Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 25*x^26 + 411*x^24 - 3816*x^22 + 25427*x^20 - 105124*x^18 + 315729*x^16 - 623516*x^14 + 888182*x^12 - 737996*x^10 + 406547*x^8 - 33970*x^6 + 2123*x^4 - 53*x^2 + 1)
gp: K = bnfinit(y^28 - 25*y^26 + 411*y^24 - 3816*y^22 + 25427*y^20 - 105124*y^18 + 315729*y^16 - 623516*y^14 + 888182*y^12 - 737996*y^10 + 406547*y^8 - 33970*y^6 + 2123*y^4 - 53*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 25*x^26 + 411*x^24 - 3816*x^22 + 25427*x^20 - 105124*x^18 + 315729*x^16 - 623516*x^14 + 888182*x^12 - 737996*x^10 + 406547*x^8 - 33970*x^6 + 2123*x^4 - 53*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 25*x^26 + 411*x^24 - 3816*x^22 + 25427*x^20 - 105124*x^18 + 315729*x^16 - 623516*x^14 + 888182*x^12 - 737996*x^10 + 406547*x^8 - 33970*x^6 + 2123*x^4 - 53*x^2 + 1)
\( x^{28} - 25 x^{26} + 411 x^{24} - 3816 x^{22} + 25427 x^{20} - 105124 x^{18} + 315729 x^{16} - 623516 x^{14} + \cdots + 1 \)
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: |
\(160727205638753900965518547139872645824181262352384\)
\(\medspace = 2^{28}\cdot 3^{14}\cdot 29^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(62.10\) | 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: | $2\cdot 3^{1/2}29^{6/7}\approx 62.09757221780233$ | ||
| Ramified primes: |
\(2\), \(3\), \(29\)
| 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: | \(348=2^{2}\cdot 3\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{348}(1,·)$, $\chi_{348}(343,·)$, $\chi_{348}(107,·)$, $\chi_{348}(197,·)$, $\chi_{348}(7,·)$, $\chi_{348}(139,·)$, $\chi_{348}(335,·)$, $\chi_{348}(83,·)$, $\chi_{348}(277,·)$, $\chi_{348}(23,·)$, $\chi_{348}(25,·)$, $\chi_{348}(239,·)$, $\chi_{348}(223,·)$, $\chi_{348}(161,·)$, $\chi_{348}(227,·)$, $\chi_{348}(65,·)$, $\chi_{348}(103,·)$, $\chi_{348}(169,·)$, $\chi_{348}(199,·)$, $\chi_{348}(257,·)$, $\chi_{348}(175,·)$, $\chi_{348}(49,·)$, $\chi_{348}(53,·)$, $\chi_{348}(233,·)$, $\chi_{348}(313,·)$, $\chi_{348}(59,·)$, $\chi_{348}(281,·)$, $\chi_{348}(181,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{8192}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{17}a^{20}-\frac{7}{17}a^{18}+\frac{8}{17}a^{16}-\frac{3}{17}a^{14}-\frac{2}{17}a^{12}-\frac{5}{17}a^{10}-\frac{4}{17}a^{8}+\frac{2}{17}a^{6}+\frac{6}{17}a^{4}-\frac{2}{17}a^{2}-\frac{1}{17}$, $\frac{1}{17}a^{21}-\frac{7}{17}a^{19}+\frac{8}{17}a^{17}-\frac{3}{17}a^{15}-\frac{2}{17}a^{13}-\frac{5}{17}a^{11}-\frac{4}{17}a^{9}+\frac{2}{17}a^{7}+\frac{6}{17}a^{5}-\frac{2}{17}a^{3}-\frac{1}{17}a$, $\frac{1}{17}a^{22}-\frac{7}{17}a^{18}+\frac{2}{17}a^{16}-\frac{6}{17}a^{14}-\frac{2}{17}a^{12}-\frac{5}{17}a^{10}+\frac{8}{17}a^{8}+\frac{3}{17}a^{6}+\frac{6}{17}a^{4}+\frac{2}{17}a^{2}-\frac{7}{17}$, $\frac{1}{17}a^{23}-\frac{7}{17}a^{19}+\frac{2}{17}a^{17}-\frac{6}{17}a^{15}-\frac{2}{17}a^{13}-\frac{5}{17}a^{11}+\frac{8}{17}a^{9}+\frac{3}{17}a^{7}+\frac{6}{17}a^{5}+\frac{2}{17}a^{3}-\frac{7}{17}a$, $\frac{1}{11849}a^{24}+\frac{52}{11849}a^{22}-\frac{120}{11849}a^{20}-\frac{47}{11849}a^{18}-\frac{2744}{11849}a^{16}+\frac{3374}{11849}a^{14}-\frac{2484}{11849}a^{12}-\frac{1234}{11849}a^{10}-\frac{4178}{11849}a^{8}-\frac{1679}{11849}a^{6}+\frac{5416}{11849}a^{4}-\frac{44}{697}a^{2}+\frac{5189}{11849}$, $\frac{1}{11849}a^{25}+\frac{52}{11849}a^{23}-\frac{120}{11849}a^{21}-\frac{47}{11849}a^{19}-\frac{2744}{11849}a^{17}+\frac{3374}{11849}a^{15}-\frac{2484}{11849}a^{13}-\frac{1234}{11849}a^{11}-\frac{4178}{11849}a^{9}-\frac{1679}{11849}a^{7}+\frac{5416}{11849}a^{5}-\frac{44}{697}a^{3}+\frac{5189}{11849}a$, $\frac{1}{84\!\cdots\!19}a^{26}-\frac{77\!\cdots\!56}{84\!\cdots\!19}a^{24}-\frac{17\!\cdots\!43}{84\!\cdots\!19}a^{22}+\frac{16\!\cdots\!81}{84\!\cdots\!19}a^{20}-\frac{35\!\cdots\!56}{84\!\cdots\!19}a^{18}+\frac{18\!\cdots\!57}{84\!\cdots\!19}a^{16}-\frac{16\!\cdots\!95}{84\!\cdots\!19}a^{14}+\frac{15\!\cdots\!38}{84\!\cdots\!19}a^{12}-\frac{53\!\cdots\!23}{49\!\cdots\!07}a^{10}+\frac{56\!\cdots\!27}{84\!\cdots\!19}a^{8}-\frac{28\!\cdots\!15}{84\!\cdots\!19}a^{6}-\frac{36\!\cdots\!72}{84\!\cdots\!19}a^{4}+\frac{26\!\cdots\!80}{84\!\cdots\!19}a^{2}-\frac{36\!\cdots\!82}{84\!\cdots\!19}$, $\frac{1}{84\!\cdots\!19}a^{27}-\frac{77\!\cdots\!56}{84\!\cdots\!19}a^{25}-\frac{17\!\cdots\!43}{84\!\cdots\!19}a^{23}+\frac{16\!\cdots\!81}{84\!\cdots\!19}a^{21}-\frac{35\!\cdots\!56}{84\!\cdots\!19}a^{19}+\frac{18\!\cdots\!57}{84\!\cdots\!19}a^{17}-\frac{16\!\cdots\!95}{84\!\cdots\!19}a^{15}+\frac{15\!\cdots\!38}{84\!\cdots\!19}a^{13}-\frac{53\!\cdots\!23}{49\!\cdots\!07}a^{11}+\frac{56\!\cdots\!27}{84\!\cdots\!19}a^{9}-\frac{28\!\cdots\!15}{84\!\cdots\!19}a^{7}-\frac{36\!\cdots\!72}{84\!\cdots\!19}a^{5}+\frac{26\!\cdots\!80}{84\!\cdots\!19}a^{3}-\frac{36\!\cdots\!82}{84\!\cdots\!19}a$
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
$C_{4}\times C_{4}\times C_{28}$, which has order $448$ (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: |
\( -\frac{288590093062887570060361338540}{8488749944424829218280470588419} a^{27} + \frac{7192302348233041285334129907620}{8488749944424829218280470588419} a^{25} - \frac{118049944760298021357460582714080}{8488749944424829218280470588419} a^{23} + \frac{1092049354853136431580153280231920}{8488749944424829218280470588419} a^{21} - \frac{7252581220191819876020875790151593}{8488749944424829218280470588419} a^{19} + \frac{29769390625439443012468847663524410}{8488749944424829218280470588419} a^{17} - \frac{88772612343268179347793001574123310}{8488749944424829218280470588419} a^{15} + \frac{172918802109188044818442110393529645}{8488749944424829218280470588419} a^{13} - \frac{242518112588426636198495167467734970}{8488749944424829218280470588419} a^{11} + \frac{193410159954734900696486829247578240}{8488749944424829218280470588419} a^{9} - \frac{2469852703732143960323866927526123}{207042681571337298006840746059} a^{7} + \frac{1049219412973485518423970884632950}{8488749944424829218280470588419} a^{5} - \frac{26295418437133815605295921542250}{8488749944424829218280470588419} a^{3} - \frac{50965411896753748653089392725391}{8488749944424829218280470588419} a \)
(order $12$)
| 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: |
$\frac{28\!\cdots\!40}{84\!\cdots\!19}a^{27}-\frac{71\!\cdots\!20}{84\!\cdots\!19}a^{25}+\frac{11\!\cdots\!80}{84\!\cdots\!19}a^{23}-\frac{10\!\cdots\!20}{84\!\cdots\!19}a^{21}+\frac{72\!\cdots\!93}{84\!\cdots\!19}a^{19}-\frac{29\!\cdots\!10}{84\!\cdots\!19}a^{17}+\frac{88\!\cdots\!10}{84\!\cdots\!19}a^{15}-\frac{17\!\cdots\!45}{84\!\cdots\!19}a^{13}+\frac{24\!\cdots\!70}{84\!\cdots\!19}a^{11}-\frac{19\!\cdots\!40}{84\!\cdots\!19}a^{9}+\frac{24\!\cdots\!23}{20\!\cdots\!59}a^{7}-\frac{10\!\cdots\!50}{84\!\cdots\!19}a^{5}+\frac{26\!\cdots\!50}{84\!\cdots\!19}a^{3}+\frac{50\!\cdots\!91}{84\!\cdots\!19}a+1$, $\frac{10\!\cdots\!30}{84\!\cdots\!19}a^{26}-\frac{25\!\cdots\!78}{84\!\cdots\!19}a^{24}+\frac{42\!\cdots\!42}{84\!\cdots\!19}a^{22}-\frac{39\!\cdots\!94}{84\!\cdots\!19}a^{20}+\frac{26\!\cdots\!62}{84\!\cdots\!19}a^{18}-\frac{10\!\cdots\!08}{84\!\cdots\!19}a^{16}+\frac{32\!\cdots\!90}{84\!\cdots\!19}a^{14}-\frac{64\!\cdots\!82}{84\!\cdots\!19}a^{12}+\frac{91\!\cdots\!34}{84\!\cdots\!19}a^{10}-\frac{74\!\cdots\!45}{84\!\cdots\!19}a^{8}+\frac{40\!\cdots\!28}{84\!\cdots\!19}a^{6}-\frac{27\!\cdots\!32}{84\!\cdots\!19}a^{4}+\frac{12\!\cdots\!56}{84\!\cdots\!19}a^{2}-\frac{13\!\cdots\!72}{84\!\cdots\!19}$, $\frac{12\!\cdots\!82}{84\!\cdots\!19}a^{26}-\frac{31\!\cdots\!47}{84\!\cdots\!19}a^{24}+\frac{52\!\cdots\!27}{84\!\cdots\!19}a^{22}-\frac{48\!\cdots\!39}{84\!\cdots\!19}a^{20}+\frac{32\!\cdots\!51}{84\!\cdots\!19}a^{18}-\frac{13\!\cdots\!54}{84\!\cdots\!19}a^{16}+\frac{40\!\cdots\!90}{84\!\cdots\!19}a^{14}-\frac{79\!\cdots\!19}{84\!\cdots\!19}a^{12}+\frac{11\!\cdots\!57}{84\!\cdots\!19}a^{10}-\frac{93\!\cdots\!61}{84\!\cdots\!19}a^{8}+\frac{50\!\cdots\!64}{84\!\cdots\!19}a^{6}-\frac{34\!\cdots\!32}{84\!\cdots\!19}a^{4}-\frac{31\!\cdots\!40}{84\!\cdots\!19}a^{2}-\frac{16\!\cdots\!98}{84\!\cdots\!19}$, $\frac{22\!\cdots\!80}{84\!\cdots\!19}a^{26}-\frac{56\!\cdots\!60}{84\!\cdots\!19}a^{24}+\frac{92\!\cdots\!20}{84\!\cdots\!19}a^{22}-\frac{85\!\cdots\!87}{84\!\cdots\!19}a^{20}+\frac{56\!\cdots\!50}{84\!\cdots\!19}a^{18}-\frac{23\!\cdots\!50}{84\!\cdots\!19}a^{16}+\frac{70\!\cdots\!95}{84\!\cdots\!19}a^{14}-\frac{13\!\cdots\!10}{84\!\cdots\!19}a^{12}+\frac{19\!\cdots\!00}{84\!\cdots\!19}a^{10}-\frac{16\!\cdots\!37}{84\!\cdots\!19}a^{8}+\frac{87\!\cdots\!50}{84\!\cdots\!19}a^{6}-\frac{58\!\cdots\!70}{84\!\cdots\!19}a^{4}+\frac{66\!\cdots\!11}{84\!\cdots\!19}a^{2}-\frac{28\!\cdots\!40}{84\!\cdots\!19}$, $\frac{67\!\cdots\!00}{84\!\cdots\!19}a^{26}-\frac{16\!\cdots\!32}{84\!\cdots\!19}a^{24}+\frac{27\!\cdots\!75}{84\!\cdots\!19}a^{22}-\frac{25\!\cdots\!65}{84\!\cdots\!19}a^{20}+\frac{17\!\cdots\!15}{84\!\cdots\!19}a^{18}-\frac{69\!\cdots\!80}{84\!\cdots\!19}a^{16}+\frac{20\!\cdots\!45}{84\!\cdots\!19}a^{14}-\frac{40\!\cdots\!02}{84\!\cdots\!19}a^{12}+\frac{56\!\cdots\!60}{84\!\cdots\!19}a^{10}-\frac{45\!\cdots\!35}{84\!\cdots\!19}a^{8}+\frac{57\!\cdots\!91}{20\!\cdots\!59}a^{6}-\frac{24\!\cdots\!45}{84\!\cdots\!19}a^{4}+\frac{61\!\cdots\!65}{84\!\cdots\!19}a^{2}+\frac{13\!\cdots\!70}{84\!\cdots\!19}$, $\frac{96\!\cdots\!94}{84\!\cdots\!19}a^{26}-\frac{24\!\cdots\!22}{84\!\cdots\!19}a^{24}+\frac{39\!\cdots\!06}{84\!\cdots\!19}a^{22}-\frac{36\!\cdots\!78}{84\!\cdots\!19}a^{20}+\frac{24\!\cdots\!00}{84\!\cdots\!19}a^{18}-\frac{10\!\cdots\!04}{84\!\cdots\!19}a^{16}+\frac{10\!\cdots\!04}{29\!\cdots\!71}a^{14}-\frac{60\!\cdots\!20}{84\!\cdots\!19}a^{12}+\frac{86\!\cdots\!38}{84\!\cdots\!19}a^{10}-\frac{72\!\cdots\!34}{84\!\cdots\!19}a^{8}+\frac{40\!\cdots\!44}{84\!\cdots\!19}a^{6}-\frac{39\!\cdots\!79}{84\!\cdots\!19}a^{4}+\frac{21\!\cdots\!50}{84\!\cdots\!19}a^{2}-\frac{52\!\cdots\!18}{84\!\cdots\!19}$, $\frac{14\!\cdots\!90}{84\!\cdots\!19}a^{26}-\frac{36\!\cdots\!86}{84\!\cdots\!19}a^{24}+\frac{59\!\cdots\!29}{84\!\cdots\!19}a^{22}-\frac{55\!\cdots\!62}{84\!\cdots\!19}a^{20}+\frac{37\!\cdots\!68}{84\!\cdots\!19}a^{18}-\frac{15\!\cdots\!54}{84\!\cdots\!19}a^{16}+\frac{27\!\cdots\!22}{49\!\cdots\!07}a^{14}-\frac{93\!\cdots\!74}{84\!\cdots\!19}a^{12}+\frac{13\!\cdots\!83}{84\!\cdots\!19}a^{10}-\frac{11\!\cdots\!28}{84\!\cdots\!19}a^{8}+\frac{64\!\cdots\!36}{84\!\cdots\!19}a^{6}-\frac{80\!\cdots\!77}{84\!\cdots\!19}a^{4}+\frac{34\!\cdots\!80}{84\!\cdots\!19}a^{2}-\frac{85\!\cdots\!80}{84\!\cdots\!19}$, $\frac{15\!\cdots\!14}{84\!\cdots\!19}a^{27}+\frac{10\!\cdots\!68}{84\!\cdots\!19}a^{26}-\frac{39\!\cdots\!44}{84\!\cdots\!19}a^{25}-\frac{26\!\cdots\!10}{84\!\cdots\!19}a^{24}+\frac{65\!\cdots\!92}{84\!\cdots\!19}a^{23}+\frac{44\!\cdots\!06}{84\!\cdots\!19}a^{22}-\frac{60\!\cdots\!60}{84\!\cdots\!19}a^{21}-\frac{41\!\cdots\!43}{84\!\cdots\!19}a^{20}+\frac{40\!\cdots\!96}{84\!\cdots\!19}a^{19}+\frac{27\!\cdots\!73}{84\!\cdots\!19}a^{18}-\frac{16\!\cdots\!17}{84\!\cdots\!19}a^{17}-\frac{11\!\cdots\!28}{84\!\cdots\!19}a^{16}+\frac{50\!\cdots\!51}{84\!\cdots\!19}a^{15}+\frac{19\!\cdots\!33}{49\!\cdots\!07}a^{14}-\frac{24\!\cdots\!79}{20\!\cdots\!59}a^{13}-\frac{65\!\cdots\!04}{84\!\cdots\!19}a^{12}+\frac{14\!\cdots\!85}{84\!\cdots\!19}a^{11}+\frac{92\!\cdots\!26}{84\!\cdots\!19}a^{10}-\frac{11\!\cdots\!34}{84\!\cdots\!19}a^{9}-\frac{75\!\cdots\!41}{84\!\cdots\!19}a^{8}+\frac{64\!\cdots\!74}{84\!\cdots\!19}a^{7}+\frac{40\!\cdots\!99}{84\!\cdots\!19}a^{6}-\frac{52\!\cdots\!99}{84\!\cdots\!19}a^{5}-\frac{19\!\cdots\!94}{84\!\cdots\!19}a^{4}+\frac{25\!\cdots\!17}{84\!\cdots\!19}a^{3}+\frac{21\!\cdots\!80}{84\!\cdots\!19}a^{2}-\frac{44\!\cdots\!87}{84\!\cdots\!19}a-\frac{52\!\cdots\!46}{84\!\cdots\!19}$, $\frac{10\!\cdots\!94}{84\!\cdots\!19}a^{27}+\frac{10\!\cdots\!10}{84\!\cdots\!19}a^{26}-\frac{25\!\cdots\!51}{84\!\cdots\!19}a^{25}-\frac{27\!\cdots\!54}{84\!\cdots\!19}a^{24}+\frac{41\!\cdots\!11}{84\!\cdots\!19}a^{23}+\frac{45\!\cdots\!37}{84\!\cdots\!19}a^{22}-\frac{38\!\cdots\!91}{84\!\cdots\!19}a^{21}-\frac{41\!\cdots\!58}{84\!\cdots\!19}a^{20}+\frac{25\!\cdots\!79}{84\!\cdots\!19}a^{19}+\frac{27\!\cdots\!60}{84\!\cdots\!19}a^{18}-\frac{10\!\cdots\!11}{84\!\cdots\!19}a^{17}-\frac{11\!\cdots\!81}{84\!\cdots\!19}a^{16}+\frac{31\!\cdots\!24}{84\!\cdots\!19}a^{15}+\frac{20\!\cdots\!26}{49\!\cdots\!07}a^{14}-\frac{15\!\cdots\!28}{20\!\cdots\!59}a^{13}-\frac{68\!\cdots\!30}{84\!\cdots\!19}a^{12}+\frac{88\!\cdots\!27}{84\!\cdots\!19}a^{11}+\frac{98\!\cdots\!05}{84\!\cdots\!19}a^{10}-\frac{73\!\cdots\!37}{84\!\cdots\!19}a^{9}-\frac{82\!\cdots\!44}{84\!\cdots\!19}a^{8}+\frac{40\!\cdots\!36}{84\!\cdots\!19}a^{7}+\frac{45\!\cdots\!96}{84\!\cdots\!19}a^{6}-\frac{31\!\cdots\!12}{84\!\cdots\!19}a^{5}-\frac{42\!\cdots\!76}{84\!\cdots\!19}a^{4}+\frac{16\!\cdots\!88}{84\!\cdots\!19}a^{3}+\frac{23\!\cdots\!24}{84\!\cdots\!19}a^{2}-\frac{28\!\cdots\!58}{84\!\cdots\!19}a-\frac{59\!\cdots\!68}{84\!\cdots\!19}$, $\frac{10\!\cdots\!46}{20\!\cdots\!59}a^{27}-\frac{20\!\cdots\!68}{84\!\cdots\!19}a^{26}-\frac{25\!\cdots\!16}{20\!\cdots\!59}a^{25}+\frac{51\!\cdots\!78}{84\!\cdots\!19}a^{24}+\frac{41\!\cdots\!03}{20\!\cdots\!59}a^{23}-\frac{83\!\cdots\!96}{84\!\cdots\!19}a^{22}-\frac{38\!\cdots\!53}{20\!\cdots\!59}a^{21}+\frac{75\!\cdots\!80}{84\!\cdots\!19}a^{20}+\frac{25\!\cdots\!37}{20\!\cdots\!59}a^{19}-\frac{49\!\cdots\!36}{84\!\cdots\!19}a^{18}-\frac{10\!\cdots\!72}{20\!\cdots\!59}a^{17}+\frac{19\!\cdots\!64}{84\!\cdots\!19}a^{16}+\frac{18\!\cdots\!61}{12\!\cdots\!27}a^{15}-\frac{56\!\cdots\!90}{84\!\cdots\!19}a^{14}-\frac{62\!\cdots\!96}{20\!\cdots\!59}a^{13}+\frac{10\!\cdots\!70}{84\!\cdots\!19}a^{12}+\frac{88\!\cdots\!60}{20\!\cdots\!59}a^{11}-\frac{13\!\cdots\!62}{84\!\cdots\!19}a^{10}-\frac{72\!\cdots\!67}{20\!\cdots\!59}a^{9}+\frac{79\!\cdots\!04}{84\!\cdots\!19}a^{8}+\frac{39\!\cdots\!31}{20\!\cdots\!59}a^{7}-\frac{24\!\cdots\!26}{84\!\cdots\!19}a^{6}-\frac{25\!\cdots\!70}{20\!\cdots\!59}a^{5}-\frac{25\!\cdots\!94}{84\!\cdots\!19}a^{4}+\frac{20\!\cdots\!14}{20\!\cdots\!59}a^{3}+\frac{10\!\cdots\!15}{84\!\cdots\!19}a^{2}-\frac{51\!\cdots\!00}{20\!\cdots\!59}a-\frac{41\!\cdots\!04}{84\!\cdots\!19}$, $\frac{79\!\cdots\!11}{84\!\cdots\!19}a^{27}+\frac{20\!\cdots\!38}{20\!\cdots\!59}a^{26}-\frac{19\!\cdots\!66}{84\!\cdots\!19}a^{25}-\frac{50\!\cdots\!55}{20\!\cdots\!59}a^{24}+\frac{32\!\cdots\!78}{84\!\cdots\!19}a^{23}+\frac{82\!\cdots\!61}{20\!\cdots\!59}a^{22}-\frac{29\!\cdots\!75}{84\!\cdots\!19}a^{21}-\frac{76\!\cdots\!31}{20\!\cdots\!59}a^{20}+\frac{19\!\cdots\!93}{84\!\cdots\!19}a^{19}+\frac{50\!\cdots\!31}{20\!\cdots\!59}a^{18}-\frac{76\!\cdots\!31}{84\!\cdots\!19}a^{17}-\frac{20\!\cdots\!01}{20\!\cdots\!59}a^{16}+\frac{22\!\cdots\!43}{84\!\cdots\!19}a^{15}+\frac{62\!\cdots\!58}{20\!\cdots\!59}a^{14}-\frac{40\!\cdots\!85}{84\!\cdots\!19}a^{13}-\frac{12\!\cdots\!20}{20\!\cdots\!59}a^{12}+\frac{53\!\cdots\!48}{84\!\cdots\!19}a^{11}+\frac{16\!\cdots\!17}{20\!\cdots\!59}a^{10}-\frac{33\!\cdots\!69}{84\!\cdots\!19}a^{9}-\frac{13\!\cdots\!21}{20\!\cdots\!59}a^{8}+\frac{12\!\cdots\!01}{84\!\cdots\!19}a^{7}+\frac{70\!\cdots\!48}{20\!\cdots\!59}a^{6}+\frac{81\!\cdots\!47}{84\!\cdots\!19}a^{5}-\frac{73\!\cdots\!06}{20\!\cdots\!59}a^{4}-\frac{32\!\cdots\!28}{84\!\cdots\!19}a^{3}+\frac{18\!\cdots\!92}{20\!\cdots\!59}a^{2}+\frac{12\!\cdots\!00}{84\!\cdots\!19}a+\frac{26\!\cdots\!96}{20\!\cdots\!59}$, $\frac{13\!\cdots\!52}{84\!\cdots\!19}a^{27}+\frac{36\!\cdots\!02}{84\!\cdots\!19}a^{26}-\frac{34\!\cdots\!64}{84\!\cdots\!19}a^{25}-\frac{91\!\cdots\!37}{84\!\cdots\!19}a^{24}+\frac{56\!\cdots\!52}{84\!\cdots\!19}a^{23}+\frac{15\!\cdots\!94}{84\!\cdots\!19}a^{22}-\frac{51\!\cdots\!50}{84\!\cdots\!19}a^{21}-\frac{14\!\cdots\!65}{84\!\cdots\!19}a^{20}+\frac{34\!\cdots\!99}{84\!\cdots\!19}a^{19}+\frac{97\!\cdots\!76}{84\!\cdots\!19}a^{18}-\frac{14\!\cdots\!37}{84\!\cdots\!19}a^{17}-\frac{41\!\cdots\!05}{84\!\cdots\!19}a^{16}+\frac{42\!\cdots\!66}{84\!\cdots\!19}a^{15}+\frac{12\!\cdots\!10}{84\!\cdots\!19}a^{14}-\frac{82\!\cdots\!34}{84\!\cdots\!19}a^{13}-\frac{26\!\cdots\!63}{84\!\cdots\!19}a^{12}+\frac{11\!\cdots\!62}{84\!\cdots\!19}a^{11}+\frac{40\!\cdots\!20}{84\!\cdots\!19}a^{10}-\frac{93\!\cdots\!03}{84\!\cdots\!19}a^{9}-\frac{38\!\cdots\!91}{84\!\cdots\!19}a^{8}+\frac{49\!\cdots\!86}{84\!\cdots\!19}a^{7}+\frac{24\!\cdots\!84}{84\!\cdots\!19}a^{6}-\frac{14\!\cdots\!25}{84\!\cdots\!19}a^{5}-\frac{64\!\cdots\!16}{84\!\cdots\!19}a^{4}+\frac{12\!\cdots\!54}{84\!\cdots\!19}a^{3}+\frac{27\!\cdots\!29}{84\!\cdots\!19}a^{2}-\frac{11\!\cdots\!87}{84\!\cdots\!19}a-\frac{31\!\cdots\!44}{84\!\cdots\!19}$, $\frac{47\!\cdots\!90}{84\!\cdots\!19}a^{27}+\frac{99\!\cdots\!51}{84\!\cdots\!19}a^{26}-\frac{11\!\cdots\!21}{84\!\cdots\!19}a^{25}-\frac{24\!\cdots\!64}{84\!\cdots\!19}a^{24}+\frac{19\!\cdots\!32}{84\!\cdots\!19}a^{23}+\frac{41\!\cdots\!87}{84\!\cdots\!19}a^{22}-\frac{18\!\cdots\!54}{84\!\cdots\!19}a^{21}-\frac{92\!\cdots\!76}{20\!\cdots\!59}a^{20}+\frac{12\!\cdots\!98}{84\!\cdots\!19}a^{19}+\frac{25\!\cdots\!82}{84\!\cdots\!19}a^{18}-\frac{49\!\cdots\!99}{84\!\cdots\!19}a^{17}-\frac{10\!\cdots\!43}{84\!\cdots\!19}a^{16}+\frac{87\!\cdots\!27}{49\!\cdots\!07}a^{15}+\frac{76\!\cdots\!84}{20\!\cdots\!59}a^{14}-\frac{29\!\cdots\!11}{84\!\cdots\!19}a^{13}-\frac{61\!\cdots\!46}{84\!\cdots\!19}a^{12}+\frac{41\!\cdots\!51}{84\!\cdots\!19}a^{11}+\frac{87\!\cdots\!21}{84\!\cdots\!19}a^{10}-\frac{34\!\cdots\!71}{84\!\cdots\!19}a^{9}-\frac{72\!\cdots\!80}{84\!\cdots\!19}a^{8}+\frac{18\!\cdots\!56}{84\!\cdots\!19}a^{7}+\frac{39\!\cdots\!89}{84\!\cdots\!19}a^{6}-\frac{11\!\cdots\!78}{84\!\cdots\!19}a^{5}-\frac{27\!\cdots\!54}{84\!\cdots\!19}a^{4}+\frac{13\!\cdots\!64}{20\!\cdots\!59}a^{3}+\frac{14\!\cdots\!21}{84\!\cdots\!19}a^{2}-\frac{30\!\cdots\!28}{84\!\cdots\!19}a-\frac{22\!\cdots\!24}{84\!\cdots\!19}$
| 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: | \( 266164993237.31995 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 266164993237.31995 \cdot 448}{12\cdot\sqrt{160727205638753900965518547139872645824181262352384}}\cr\approx \mathstrut & 0.117144300415993 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 - 25*x^26 + 411*x^24 - 3816*x^22 + 25427*x^20 - 105124*x^18 + 315729*x^16 - 623516*x^14 + 888182*x^12 - 737996*x^10 + 406547*x^8 - 33970*x^6 + 2123*x^4 - 53*x^2 + 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 - 25*x^26 + 411*x^24 - 3816*x^22 + 25427*x^20 - 105124*x^18 + 315729*x^16 - 623516*x^14 + 888182*x^12 - 737996*x^10 + 406547*x^8 - 33970*x^6 + 2123*x^4 - 53*x^2 + 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 - 25*x^26 + 411*x^24 - 3816*x^22 + 25427*x^20 - 105124*x^18 + 315729*x^16 - 623516*x^14 + 888182*x^12 - 737996*x^10 + 406547*x^8 - 33970*x^6 + 2123*x^4 - 53*x^2 + 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 - 25*x^26 + 411*x^24 - 3816*x^22 + 25427*x^20 - 105124*x^18 + 315729*x^16 - 623516*x^14 + 888182*x^12 - 737996*x^10 + 406547*x^8 - 33970*x^6 + 2123*x^4 - 53*x^2 + 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 $C_2\times C_{14}$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 7.7.594823321.1, 14.14.12677823379379991227056128.1, 14.0.773792930870360792667.1, 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 | R | ${\href{/padicField/5.14.0.1}{14} }^{2}$ | ${\href{/padicField/7.14.0.1}{14} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{14}$ | ${\href{/padicField/19.14.0.1}{14} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | R | ${\href{/padicField/31.14.0.1}{14} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{14}$ | ${\href{/padicField/43.14.0.1}{14} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{14}$ |
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 $28$ | $2$ | $14$ | $28$ | |||
|
\(3\)
| Deg $28$ | $2$ | $14$ | $14$ | |||
|
\(29\)
| 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 | $x^{14} + 168 x^{13} + 12110 x^{12} + 485856 x^{11} + 11733204 x^{10} + 171095904 x^{9} + 1407877912 x^{8} + 5266970938 x^{7} + 2815760696 x^{6} + 684731964 x^{5} + 107750832 x^{4} + 339857336 x^{3} + 4765729696 x^{2} + 37989914704 x + 129797258121$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ |