!63}{36\!\cdots\!81}a^{15}-\frac{26\!\cdots\!22}{61\!\cdots\!77}a^{11}+\frac{14\!\cdots\!97}{61\!\cdots\!77}a^{7}-\frac{21\!\cdots\!64}{61\!\cdots\!77}a^{3}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $17$ |
Class group and class number
$C_{4}\times C_{4}\times C_{1204}$, which has order $19264$ (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: |
\( \frac{120147492678582}{61692893418446977} a^{25} + \frac{23669843847044992}{61692893418446977} a^{21} + \frac{1181686024924130293}{61692893418446977} a^{17} + \frac{788139880767185850}{3628993730496881} a^{13} + \frac{47662821025571406002}{61692893418446977} a^{9} + \frac{41841580176291853913}{61692893418446977} a^{5} + \frac{394774270161398293}{61692893418446977} a \)
(order $8$)
| 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: |
$\frac{91\!\cdots\!59}{61\!\cdots\!77}a^{27}+\frac{1729821150483}{88512042207241}a^{26}+\frac{17\!\cdots\!97}{61\!\cdots\!77}a^{23}+\frac{340777446473649}{88512042207241}a^{22}+\frac{89\!\cdots\!45}{61\!\cdots\!77}a^{19}+\frac{17\!\cdots\!63}{88512042207241}a^{18}+\frac{59\!\cdots\!29}{36\!\cdots\!81}a^{15}+\frac{11\!\cdots\!88}{5206590718073}a^{14}+\frac{36\!\cdots\!73}{61\!\cdots\!77}a^{11}+\frac{68\!\cdots\!06}{88512042207241}a^{10}+\frac{31\!\cdots\!76}{61\!\cdots\!77}a^{7}+\frac{59\!\cdots\!54}{88512042207241}a^{6}+\frac{14\!\cdots\!83}{61\!\cdots\!77}a^{3}+\frac{35\!\cdots\!26}{88512042207241}a^{2}-1$, $\frac{142017612666544}{61\!\cdots\!77}a^{24}+\frac{27\!\cdots\!30}{61\!\cdots\!77}a^{20}+\frac{13\!\cdots\!17}{61\!\cdots\!77}a^{16}+\frac{93\!\cdots\!14}{36\!\cdots\!81}a^{12}+\frac{56\!\cdots\!56}{61\!\cdots\!77}a^{8}+\frac{48\!\cdots\!16}{61\!\cdots\!77}a^{4}-\frac{46\!\cdots\!94}{61\!\cdots\!77}$, $\frac{205988713259090}{61\!\cdots\!77}a^{24}+\frac{40\!\cdots\!90}{61\!\cdots\!77}a^{20}+\frac{20\!\cdots\!38}{61\!\cdots\!77}a^{16}+\frac{13\!\cdots\!14}{36\!\cdots\!81}a^{12}+\frac{81\!\cdots\!69}{61\!\cdots\!77}a^{8}+\frac{71\!\cdots\!64}{61\!\cdots\!77}a^{4}+\frac{16\!\cdots\!72}{61\!\cdots\!77}$, $\frac{5244394393974}{61\!\cdots\!77}a^{24}+\frac{10\!\cdots\!39}{61\!\cdots\!77}a^{20}+\frac{51\!\cdots\!84}{61\!\cdots\!77}a^{16}+\frac{34\!\cdots\!27}{36\!\cdots\!81}a^{12}+\frac{20\!\cdots\!82}{61\!\cdots\!77}a^{8}+\frac{17\!\cdots\!00}{61\!\cdots\!77}a^{4}-\frac{91\!\cdots\!59}{61\!\cdots\!77}$, $\frac{5734972621481}{36\!\cdots\!81}a^{24}+\frac{11\!\cdots\!03}{36\!\cdots\!81}a^{20}+\frac{56\!\cdots\!65}{36\!\cdots\!81}a^{16}+\frac{37\!\cdots\!70}{213470219440993}a^{12}+\frac{22\!\cdots\!86}{36\!\cdots\!81}a^{8}+\frac{19\!\cdots\!22}{36\!\cdots\!81}a^{4}+\frac{51\!\cdots\!06}{36\!\cdots\!81}$, $\frac{262165105345126}{61\!\cdots\!77}a^{24}+\frac{51\!\cdots\!22}{61\!\cdots\!77}a^{20}+\frac{25\!\cdots\!10}{61\!\cdots\!77}a^{16}+\frac{17\!\cdots\!64}{36\!\cdots\!81}a^{12}+\frac{10\!\cdots\!58}{61\!\cdots\!77}a^{8}+\frac{90\!\cdots\!29}{61\!\cdots\!77}a^{4}+\frac{28\!\cdots\!22}{61\!\cdots\!77}$, $\frac{108494178693913}{61\!\cdots\!77}a^{24}+\frac{21\!\cdots\!39}{61\!\cdots\!77}a^{20}+\frac{10\!\cdots\!33}{61\!\cdots\!77}a^{16}+\frac{71\!\cdots\!24}{36\!\cdots\!81}a^{12}+\frac{43\!\cdots\!07}{61\!\cdots\!77}a^{8}+\frac{37\!\cdots\!90}{61\!\cdots\!77}a^{4}+\frac{18\!\cdots\!93}{61\!\cdots\!77}$, $\frac{48\!\cdots\!47}{61\!\cdots\!77}a^{27}-\frac{87\!\cdots\!45}{61\!\cdots\!77}a^{26}+\frac{313800418303573}{61\!\cdots\!77}a^{24}+\frac{96\!\cdots\!66}{61\!\cdots\!77}a^{23}-\frac{17\!\cdots\!53}{61\!\cdots\!77}a^{22}+\frac{61\!\cdots\!17}{61\!\cdots\!77}a^{20}+\frac{47\!\cdots\!57}{61\!\cdots\!77}a^{19}-\frac{86\!\cdots\!77}{61\!\cdots\!77}a^{18}+\frac{30\!\cdots\!17}{61\!\cdots\!77}a^{16}+\frac{31\!\cdots\!96}{36\!\cdots\!81}a^{15}-\frac{57\!\cdots\!30}{36\!\cdots\!81}a^{14}+\frac{20\!\cdots\!23}{36\!\cdots\!81}a^{12}+\frac{19\!\cdots\!00}{61\!\cdots\!77}a^{11}-\frac{34\!\cdots\!11}{61\!\cdots\!77}a^{10}+\frac{12\!\cdots\!06}{61\!\cdots\!77}a^{8}+\frac{16\!\cdots\!93}{61\!\cdots\!77}a^{7}-\frac{30\!\cdots\!23}{61\!\cdots\!77}a^{6}+\frac{10\!\cdots\!06}{61\!\cdots\!77}a^{4}+\frac{59\!\cdots\!30}{61\!\cdots\!77}a^{3}-\frac{83\!\cdots\!62}{61\!\cdots\!77}a^{2}+\frac{26\!\cdots\!70}{61\!\cdots\!77}$, $\frac{12\!\cdots\!68}{61\!\cdots\!77}a^{27}+\frac{716576809761432}{61\!\cdots\!77}a^{26}-\frac{142017612666544}{61\!\cdots\!77}a^{24}+\frac{24\!\cdots\!08}{61\!\cdots\!77}a^{23}+\frac{14\!\cdots\!79}{61\!\cdots\!77}a^{22}-\frac{27\!\cdots\!30}{61\!\cdots\!77}a^{20}+\frac{12\!\cdots\!51}{61\!\cdots\!77}a^{19}+\frac{70\!\cdots\!27}{61\!\cdots\!77}a^{18}-\frac{13\!\cdots\!17}{61\!\cdots\!77}a^{16}+\frac{80\!\cdots\!70}{36\!\cdots\!81}a^{15}+\frac{46\!\cdots\!13}{36\!\cdots\!81}a^{14}-\frac{93\!\cdots\!14}{36\!\cdots\!81}a^{12}+\frac{48\!\cdots\!46}{61\!\cdots\!77}a^{11}+\frac{28\!\cdots\!79}{61\!\cdots\!77}a^{10}-\frac{56\!\cdots\!56}{61\!\cdots\!77}a^{8}+\frac{41\!\cdots\!83}{61\!\cdots\!77}a^{7}+\frac{24\!\cdots\!51}{61\!\cdots\!77}a^{6}-\frac{48\!\cdots\!16}{61\!\cdots\!77}a^{4}-\frac{75\!\cdots\!97}{61\!\cdots\!77}a^{3}-\frac{28\!\cdots\!34}{61\!\cdots\!77}a^{2}+\frac{46\!\cdots\!94}{61\!\cdots\!77}$, $\frac{66\!\cdots\!32}{61\!\cdots\!77}a^{27}-\frac{86\!\cdots\!40}{61\!\cdots\!77}a^{26}+\frac{262165105345126}{61\!\cdots\!77}a^{24}+\frac{13\!\cdots\!42}{61\!\cdots\!77}a^{23}-\frac{16\!\cdots\!23}{61\!\cdots\!77}a^{22}+\frac{51\!\cdots\!22}{61\!\cdots\!77}a^{20}+\frac{65\!\cdots\!95}{61\!\cdots\!77}a^{19}-\frac{84\!\cdots\!61}{61\!\cdots\!77}a^{18}+\frac{25\!\cdots\!10}{61\!\cdots\!77}a^{16}+\frac{43\!\cdots\!32}{36\!\cdots\!81}a^{15}-\frac{56\!\cdots\!06}{36\!\cdots\!81}a^{14}+\frac{17\!\cdots\!64}{36\!\cdots\!81}a^{12}+\frac{26\!\cdots\!81}{61\!\cdots\!77}a^{11}-\frac{34\!\cdots\!70}{61\!\cdots\!77}a^{10}+\frac{10\!\cdots\!58}{61\!\cdots\!77}a^{8}+\frac{22\!\cdots\!05}{61\!\cdots\!77}a^{7}-\frac{29\!\cdots\!89}{61\!\cdots\!77}a^{6}+\frac{90\!\cdots\!29}{61\!\cdots\!77}a^{4}+\frac{77\!\cdots\!19}{61\!\cdots\!77}a^{3}-\frac{95\!\cdots\!27}{61\!\cdots\!77}a^{2}+\frac{28\!\cdots\!22}{61\!\cdots\!77}$, $\frac{1729821150483}{88512042207241}a^{27}+\frac{142017612666544}{61\!\cdots\!77}a^{24}+\frac{340777446473649}{88512042207241}a^{23}+\frac{27\!\cdots\!30}{61\!\cdots\!77}a^{20}+\frac{17\!\cdots\!63}{88512042207241}a^{19}+\frac{13\!\cdots\!17}{61\!\cdots\!77}a^{16}+\frac{11\!\cdots\!88}{5206590718073}a^{15}+\frac{93\!\cdots\!14}{36\!\cdots\!81}a^{12}+\frac{68\!\cdots\!06}{88512042207241}a^{11}+\frac{56\!\cdots\!56}{61\!\cdots\!77}a^{8}+\frac{59\!\cdots\!54}{88512042207241}a^{7}+\frac{48\!\cdots\!16}{61\!\cdots\!77}a^{4}+\frac{35\!\cdots\!26}{88512042207241}a^{3}-a+\frac{76\!\cdots\!60}{61\!\cdots\!77}$, $\frac{57\!\cdots\!49}{61\!\cdots\!77}a^{27}+\frac{62\!\cdots\!38}{61\!\cdots\!77}a^{26}+\frac{427920678462738}{61\!\cdots\!77}a^{25}+\frac{11\!\cdots\!96}{61\!\cdots\!77}a^{23}+\frac{12\!\cdots\!17}{61\!\cdots\!77}a^{22}+\frac{84\!\cdots\!67}{61\!\cdots\!77}a^{21}+\frac{56\!\cdots\!14}{61\!\cdots\!77}a^{19}+\frac{61\!\cdots\!39}{61\!\cdots\!77}a^{18}+\frac{42\!\cdots\!62}{61\!\cdots\!77}a^{17}+\frac{37\!\cdots\!47}{36\!\cdots\!81}a^{15}+\frac{40\!\cdots\!34}{36\!\cdots\!81}a^{14}+\frac{28\!\cdots\!50}{36\!\cdots\!81}a^{13}+\frac{22\!\cdots\!83}{61\!\cdots\!77}a^{11}+\frac{24\!\cdots\!06}{61\!\cdots\!77}a^{10}+\frac{16\!\cdots\!40}{61\!\cdots\!77}a^{9}+\frac{19\!\cdots\!67}{61\!\cdots\!77}a^{7}+\frac{21\!\cdots\!13}{61\!\cdots\!77}a^{6}+\frac{14\!\cdots\!83}{61\!\cdots\!77}a^{5}-\frac{11\!\cdots\!13}{61\!\cdots\!77}a^{3}+\frac{46\!\cdots\!83}{61\!\cdots\!77}a^{2}+\frac{43\!\cdots\!98}{61\!\cdots\!77}a$, $\frac{30\!\cdots\!86}{61\!\cdots\!77}a^{27}-\frac{796881717909375}{61\!\cdots\!77}a^{26}+\frac{13\!\cdots\!40}{61\!\cdots\!77}a^{25}+\frac{59\!\cdots\!02}{61\!\cdots\!77}a^{23}-\frac{15\!\cdots\!49}{61\!\cdots\!77}a^{22}+\frac{25\!\cdots\!35}{61\!\cdots\!77}a^{21}+\frac{29\!\cdots\!41}{61\!\cdots\!77}a^{19}-\frac{78\!\cdots\!53}{61\!\cdots\!77}a^{18}+\frac{12\!\cdots\!60}{61\!\cdots\!77}a^{17}+\frac{19\!\cdots\!28}{36\!\cdots\!81}a^{15}-\frac{52\!\cdots\!23}{36\!\cdots\!81}a^{14}+\frac{86\!\cdots\!75}{36\!\cdots\!81}a^{13}+\frac{12\!\cdots\!92}{61\!\cdots\!77}a^{11}-\frac{31\!\cdots\!41}{61\!\cdots\!77}a^{10}+\frac{52\!\cdots\!80}{61\!\cdots\!77}a^{9}+\frac{10\!\cdots\!35}{61\!\cdots\!77}a^{7}-\frac{27\!\cdots\!57}{61\!\cdots\!77}a^{6}+\frac{45\!\cdots\!72}{61\!\cdots\!77}a^{5}+\frac{22\!\cdots\!69}{61\!\cdots\!77}a^{3}-\frac{48\!\cdots\!61}{61\!\cdots\!77}a^{2}+\frac{78\!\cdots\!55}{61\!\cdots\!77}a$
| 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: | \( 297452739458.56445 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 297452739458.56445 \cdot 19264}{8\cdot\sqrt{9020522762586308771279473372699929575905800975548416}}\cr\approx \mathstrut & 1.12713713712242 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 + 197*x^24 + 9834*x^20 + 111451*x^16 + 395946*x^12 + 345566*x^8 + 1437*x^4 + 1)
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 + 197*x^24 + 9834*x^20 + 111451*x^16 + 395946*x^12 + 345566*x^8 + 1437*x^4 + 1, 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 + 197*x^24 + 9834*x^20 + 111451*x^16 + 395946*x^12 + 345566*x^8 + 1437*x^4 + 1);
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 + 197*x^24 + 9834*x^20 + 111451*x^16 + 395946*x^12 + 345566*x^8 + 1437*x^4 + 1);
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
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{8})\), 7.7.594823321.1, 14.14.742003380228915810271232.1, 14.0.742003380228915810271232.1, 14.0.5796901408038404767744.1 |
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 | ${\href{/padicField/3.14.0.1}{14} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }^{2}$ | ${\href{/padicField/17.1.0.1}{1} }^{28}$ | ${\href{/padicField/19.14.0.1}{14} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | R | ${\href{/padicField/31.14.0.1}{14} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{28}$ | ${\href{/padicField/43.14.0.1}{14} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{14}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $28$ | $4$ | $7$ | $56$ | |||
|
\(29\)
| 29.14.12.1 | $x^{14} + 168 x^{13} + 12110 x^{12} + 485856 x^{11} + 11733204 x^{10} + 171095904 x^{9} + 1407877912 x^{8} + 5266970938 x^{7} + 2815760696 x^{6} + 684731964 x^{5} + 107750832 x^{4} + 339857336 x^{3} + 4765729696 x^{2} + 37989914704 x + 129797258121$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ |
| 29.14.12.1 | $x^{14} + 168 x^{13} + 12110 x^{12} + 485856 x^{11} + 11733204 x^{10} + 171095904 x^{9} + 1407877912 x^{8} + 5266970938 x^{7} + 2815760696 x^{6} + 684731964 x^{5} + 107750832 x^{4} + 339857336 x^{3} + 4765729696 x^{2} + 37989914704 x + 129797258121$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ |