Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 249*x^14 + 1468*x^13 + 14951*x^12 - 127824*x^11 - 146116*x^10 + 3524299*x^9 - 6618428*x^8 - 25092711*x^7 + 111980624*x^6 - 122939949*x^5 - 71984014*x^4 + 233617078*x^3 - 109771874*x^2 - 52840783*x + 40491571)
gp: K = bnfinit(y^16 - 3*y^15 - 249*y^14 + 1468*y^13 + 14951*y^12 - 127824*y^11 - 146116*y^10 + 3524299*y^9 - 6618428*y^8 - 25092711*y^7 + 111980624*y^6 - 122939949*y^5 - 71984014*y^4 + 233617078*y^3 - 109771874*y^2 - 52840783*y + 40491571, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 - 249*x^14 + 1468*x^13 + 14951*x^12 - 127824*x^11 - 146116*x^10 + 3524299*x^9 - 6618428*x^8 - 25092711*x^7 + 111980624*x^6 - 122939949*x^5 - 71984014*x^4 + 233617078*x^3 - 109771874*x^2 - 52840783*x + 40491571);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 - 249*x^14 + 1468*x^13 + 14951*x^12 - 127824*x^11 - 146116*x^10 + 3524299*x^9 - 6618428*x^8 - 25092711*x^7 + 111980624*x^6 - 122939949*x^5 - 71984014*x^4 + 233617078*x^3 - 109771874*x^2 - 52840783*x + 40491571)
\( x^{16} - 3 x^{15} - 249 x^{14} + 1468 x^{13} + 14951 x^{12} - 127824 x^{11} - 146116 x^{10} + \cdots + 40491571 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(54377966460580450275755400738525390625\)
\(\medspace = 5^{14}\cdot 73^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(228.28\) | 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: | $5^{7/8}73^{15/16}\approx 228.2779349400531$ | ||
| Ramified primes: |
\(5\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{73}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{10}-\frac{1}{5}a^{5}+\frac{1}{5}$, $\frac{1}{5}a^{11}-\frac{1}{5}a^{6}+\frac{1}{5}a$, $\frac{1}{15}a^{12}-\frac{1}{15}a^{11}-\frac{1}{15}a^{10}-\frac{1}{15}a^{9}-\frac{1}{15}a^{8}+\frac{4}{15}a^{7}+\frac{7}{15}a^{6}+\frac{2}{15}a^{5}+\frac{2}{5}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{7}{15}a-\frac{1}{15}$, $\frac{1}{15}a^{13}+\frac{1}{15}a^{11}+\frac{1}{15}a^{10}+\frac{1}{15}a^{9}+\frac{1}{3}a^{7}+\frac{1}{5}a^{6}-\frac{7}{15}a^{5}+\frac{4}{15}a^{4}-\frac{4}{15}a^{3}+\frac{2}{5}a^{2}+\frac{7}{15}a-\frac{4}{15}$, $\frac{1}{1635}a^{14}+\frac{2}{327}a^{13}-\frac{5}{327}a^{12}+\frac{121}{1635}a^{11}+\frac{4}{1635}a^{10}+\frac{41}{545}a^{9}-\frac{152}{1635}a^{8}+\frac{253}{545}a^{7}+\frac{43}{109}a^{6}-\frac{532}{1635}a^{5}-\frac{74}{545}a^{4}+\frac{47}{545}a^{3}+\frac{389}{1635}a^{2}-\frac{661}{1635}a+\frac{133}{1635}$, $\frac{1}{26\!\cdots\!05}a^{15}+\frac{22\!\cdots\!52}{26\!\cdots\!05}a^{14}+\frac{10\!\cdots\!26}{87\!\cdots\!35}a^{13}+\frac{40\!\cdots\!36}{26\!\cdots\!05}a^{12}-\frac{59\!\cdots\!17}{87\!\cdots\!35}a^{11}-\frac{26\!\cdots\!64}{26\!\cdots\!05}a^{10}+\frac{19\!\cdots\!49}{26\!\cdots\!05}a^{9}-\frac{92\!\cdots\!35}{52\!\cdots\!61}a^{8}+\frac{61\!\cdots\!51}{26\!\cdots\!05}a^{7}-\frac{10\!\cdots\!68}{52\!\cdots\!61}a^{6}+\frac{27\!\cdots\!92}{87\!\cdots\!35}a^{5}-\frac{60\!\cdots\!17}{87\!\cdots\!35}a^{4}-\frac{12\!\cdots\!14}{26\!\cdots\!05}a^{3}-\frac{92\!\cdots\!66}{87\!\cdots\!35}a^{2}+\frac{97\!\cdots\!52}{26\!\cdots\!05}a+\frac{26\!\cdots\!64}{10\!\cdots\!55}$
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: | $15$ | 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: |
$\frac{23\!\cdots\!27}{52\!\cdots\!61}a^{15}-\frac{21\!\cdots\!72}{26\!\cdots\!05}a^{14}-\frac{99\!\cdots\!82}{87\!\cdots\!35}a^{13}+\frac{46\!\cdots\!67}{87\!\cdots\!35}a^{12}+\frac{19\!\cdots\!59}{26\!\cdots\!05}a^{11}-\frac{12\!\cdots\!16}{26\!\cdots\!05}a^{10}-\frac{32\!\cdots\!58}{26\!\cdots\!05}a^{9}+\frac{37\!\cdots\!31}{26\!\cdots\!05}a^{8}-\frac{10\!\cdots\!67}{87\!\cdots\!35}a^{7}-\frac{33\!\cdots\!99}{26\!\cdots\!05}a^{6}+\frac{92\!\cdots\!46}{26\!\cdots\!05}a^{5}-\frac{11\!\cdots\!29}{87\!\cdots\!35}a^{4}-\frac{12\!\cdots\!71}{26\!\cdots\!05}a^{3}+\frac{12\!\cdots\!96}{26\!\cdots\!05}a^{2}+\frac{62\!\cdots\!73}{87\!\cdots\!35}a-\frac{15\!\cdots\!86}{10\!\cdots\!55}$, $\frac{13\!\cdots\!19}{26\!\cdots\!05}a^{15}-\frac{21\!\cdots\!99}{26\!\cdots\!05}a^{14}-\frac{33\!\cdots\!82}{26\!\cdots\!05}a^{13}+\frac{50\!\cdots\!21}{87\!\cdots\!35}a^{12}+\frac{74\!\cdots\!34}{87\!\cdots\!35}a^{11}-\frac{14\!\cdots\!94}{26\!\cdots\!05}a^{10}-\frac{39\!\cdots\!91}{26\!\cdots\!05}a^{9}+\frac{42\!\cdots\!17}{26\!\cdots\!05}a^{8}-\frac{31\!\cdots\!83}{26\!\cdots\!05}a^{7}-\frac{38\!\cdots\!42}{26\!\cdots\!05}a^{6}+\frac{98\!\cdots\!94}{26\!\cdots\!05}a^{5}-\frac{29\!\cdots\!94}{26\!\cdots\!05}a^{4}-\frac{13\!\cdots\!67}{26\!\cdots\!05}a^{3}+\frac{12\!\cdots\!53}{26\!\cdots\!05}a^{2}+\frac{23\!\cdots\!92}{26\!\cdots\!05}a-\frac{10\!\cdots\!92}{69\!\cdots\!37}$, $\frac{89\!\cdots\!36}{17\!\cdots\!87}a^{15}-\frac{78\!\cdots\!37}{87\!\cdots\!35}a^{14}-\frac{67\!\cdots\!64}{52\!\cdots\!61}a^{13}+\frac{51\!\cdots\!14}{87\!\cdots\!35}a^{12}+\frac{21\!\cdots\!01}{26\!\cdots\!05}a^{11}-\frac{28\!\cdots\!26}{52\!\cdots\!61}a^{10}-\frac{37\!\cdots\!39}{26\!\cdots\!05}a^{9}+\frac{28\!\cdots\!52}{17\!\cdots\!87}a^{8}-\frac{35\!\cdots\!72}{26\!\cdots\!05}a^{7}-\frac{12\!\cdots\!67}{87\!\cdots\!35}a^{6}+\frac{20\!\cdots\!54}{52\!\cdots\!61}a^{5}-\frac{37\!\cdots\!51}{26\!\cdots\!05}a^{4}-\frac{28\!\cdots\!58}{52\!\cdots\!61}a^{3}+\frac{45\!\cdots\!49}{87\!\cdots\!35}a^{2}+\frac{21\!\cdots\!91}{26\!\cdots\!05}a-\frac{34\!\cdots\!54}{20\!\cdots\!11}$, $\frac{20\!\cdots\!82}{52\!\cdots\!61}a^{15}-\frac{20\!\cdots\!93}{26\!\cdots\!05}a^{14}-\frac{52\!\cdots\!61}{52\!\cdots\!61}a^{13}+\frac{42\!\cdots\!33}{87\!\cdots\!35}a^{12}+\frac{33\!\cdots\!86}{52\!\cdots\!61}a^{11}-\frac{38\!\cdots\!96}{87\!\cdots\!35}a^{10}-\frac{89\!\cdots\!69}{87\!\cdots\!35}a^{9}+\frac{68\!\cdots\!09}{52\!\cdots\!61}a^{8}-\frac{34\!\cdots\!94}{26\!\cdots\!05}a^{7}-\frac{59\!\cdots\!42}{52\!\cdots\!61}a^{6}+\frac{28\!\cdots\!31}{87\!\cdots\!35}a^{5}-\frac{39\!\cdots\!73}{26\!\cdots\!05}a^{4}-\frac{78\!\cdots\!24}{17\!\cdots\!87}a^{3}+\frac{12\!\cdots\!04}{26\!\cdots\!05}a^{2}+\frac{30\!\cdots\!11}{52\!\cdots\!61}a-\frac{16\!\cdots\!23}{10\!\cdots\!55}$, $\frac{39\!\cdots\!66}{17\!\cdots\!87}a^{15}-\frac{71\!\cdots\!91}{17\!\cdots\!87}a^{14}-\frac{29\!\cdots\!82}{52\!\cdots\!61}a^{13}+\frac{23\!\cdots\!48}{87\!\cdots\!35}a^{12}+\frac{96\!\cdots\!92}{26\!\cdots\!05}a^{11}-\frac{64\!\cdots\!12}{26\!\cdots\!05}a^{10}-\frac{32\!\cdots\!54}{52\!\cdots\!61}a^{9}+\frac{12\!\cdots\!09}{17\!\cdots\!87}a^{8}-\frac{16\!\cdots\!54}{26\!\cdots\!05}a^{7}-\frac{56\!\cdots\!94}{87\!\cdots\!35}a^{6}+\frac{46\!\cdots\!67}{26\!\cdots\!05}a^{5}-\frac{34\!\cdots\!13}{52\!\cdots\!61}a^{4}-\frac{12\!\cdots\!41}{52\!\cdots\!61}a^{3}+\frac{20\!\cdots\!93}{87\!\cdots\!35}a^{2}+\frac{94\!\cdots\!22}{26\!\cdots\!05}a-\frac{79\!\cdots\!72}{10\!\cdots\!55}$, $\frac{73\!\cdots\!66}{87\!\cdots\!35}a^{15}-\frac{13\!\cdots\!56}{87\!\cdots\!35}a^{14}-\frac{11\!\cdots\!43}{52\!\cdots\!61}a^{13}+\frac{17\!\cdots\!27}{17\!\cdots\!87}a^{12}+\frac{36\!\cdots\!56}{26\!\cdots\!05}a^{11}-\frac{47\!\cdots\!02}{52\!\cdots\!61}a^{10}-\frac{61\!\cdots\!97}{26\!\cdots\!05}a^{9}+\frac{47\!\cdots\!78}{17\!\cdots\!87}a^{8}-\frac{12\!\cdots\!06}{52\!\cdots\!61}a^{7}-\frac{21\!\cdots\!12}{87\!\cdots\!35}a^{6}+\frac{34\!\cdots\!20}{52\!\cdots\!61}a^{5}-\frac{63\!\cdots\!08}{26\!\cdots\!05}a^{4}-\frac{47\!\cdots\!91}{52\!\cdots\!61}a^{3}+\frac{15\!\cdots\!09}{17\!\cdots\!87}a^{2}+\frac{35\!\cdots\!61}{26\!\cdots\!05}a-\frac{29\!\cdots\!87}{10\!\cdots\!55}$, $\frac{69\!\cdots\!81}{26\!\cdots\!05}a^{15}-\frac{12\!\cdots\!92}{26\!\cdots\!05}a^{14}-\frac{58\!\cdots\!11}{87\!\cdots\!35}a^{13}+\frac{53\!\cdots\!77}{17\!\cdots\!87}a^{12}+\frac{11\!\cdots\!79}{26\!\cdots\!05}a^{11}-\frac{75\!\cdots\!07}{26\!\cdots\!05}a^{10}-\frac{19\!\cdots\!87}{26\!\cdots\!05}a^{9}+\frac{44\!\cdots\!48}{52\!\cdots\!61}a^{8}-\frac{64\!\cdots\!58}{87\!\cdots\!35}a^{7}-\frac{39\!\cdots\!06}{52\!\cdots\!61}a^{6}+\frac{10\!\cdots\!66}{52\!\cdots\!61}a^{5}-\frac{68\!\cdots\!48}{87\!\cdots\!35}a^{4}-\frac{74\!\cdots\!09}{26\!\cdots\!05}a^{3}+\frac{72\!\cdots\!16}{26\!\cdots\!05}a^{2}+\frac{36\!\cdots\!32}{87\!\cdots\!35}a-\frac{18\!\cdots\!03}{20\!\cdots\!11}$, $\frac{97\!\cdots\!39}{87\!\cdots\!35}a^{15}-\frac{52\!\cdots\!04}{26\!\cdots\!05}a^{14}-\frac{24\!\cdots\!18}{87\!\cdots\!35}a^{13}+\frac{68\!\cdots\!33}{52\!\cdots\!61}a^{12}+\frac{47\!\cdots\!68}{26\!\cdots\!05}a^{11}-\frac{31\!\cdots\!09}{26\!\cdots\!05}a^{10}-\frac{26\!\cdots\!16}{87\!\cdots\!35}a^{9}+\frac{93\!\cdots\!29}{26\!\cdots\!05}a^{8}-\frac{27\!\cdots\!61}{87\!\cdots\!35}a^{7}-\frac{83\!\cdots\!56}{26\!\cdots\!05}a^{6}+\frac{22\!\cdots\!74}{26\!\cdots\!05}a^{5}-\frac{85\!\cdots\!94}{26\!\cdots\!05}a^{4}-\frac{10\!\cdots\!32}{87\!\cdots\!35}a^{3}+\frac{30\!\cdots\!53}{26\!\cdots\!05}a^{2}+\frac{46\!\cdots\!63}{26\!\cdots\!05}a-\frac{26\!\cdots\!78}{69\!\cdots\!37}$, $\frac{64\!\cdots\!21}{26\!\cdots\!05}a^{15}-\frac{38\!\cdots\!69}{87\!\cdots\!35}a^{14}-\frac{16\!\cdots\!78}{26\!\cdots\!05}a^{13}+\frac{24\!\cdots\!96}{87\!\cdots\!35}a^{12}+\frac{10\!\cdots\!27}{26\!\cdots\!05}a^{11}-\frac{69\!\cdots\!06}{26\!\cdots\!05}a^{10}-\frac{59\!\cdots\!17}{87\!\cdots\!35}a^{9}+\frac{20\!\cdots\!91}{26\!\cdots\!05}a^{8}-\frac{17\!\cdots\!72}{26\!\cdots\!05}a^{7}-\frac{60\!\cdots\!73}{87\!\cdots\!35}a^{6}+\frac{16\!\cdots\!56}{87\!\cdots\!35}a^{5}-\frac{12\!\cdots\!46}{17\!\cdots\!87}a^{4}-\frac{22\!\cdots\!73}{87\!\cdots\!35}a^{3}+\frac{13\!\cdots\!94}{52\!\cdots\!61}a^{2}+\frac{20\!\cdots\!61}{52\!\cdots\!61}a-\frac{86\!\cdots\!04}{10\!\cdots\!55}$, $\frac{30\!\cdots\!82}{17\!\cdots\!87}a^{15}-\frac{27\!\cdots\!88}{87\!\cdots\!35}a^{14}-\frac{11\!\cdots\!62}{26\!\cdots\!05}a^{13}+\frac{53\!\cdots\!16}{26\!\cdots\!05}a^{12}+\frac{74\!\cdots\!16}{26\!\cdots\!05}a^{11}-\frac{49\!\cdots\!58}{26\!\cdots\!05}a^{10}-\frac{12\!\cdots\!57}{26\!\cdots\!05}a^{9}+\frac{14\!\cdots\!02}{26\!\cdots\!05}a^{8}-\frac{25\!\cdots\!48}{52\!\cdots\!61}a^{7}-\frac{25\!\cdots\!33}{52\!\cdots\!61}a^{6}+\frac{11\!\cdots\!51}{87\!\cdots\!35}a^{5}-\frac{13\!\cdots\!69}{26\!\cdots\!05}a^{4}-\frac{48\!\cdots\!18}{26\!\cdots\!05}a^{3}+\frac{95\!\cdots\!94}{52\!\cdots\!61}a^{2}+\frac{72\!\cdots\!71}{26\!\cdots\!05}a-\frac{20\!\cdots\!54}{34\!\cdots\!85}$, $\frac{10\!\cdots\!46}{26\!\cdots\!05}a^{15}-\frac{12\!\cdots\!41}{17\!\cdots\!87}a^{14}-\frac{26\!\cdots\!23}{26\!\cdots\!05}a^{13}+\frac{41\!\cdots\!09}{87\!\cdots\!35}a^{12}+\frac{57\!\cdots\!96}{87\!\cdots\!35}a^{11}-\frac{22\!\cdots\!25}{52\!\cdots\!61}a^{10}-\frac{29\!\cdots\!36}{26\!\cdots\!05}a^{9}+\frac{33\!\cdots\!14}{26\!\cdots\!05}a^{8}-\frac{19\!\cdots\!58}{17\!\cdots\!87}a^{7}-\frac{10\!\cdots\!44}{87\!\cdots\!35}a^{6}+\frac{82\!\cdots\!94}{26\!\cdots\!05}a^{5}-\frac{31\!\cdots\!56}{26\!\cdots\!05}a^{4}-\frac{22\!\cdots\!66}{52\!\cdots\!61}a^{3}+\frac{11\!\cdots\!01}{26\!\cdots\!05}a^{2}+\frac{16\!\cdots\!24}{26\!\cdots\!05}a-\frac{47\!\cdots\!99}{34\!\cdots\!85}$, $\frac{62\!\cdots\!46}{26\!\cdots\!05}a^{15}-\frac{22\!\cdots\!11}{52\!\cdots\!61}a^{14}-\frac{52\!\cdots\!96}{87\!\cdots\!35}a^{13}+\frac{73\!\cdots\!13}{26\!\cdots\!05}a^{12}+\frac{20\!\cdots\!74}{52\!\cdots\!61}a^{11}-\frac{22\!\cdots\!43}{87\!\cdots\!35}a^{10}-\frac{57\!\cdots\!64}{87\!\cdots\!35}a^{9}+\frac{66\!\cdots\!87}{87\!\cdots\!35}a^{8}-\frac{17\!\cdots\!88}{26\!\cdots\!05}a^{7}-\frac{59\!\cdots\!71}{87\!\cdots\!35}a^{6}+\frac{48\!\cdots\!71}{26\!\cdots\!05}a^{5}-\frac{61\!\cdots\!11}{87\!\cdots\!35}a^{4}-\frac{22\!\cdots\!86}{87\!\cdots\!35}a^{3}+\frac{65\!\cdots\!26}{26\!\cdots\!05}a^{2}+\frac{10\!\cdots\!63}{26\!\cdots\!05}a-\frac{28\!\cdots\!96}{34\!\cdots\!85}$, $\frac{86\!\cdots\!12}{52\!\cdots\!61}a^{15}-\frac{71\!\cdots\!29}{26\!\cdots\!05}a^{14}-\frac{10\!\cdots\!79}{26\!\cdots\!05}a^{13}+\frac{16\!\cdots\!54}{87\!\cdots\!35}a^{12}+\frac{71\!\cdots\!02}{26\!\cdots\!05}a^{11}-\frac{45\!\cdots\!18}{26\!\cdots\!05}a^{10}-\frac{12\!\cdots\!27}{26\!\cdots\!05}a^{9}+\frac{27\!\cdots\!23}{52\!\cdots\!61}a^{8}-\frac{21\!\cdots\!39}{52\!\cdots\!61}a^{7}-\frac{12\!\cdots\!71}{26\!\cdots\!05}a^{6}+\frac{32\!\cdots\!42}{26\!\cdots\!05}a^{5}-\frac{75\!\cdots\!62}{17\!\cdots\!87}a^{4}-\frac{44\!\cdots\!31}{26\!\cdots\!05}a^{3}+\frac{42\!\cdots\!86}{26\!\cdots\!05}a^{2}+\frac{69\!\cdots\!18}{26\!\cdots\!05}a-\frac{54\!\cdots\!61}{10\!\cdots\!55}$, $\frac{14\!\cdots\!69}{26\!\cdots\!05}a^{15}-\frac{26\!\cdots\!88}{26\!\cdots\!05}a^{14}-\frac{74\!\cdots\!63}{52\!\cdots\!61}a^{13}+\frac{17\!\cdots\!72}{26\!\cdots\!05}a^{12}+\frac{80\!\cdots\!19}{87\!\cdots\!35}a^{11}-\frac{15\!\cdots\!32}{26\!\cdots\!05}a^{10}-\frac{40\!\cdots\!54}{26\!\cdots\!05}a^{9}+\frac{15\!\cdots\!88}{87\!\cdots\!35}a^{8}-\frac{40\!\cdots\!86}{26\!\cdots\!05}a^{7}-\frac{27\!\cdots\!07}{17\!\cdots\!87}a^{6}+\frac{11\!\cdots\!48}{26\!\cdots\!05}a^{5}-\frac{43\!\cdots\!77}{26\!\cdots\!05}a^{4}-\frac{15\!\cdots\!56}{26\!\cdots\!05}a^{3}+\frac{30\!\cdots\!64}{52\!\cdots\!61}a^{2}+\frac{23\!\cdots\!21}{26\!\cdots\!05}a-\frac{19\!\cdots\!97}{10\!\cdots\!55}$, $\frac{42\!\cdots\!92}{26\!\cdots\!05}a^{15}-\frac{25\!\cdots\!28}{87\!\cdots\!35}a^{14}-\frac{21\!\cdots\!96}{52\!\cdots\!61}a^{13}+\frac{49\!\cdots\!48}{26\!\cdots\!05}a^{12}+\frac{68\!\cdots\!52}{26\!\cdots\!05}a^{11}-\frac{30\!\cdots\!78}{17\!\cdots\!87}a^{10}-\frac{11\!\cdots\!21}{26\!\cdots\!05}a^{9}+\frac{26\!\cdots\!52}{52\!\cdots\!61}a^{8}-\frac{11\!\cdots\!23}{26\!\cdots\!05}a^{7}-\frac{11\!\cdots\!12}{26\!\cdots\!05}a^{6}+\frac{21\!\cdots\!40}{17\!\cdots\!87}a^{5}-\frac{12\!\cdots\!94}{26\!\cdots\!05}a^{4}-\frac{90\!\cdots\!47}{52\!\cdots\!61}a^{3}+\frac{14\!\cdots\!11}{87\!\cdots\!35}a^{2}+\frac{22\!\cdots\!69}{87\!\cdots\!35}a-\frac{56\!\cdots\!28}{10\!\cdots\!55}$
| 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: | \( 17978470399500 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 17978470399500 \cdot 2}{2\cdot\sqrt{54377966460580450275755400738525390625}}\cr\approx \mathstrut & 0.159779548184107 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 249*x^14 + 1468*x^13 + 14951*x^12 - 127824*x^11 - 146116*x^10 + 3524299*x^9 - 6618428*x^8 - 25092711*x^7 + 111980624*x^6 - 122939949*x^5 - 71984014*x^4 + 233617078*x^3 - 109771874*x^2 - 52840783*x + 40491571)
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^16 - 3*x^15 - 249*x^14 + 1468*x^13 + 14951*x^12 - 127824*x^11 - 146116*x^10 + 3524299*x^9 - 6618428*x^8 - 25092711*x^7 + 111980624*x^6 - 122939949*x^5 - 71984014*x^4 + 233617078*x^3 - 109771874*x^2 - 52840783*x + 40491571, 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^16 - 3*x^15 - 249*x^14 + 1468*x^13 + 14951*x^12 - 127824*x^11 - 146116*x^10 + 3524299*x^9 - 6618428*x^8 - 25092711*x^7 + 111980624*x^6 - 122939949*x^5 - 71984014*x^4 + 233617078*x^3 - 109771874*x^2 - 52840783*x + 40491571);
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^16 - 3*x^15 - 249*x^14 + 1468*x^13 + 14951*x^12 - 127824*x^11 - 146116*x^10 + 3524299*x^9 - 6618428*x^8 - 25092711*x^7 + 111980624*x^6 - 122939949*x^5 - 71984014*x^4 + 233617078*x^3 - 109771874*x^2 - 52840783*x + 40491571);
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
$\OD_{32}$ (as 16T22):
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 solvable group of order 32 |
| The 20 conjugacy class representatives for $C_{16} : C_2$ |
| Character table for $C_{16} : C_2$ |
Intermediate fields
| \(\Q(\sqrt{73}) \), 4.4.9725425.1, 8.8.172615601860890625.2 |
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
| Galois closure: | deg 32 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | $16$ | $16$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | $16$ | $16$ | $16$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.16.14.5 | $x^{16} - 20 x^{8} + 50$ | $8$ | $2$ | $14$ | $C_{16} : C_2$ | $[\ ]_{8}^{4}$ |
|
\(73\)
| 73.16.15.2 | $x^{16} + 219$ | $16$ | $1$ | $15$ | $C_{16} : C_2$ | $[\ ]_{16}^{2}$ |