Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 43*x^18 - 74*x^17 - 105*x^16 + 1038*x^15 - 3587*x^14 + 9035*x^13 - 17355*x^12 + 24617*x^11 - 15587*x^10 - 28001*x^9 + 145716*x^8 - 311207*x^7 + 514366*x^6 - 473357*x^5 + 360299*x^4 + 378181*x^3 - 556830*x^2 + 950849*x - 397561)
gp: K = bnfinit(y^20 - 10*y^19 + 43*y^18 - 74*y^17 - 105*y^16 + 1038*y^15 - 3587*y^14 + 9035*y^13 - 17355*y^12 + 24617*y^11 - 15587*y^10 - 28001*y^9 + 145716*y^8 - 311207*y^7 + 514366*y^6 - 473357*y^5 + 360299*y^4 + 378181*y^3 - 556830*y^2 + 950849*y - 397561, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 + 43*x^18 - 74*x^17 - 105*x^16 + 1038*x^15 - 3587*x^14 + 9035*x^13 - 17355*x^12 + 24617*x^11 - 15587*x^10 - 28001*x^9 + 145716*x^8 - 311207*x^7 + 514366*x^6 - 473357*x^5 + 360299*x^4 + 378181*x^3 - 556830*x^2 + 950849*x - 397561);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 10*x^19 + 43*x^18 - 74*x^17 - 105*x^16 + 1038*x^15 - 3587*x^14 + 9035*x^13 - 17355*x^12 + 24617*x^11 - 15587*x^10 - 28001*x^9 + 145716*x^8 - 311207*x^7 + 514366*x^6 - 473357*x^5 + 360299*x^4 + 378181*x^3 - 556830*x^2 + 950849*x - 397561)
\( x^{20} - 10 x^{19} + 43 x^{18} - 74 x^{17} - 105 x^{16} + 1038 x^{15} - 3587 x^{14} + 9035 x^{13} + \cdots - 397561 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1969799822613689762148183291015625\)
\(\medspace = 5^{10}\cdot 61^{6}\cdot 397^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(46.21\) | 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: | $5^{1/2}61^{3/4}397^{3/4}\approx 4340.854794080872$ | ||
| Ramified primes: |
\(5\), \(61\), \(397\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{13}+\frac{1}{9}a^{12}-\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{2}{9}a^{8}-\frac{1}{3}a^{7}+\frac{2}{9}a^{6}-\frac{1}{9}a^{5}-\frac{1}{3}a^{4}+\frac{4}{9}a^{3}+\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{9}a^{15}-\frac{2}{9}a^{11}-\frac{1}{9}a^{9}+\frac{4}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{4}{9}a^{5}+\frac{1}{9}a^{4}+\frac{4}{9}a^{3}+\frac{1}{9}a^{2}-\frac{1}{9}a-\frac{2}{9}$, $\frac{1}{351}a^{16}-\frac{14}{351}a^{15}+\frac{16}{351}a^{14}+\frac{47}{351}a^{13}+\frac{41}{351}a^{12}+\frac{1}{9}a^{11}+\frac{28}{351}a^{10}+\frac{34}{351}a^{9}+\frac{145}{351}a^{8}+\frac{46}{117}a^{7}-\frac{157}{351}a^{6}-\frac{58}{351}a^{5}-\frac{22}{351}a^{4}+\frac{4}{39}a^{3}+\frac{40}{117}a^{2}+\frac{109}{351}a-\frac{67}{351}$, $\frac{1}{351}a^{17}+\frac{5}{117}a^{15}-\frac{2}{351}a^{14}+\frac{4}{39}a^{13}-\frac{11}{351}a^{12}-\frac{11}{351}a^{11}+\frac{38}{117}a^{10}+\frac{4}{39}a^{9}+\frac{101}{351}a^{8}+\frac{59}{351}a^{7}+\frac{28}{117}a^{6}-\frac{2}{13}a^{5}+\frac{40}{351}a^{4}-\frac{4}{9}a^{3}-\frac{122}{351}a^{2}+\frac{172}{351}a-\frac{80}{351}$, $\frac{1}{1053}a^{18}-\frac{1}{1053}a^{16}-\frac{4}{351}a^{15}+\frac{53}{1053}a^{14}-\frac{100}{1053}a^{13}-\frac{160}{1053}a^{12}+\frac{17}{117}a^{11}+\frac{17}{1053}a^{10}+\frac{64}{1053}a^{9}+\frac{118}{1053}a^{8}-\frac{5}{39}a^{7}+\frac{313}{1053}a^{6}-\frac{241}{1053}a^{5}-\frac{38}{1053}a^{4}-\frac{425}{1053}a^{3}-\frac{461}{1053}a^{2}-\frac{49}{351}a+\frac{409}{1053}$, $\frac{1}{12\!\cdots\!11}a^{19}-\frac{42\!\cdots\!66}{12\!\cdots\!11}a^{18}+\frac{84\!\cdots\!95}{12\!\cdots\!11}a^{17}-\frac{13\!\cdots\!79}{12\!\cdots\!11}a^{16}-\frac{42\!\cdots\!37}{12\!\cdots\!11}a^{15}-\frac{10\!\cdots\!73}{12\!\cdots\!11}a^{14}+\frac{94\!\cdots\!06}{12\!\cdots\!11}a^{13}-\frac{89\!\cdots\!58}{12\!\cdots\!11}a^{12}+\frac{11\!\cdots\!03}{12\!\cdots\!11}a^{11}+\frac{46\!\cdots\!99}{15\!\cdots\!31}a^{10}-\frac{33\!\cdots\!07}{12\!\cdots\!11}a^{9}+\frac{11\!\cdots\!48}{12\!\cdots\!11}a^{8}+\frac{45\!\cdots\!82}{12\!\cdots\!11}a^{7}+\frac{65\!\cdots\!71}{15\!\cdots\!31}a^{6}-\frac{11\!\cdots\!81}{46\!\cdots\!93}a^{5}+\frac{56\!\cdots\!27}{12\!\cdots\!11}a^{4}+\frac{53\!\cdots\!13}{12\!\cdots\!11}a^{3}-\frac{62\!\cdots\!78}{12\!\cdots\!11}a^{2}-\frac{44\!\cdots\!37}{12\!\cdots\!11}a-\frac{25\!\cdots\!18}{12\!\cdots\!11}$
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
$C_{2}$, which has order $2$ (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: | $11$ | 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: |
$\frac{36\!\cdots\!24}{11\!\cdots\!87}a^{19}-\frac{34\!\cdots\!66}{11\!\cdots\!87}a^{18}+\frac{13\!\cdots\!78}{11\!\cdots\!87}a^{17}-\frac{17\!\cdots\!81}{11\!\cdots\!87}a^{16}-\frac{55\!\cdots\!44}{11\!\cdots\!87}a^{15}+\frac{35\!\cdots\!14}{11\!\cdots\!87}a^{14}-\frac{10\!\cdots\!06}{11\!\cdots\!87}a^{13}+\frac{25\!\cdots\!94}{11\!\cdots\!87}a^{12}-\frac{43\!\cdots\!50}{11\!\cdots\!87}a^{11}+\frac{50\!\cdots\!56}{11\!\cdots\!87}a^{10}-\frac{12\!\cdots\!82}{11\!\cdots\!87}a^{9}-\frac{14\!\cdots\!13}{11\!\cdots\!87}a^{8}+\frac{48\!\cdots\!82}{11\!\cdots\!87}a^{7}-\frac{85\!\cdots\!14}{11\!\cdots\!87}a^{6}+\frac{12\!\cdots\!52}{11\!\cdots\!87}a^{5}-\frac{69\!\cdots\!27}{11\!\cdots\!87}a^{4}+\frac{13\!\cdots\!16}{11\!\cdots\!87}a^{3}+\frac{22\!\cdots\!21}{11\!\cdots\!87}a^{2}-\frac{17\!\cdots\!75}{11\!\cdots\!87}a+\frac{22\!\cdots\!69}{11\!\cdots\!87}$, $\frac{37\!\cdots\!96}{97\!\cdots\!17}a^{19}-\frac{31\!\cdots\!92}{97\!\cdots\!17}a^{18}+\frac{10\!\cdots\!54}{97\!\cdots\!17}a^{17}-\frac{52\!\cdots\!37}{97\!\cdots\!17}a^{16}-\frac{53\!\cdots\!86}{74\!\cdots\!09}a^{15}+\frac{99\!\cdots\!53}{32\!\cdots\!39}a^{14}-\frac{24\!\cdots\!43}{32\!\cdots\!39}a^{13}+\frac{15\!\cdots\!97}{97\!\cdots\!17}a^{12}-\frac{23\!\cdots\!87}{97\!\cdots\!17}a^{11}+\frac{20\!\cdots\!09}{97\!\cdots\!17}a^{10}+\frac{92\!\cdots\!45}{32\!\cdots\!39}a^{9}-\frac{11\!\cdots\!06}{97\!\cdots\!17}a^{8}+\frac{32\!\cdots\!95}{97\!\cdots\!17}a^{7}-\frac{44\!\cdots\!99}{97\!\cdots\!17}a^{6}+\frac{59\!\cdots\!77}{97\!\cdots\!17}a^{5}+\frac{54\!\cdots\!28}{32\!\cdots\!39}a^{4}+\frac{35\!\cdots\!25}{32\!\cdots\!39}a^{3}+\frac{19\!\cdots\!07}{97\!\cdots\!17}a^{2}-\frac{20\!\cdots\!34}{97\!\cdots\!17}a+\frac{14\!\cdots\!28}{97\!\cdots\!17}$, $\frac{25\!\cdots\!96}{97\!\cdots\!17}a^{19}-\frac{13\!\cdots\!61}{97\!\cdots\!17}a^{18}-\frac{30\!\cdots\!61}{97\!\cdots\!17}a^{17}+\frac{25\!\cdots\!77}{97\!\cdots\!17}a^{16}-\frac{61\!\cdots\!22}{74\!\cdots\!09}a^{15}+\frac{16\!\cdots\!17}{32\!\cdots\!39}a^{14}+\frac{35\!\cdots\!83}{10\!\cdots\!13}a^{13}-\frac{11\!\cdots\!47}{97\!\cdots\!17}a^{12}+\frac{29\!\cdots\!67}{97\!\cdots\!17}a^{11}-\frac{60\!\cdots\!20}{97\!\cdots\!17}a^{10}+\frac{29\!\cdots\!46}{32\!\cdots\!39}a^{9}-\frac{17\!\cdots\!45}{97\!\cdots\!17}a^{8}-\frac{13\!\cdots\!43}{97\!\cdots\!17}a^{7}+\frac{58\!\cdots\!80}{97\!\cdots\!17}a^{6}-\frac{10\!\cdots\!80}{97\!\cdots\!17}a^{5}+\frac{56\!\cdots\!04}{32\!\cdots\!39}a^{4}-\frac{42\!\cdots\!66}{32\!\cdots\!39}a^{3}+\frac{13\!\cdots\!29}{97\!\cdots\!17}a^{2}+\frac{28\!\cdots\!96}{97\!\cdots\!17}a-\frac{20\!\cdots\!38}{97\!\cdots\!17}$, $\frac{11\!\cdots\!00}{97\!\cdots\!17}a^{19}-\frac{17\!\cdots\!31}{97\!\cdots\!17}a^{18}+\frac{10\!\cdots\!15}{97\!\cdots\!17}a^{17}-\frac{31\!\cdots\!14}{97\!\cdots\!17}a^{16}+\frac{81\!\cdots\!36}{74\!\cdots\!09}a^{15}+\frac{27\!\cdots\!12}{10\!\cdots\!13}a^{14}-\frac{35\!\cdots\!92}{32\!\cdots\!39}a^{13}+\frac{26\!\cdots\!44}{97\!\cdots\!17}a^{12}-\frac{52\!\cdots\!54}{97\!\cdots\!17}a^{11}+\frac{80\!\cdots\!29}{97\!\cdots\!17}a^{10}-\frac{67\!\cdots\!67}{10\!\cdots\!13}a^{9}-\frac{97\!\cdots\!61}{97\!\cdots\!17}a^{8}+\frac{46\!\cdots\!38}{97\!\cdots\!17}a^{7}-\frac{10\!\cdots\!79}{97\!\cdots\!17}a^{6}+\frac{16\!\cdots\!57}{97\!\cdots\!17}a^{5}-\frac{17\!\cdots\!92}{10\!\cdots\!13}a^{4}+\frac{26\!\cdots\!97}{10\!\cdots\!13}a^{3}+\frac{17\!\cdots\!78}{97\!\cdots\!17}a^{2}-\frac{30\!\cdots\!30}{97\!\cdots\!17}a+\frac{34\!\cdots\!66}{97\!\cdots\!17}$, $\frac{59\!\cdots\!34}{97\!\cdots\!17}a^{19}-\frac{47\!\cdots\!95}{97\!\cdots\!17}a^{18}+\frac{12\!\cdots\!67}{97\!\cdots\!17}a^{17}+\frac{95\!\cdots\!70}{97\!\cdots\!17}a^{16}-\frac{10\!\cdots\!22}{74\!\cdots\!09}a^{15}+\frac{46\!\cdots\!59}{10\!\cdots\!13}a^{14}-\frac{24\!\cdots\!91}{32\!\cdots\!39}a^{13}+\frac{11\!\cdots\!35}{97\!\cdots\!17}a^{12}-\frac{66\!\cdots\!37}{97\!\cdots\!17}a^{11}-\frac{24\!\cdots\!09}{97\!\cdots\!17}a^{10}+\frac{11\!\cdots\!53}{10\!\cdots\!13}a^{9}-\frac{17\!\cdots\!33}{97\!\cdots\!17}a^{8}+\frac{30\!\cdots\!39}{97\!\cdots\!17}a^{7}-\frac{97\!\cdots\!92}{97\!\cdots\!17}a^{6}-\frac{23\!\cdots\!90}{97\!\cdots\!17}a^{5}+\frac{16\!\cdots\!16}{10\!\cdots\!13}a^{4}-\frac{41\!\cdots\!99}{10\!\cdots\!13}a^{3}+\frac{16\!\cdots\!82}{97\!\cdots\!17}a^{2}+\frac{19\!\cdots\!33}{97\!\cdots\!17}a-\frac{21\!\cdots\!16}{97\!\cdots\!17}$, $\frac{54\!\cdots\!04}{12\!\cdots\!11}a^{19}-\frac{18\!\cdots\!35}{12\!\cdots\!11}a^{18}+\frac{14\!\cdots\!97}{12\!\cdots\!11}a^{17}-\frac{51\!\cdots\!84}{12\!\cdots\!11}a^{16}+\frac{35\!\cdots\!62}{96\!\cdots\!47}a^{15}+\frac{10\!\cdots\!03}{41\!\cdots\!37}a^{14}-\frac{17\!\cdots\!91}{13\!\cdots\!79}a^{13}+\frac{40\!\cdots\!83}{12\!\cdots\!11}a^{12}-\frac{83\!\cdots\!11}{12\!\cdots\!11}a^{11}+\frac{13\!\cdots\!28}{12\!\cdots\!11}a^{10}-\frac{38\!\cdots\!00}{41\!\cdots\!37}a^{9}-\frac{12\!\cdots\!54}{12\!\cdots\!11}a^{8}+\frac{68\!\cdots\!15}{12\!\cdots\!11}a^{7}-\frac{16\!\cdots\!82}{12\!\cdots\!11}a^{6}+\frac{25\!\cdots\!96}{12\!\cdots\!11}a^{5}-\frac{97\!\cdots\!09}{41\!\cdots\!37}a^{4}+\frac{10\!\cdots\!94}{41\!\cdots\!37}a^{3}+\frac{21\!\cdots\!06}{12\!\cdots\!11}a^{2}-\frac{55\!\cdots\!43}{12\!\cdots\!11}a+\frac{36\!\cdots\!00}{12\!\cdots\!11}$, $\frac{19\!\cdots\!94}{41\!\cdots\!37}a^{19}-\frac{84\!\cdots\!04}{15\!\cdots\!31}a^{18}+\frac{32\!\cdots\!10}{13\!\cdots\!79}a^{17}-\frac{88\!\cdots\!96}{41\!\cdots\!37}a^{16}-\frac{18\!\cdots\!04}{10\!\cdots\!83}a^{15}+\frac{32\!\cdots\!66}{41\!\cdots\!37}a^{14}-\frac{67\!\cdots\!14}{41\!\cdots\!37}a^{13}+\frac{25\!\cdots\!36}{13\!\cdots\!79}a^{12}-\frac{10\!\cdots\!66}{46\!\cdots\!93}a^{11}-\frac{25\!\cdots\!85}{41\!\cdots\!37}a^{10}+\frac{82\!\cdots\!61}{41\!\cdots\!37}a^{9}-\frac{18\!\cdots\!45}{46\!\cdots\!93}a^{8}+\frac{68\!\cdots\!60}{15\!\cdots\!31}a^{7}+\frac{33\!\cdots\!22}{41\!\cdots\!37}a^{6}-\frac{28\!\cdots\!54}{13\!\cdots\!79}a^{5}+\frac{14\!\cdots\!29}{41\!\cdots\!37}a^{4}-\frac{15\!\cdots\!89}{41\!\cdots\!37}a^{3}-\frac{11\!\cdots\!38}{41\!\cdots\!37}a^{2}+\frac{69\!\cdots\!94}{13\!\cdots\!79}a-\frac{45\!\cdots\!60}{46\!\cdots\!93}$, $\frac{23\!\cdots\!72}{12\!\cdots\!11}a^{19}-\frac{13\!\cdots\!25}{12\!\cdots\!11}a^{18}+\frac{33\!\cdots\!89}{12\!\cdots\!11}a^{17}+\frac{21\!\cdots\!69}{12\!\cdots\!11}a^{16}-\frac{55\!\cdots\!52}{96\!\cdots\!47}a^{15}+\frac{58\!\cdots\!42}{13\!\cdots\!79}a^{14}+\frac{79\!\cdots\!30}{41\!\cdots\!37}a^{13}-\frac{82\!\cdots\!70}{12\!\cdots\!11}a^{12}+\frac{22\!\cdots\!32}{12\!\cdots\!11}a^{11}-\frac{50\!\cdots\!39}{12\!\cdots\!11}a^{10}+\frac{86\!\cdots\!67}{13\!\cdots\!79}a^{9}-\frac{24\!\cdots\!26}{12\!\cdots\!11}a^{8}-\frac{90\!\cdots\!08}{12\!\cdots\!11}a^{7}+\frac{43\!\cdots\!15}{12\!\cdots\!11}a^{6}-\frac{81\!\cdots\!75}{12\!\cdots\!11}a^{5}+\frac{14\!\cdots\!77}{13\!\cdots\!79}a^{4}-\frac{18\!\cdots\!79}{13\!\cdots\!79}a^{3}+\frac{40\!\cdots\!39}{12\!\cdots\!11}a^{2}+\frac{21\!\cdots\!15}{12\!\cdots\!11}a-\frac{21\!\cdots\!21}{12\!\cdots\!11}$, $\frac{22\!\cdots\!13}{96\!\cdots\!47}a^{19}-\frac{31\!\cdots\!98}{12\!\cdots\!11}a^{18}+\frac{14\!\cdots\!13}{12\!\cdots\!11}a^{17}-\frac{26\!\cdots\!83}{12\!\cdots\!11}a^{16}-\frac{35\!\cdots\!69}{12\!\cdots\!11}a^{15}+\frac{36\!\cdots\!30}{12\!\cdots\!11}a^{14}-\frac{12\!\cdots\!54}{12\!\cdots\!11}a^{13}+\frac{28\!\cdots\!84}{12\!\cdots\!11}a^{12}-\frac{51\!\cdots\!66}{12\!\cdots\!11}a^{11}+\frac{17\!\cdots\!42}{32\!\cdots\!49}a^{10}-\frac{25\!\cdots\!64}{12\!\cdots\!11}a^{9}-\frac{14\!\cdots\!45}{12\!\cdots\!11}a^{8}+\frac{51\!\cdots\!53}{12\!\cdots\!11}a^{7}-\frac{33\!\cdots\!04}{41\!\cdots\!37}a^{6}+\frac{43\!\cdots\!80}{41\!\cdots\!37}a^{5}-\frac{80\!\cdots\!85}{96\!\cdots\!47}a^{4}-\frac{12\!\cdots\!43}{12\!\cdots\!11}a^{3}+\frac{16\!\cdots\!44}{12\!\cdots\!11}a^{2}-\frac{22\!\cdots\!26}{12\!\cdots\!11}a+\frac{15\!\cdots\!46}{12\!\cdots\!11}$, $\frac{60\!\cdots\!20}{41\!\cdots\!37}a^{19}-\frac{67\!\cdots\!16}{41\!\cdots\!37}a^{18}+\frac{33\!\cdots\!84}{41\!\cdots\!37}a^{17}-\frac{62\!\cdots\!77}{32\!\cdots\!49}a^{16}+\frac{22\!\cdots\!20}{41\!\cdots\!37}a^{15}+\frac{60\!\cdots\!24}{41\!\cdots\!37}a^{14}-\frac{28\!\cdots\!88}{41\!\cdots\!37}a^{13}+\frac{65\!\cdots\!79}{32\!\cdots\!49}a^{12}-\frac{15\!\cdots\!18}{32\!\cdots\!49}a^{11}+\frac{35\!\cdots\!43}{41\!\cdots\!37}a^{10}-\frac{47\!\cdots\!76}{41\!\cdots\!37}a^{9}+\frac{33\!\cdots\!15}{41\!\cdots\!37}a^{8}+\frac{52\!\cdots\!54}{41\!\cdots\!37}a^{7}-\frac{24\!\cdots\!55}{41\!\cdots\!37}a^{6}+\frac{57\!\cdots\!06}{41\!\cdots\!37}a^{5}-\frac{89\!\cdots\!48}{41\!\cdots\!37}a^{4}+\frac{11\!\cdots\!80}{41\!\cdots\!37}a^{3}-\frac{34\!\cdots\!29}{13\!\cdots\!79}a^{2}+\frac{74\!\cdots\!28}{41\!\cdots\!37}a-\frac{22\!\cdots\!86}{41\!\cdots\!37}$, $\frac{36\!\cdots\!37}{12\!\cdots\!11}a^{19}-\frac{40\!\cdots\!32}{12\!\cdots\!11}a^{18}+\frac{20\!\cdots\!39}{12\!\cdots\!11}a^{17}-\frac{37\!\cdots\!18}{96\!\cdots\!47}a^{16}+\frac{13\!\cdots\!90}{12\!\cdots\!11}a^{15}+\frac{36\!\cdots\!59}{12\!\cdots\!11}a^{14}-\frac{17\!\cdots\!48}{12\!\cdots\!11}a^{13}+\frac{51\!\cdots\!68}{12\!\cdots\!11}a^{12}-\frac{11\!\cdots\!28}{12\!\cdots\!11}a^{11}+\frac{72\!\cdots\!29}{41\!\cdots\!37}a^{10}-\frac{29\!\cdots\!83}{12\!\cdots\!11}a^{9}+\frac{20\!\cdots\!57}{12\!\cdots\!11}a^{8}+\frac{31\!\cdots\!05}{12\!\cdots\!11}a^{7}-\frac{49\!\cdots\!29}{41\!\cdots\!37}a^{6}+\frac{11\!\cdots\!92}{41\!\cdots\!37}a^{5}-\frac{54\!\cdots\!54}{12\!\cdots\!11}a^{4}+\frac{71\!\cdots\!41}{12\!\cdots\!11}a^{3}-\frac{62\!\cdots\!97}{12\!\cdots\!11}a^{2}+\frac{45\!\cdots\!93}{12\!\cdots\!11}a-\frac{13\!\cdots\!38}{12\!\cdots\!11}$
| 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: | \( 229445959.72 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 229445959.72 \cdot 2}{2\cdot\sqrt{1969799822613689762148183291015625}}\cr\approx \mathstrut & 0.20092237227 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 43*x^18 - 74*x^17 - 105*x^16 + 1038*x^15 - 3587*x^14 + 9035*x^13 - 17355*x^12 + 24617*x^11 - 15587*x^10 - 28001*x^9 + 145716*x^8 - 311207*x^7 + 514366*x^6 - 473357*x^5 + 360299*x^4 + 378181*x^3 - 556830*x^2 + 950849*x - 397561)
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^20 - 10*x^19 + 43*x^18 - 74*x^17 - 105*x^16 + 1038*x^15 - 3587*x^14 + 9035*x^13 - 17355*x^12 + 24617*x^11 - 15587*x^10 - 28001*x^9 + 145716*x^8 - 311207*x^7 + 514366*x^6 - 473357*x^5 + 360299*x^4 + 378181*x^3 - 556830*x^2 + 950849*x - 397561, 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^20 - 10*x^19 + 43*x^18 - 74*x^17 - 105*x^16 + 1038*x^15 - 3587*x^14 + 9035*x^13 - 17355*x^12 + 24617*x^11 - 15587*x^10 - 28001*x^9 + 145716*x^8 - 311207*x^7 + 514366*x^6 - 473357*x^5 + 360299*x^4 + 378181*x^3 - 556830*x^2 + 950849*x - 397561);
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^20 - 10*x^19 + 43*x^18 - 74*x^17 - 105*x^16 + 1038*x^15 - 3587*x^14 + 9035*x^13 - 17355*x^12 + 24617*x^11 - 15587*x^10 - 28001*x^9 + 145716*x^8 - 311207*x^7 + 514366*x^6 - 473357*x^5 + 360299*x^4 + 378181*x^3 - 556830*x^2 + 950849*x - 397561);
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
$C_2\wr S_5$ (as 20T288):
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 non-solvable group of order 3840 |
| The 36 conjugacy class representatives for $C_2\wr S_5$ |
| Character table for $C_2\wr S_5$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1, 10.2.44382426957228125.1, 10.2.14202376626313.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.2.44382426957228125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(61\)
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.4.3.1 | $x^{4} + 183$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 61.4.3.1 | $x^{4} + 183$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(397\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $8$ | $4$ | $2$ | $6$ |