Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 40*x^14 + 5*x^13 + 496*x^12 + 365*x^11 - 2354*x^10 - 2765*x^9 + 4166*x^8 + 5870*x^7 - 3235*x^6 - 5108*x^5 + 1076*x^4 + 1825*x^3 - 100*x^2 - 200*x + 5)
gp: K = bnfinit(y^16 - 2*y^15 - 40*y^14 + 5*y^13 + 496*y^12 + 365*y^11 - 2354*y^10 - 2765*y^9 + 4166*y^8 + 5870*y^7 - 3235*y^6 - 5108*y^5 + 1076*y^4 + 1825*y^3 - 100*y^2 - 200*y + 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 40*x^14 + 5*x^13 + 496*x^12 + 365*x^11 - 2354*x^10 - 2765*x^9 + 4166*x^8 + 5870*x^7 - 3235*x^6 - 5108*x^5 + 1076*x^4 + 1825*x^3 - 100*x^2 - 200*x + 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 40*x^14 + 5*x^13 + 496*x^12 + 365*x^11 - 2354*x^10 - 2765*x^9 + 4166*x^8 + 5870*x^7 - 3235*x^6 - 5108*x^5 + 1076*x^4 + 1825*x^3 - 100*x^2 - 200*x + 5)
\( x^{16} - 2 x^{15} - 40 x^{14} + 5 x^{13} + 496 x^{12} + 365 x^{11} - 2354 x^{10} - 2765 x^{9} + 4166 x^{8} + \cdots + 5 \)
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: |
\(2643693128974804931640625\)
\(\medspace = 5^{12}\cdot 101^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.60\) | 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: | $5^{3/4}101^{1/2}\approx 33.60378443921746$ | ||
| Ramified primes: |
\(5\), \(101\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5}a^{14}+\frac{1}{5}a^{13}+\frac{2}{5}a^{12}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}-\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{4}$, $\frac{1}{20\!\cdots\!45}a^{15}+\frac{12\!\cdots\!28}{20\!\cdots\!45}a^{14}+\frac{71\!\cdots\!74}{20\!\cdots\!45}a^{13}+\frac{50\!\cdots\!89}{20\!\cdots\!45}a^{12}-\frac{95\!\cdots\!26}{20\!\cdots\!45}a^{11}+\frac{97\!\cdots\!88}{41\!\cdots\!49}a^{10}-\frac{55\!\cdots\!83}{20\!\cdots\!45}a^{9}-\frac{51\!\cdots\!12}{20\!\cdots\!45}a^{8}+\frac{84\!\cdots\!63}{20\!\cdots\!45}a^{7}+\frac{24\!\cdots\!03}{20\!\cdots\!45}a^{6}+\frac{59\!\cdots\!41}{20\!\cdots\!45}a^{5}-\frac{65\!\cdots\!03}{20\!\cdots\!45}a^{4}-\frac{54\!\cdots\!75}{41\!\cdots\!49}a^{3}+\frac{15\!\cdots\!60}{41\!\cdots\!49}a^{2}-\frac{49\!\cdots\!78}{41\!\cdots\!49}a-\frac{20\!\cdots\!88}{41\!\cdots\!49}$
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
Trivial group, which has order $1$ (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: |
$\frac{213478548758}{4030275318455}a^{15}-\frac{576115722039}{4030275318455}a^{14}-\frac{8098002410181}{4030275318455}a^{13}+\frac{6671351919321}{4030275318455}a^{12}+\frac{99697462770072}{4030275318455}a^{11}+\frac{7443764532123}{4030275318455}a^{10}-\frac{98213651642826}{806055063691}a^{9}-\frac{44555887019760}{806055063691}a^{8}+\frac{992715017519858}{4030275318455}a^{7}+\frac{436297953378457}{4030275318455}a^{6}-\frac{10\!\cdots\!92}{4030275318455}a^{5}-\frac{266215538312837}{4030275318455}a^{4}+\frac{101577842860958}{806055063691}a^{3}+\frac{5843054312469}{806055063691}a^{2}-\frac{17188348137158}{806055063691}a+\frac{919874326005}{806055063691}$, $\frac{80\!\cdots\!91}{20\!\cdots\!45}a^{15}-\frac{10\!\cdots\!81}{20\!\cdots\!45}a^{14}-\frac{70\!\cdots\!77}{41\!\cdots\!49}a^{13}-\frac{54\!\cdots\!29}{20\!\cdots\!45}a^{12}+\frac{40\!\cdots\!99}{20\!\cdots\!45}a^{11}+\frac{96\!\cdots\!44}{20\!\cdots\!45}a^{10}-\frac{13\!\cdots\!56}{20\!\cdots\!45}a^{9}-\frac{49\!\cdots\!54}{20\!\cdots\!45}a^{8}-\frac{75\!\cdots\!97}{41\!\cdots\!49}a^{7}+\frac{73\!\cdots\!72}{20\!\cdots\!45}a^{6}+\frac{29\!\cdots\!56}{20\!\cdots\!45}a^{5}-\frac{39\!\cdots\!77}{20\!\cdots\!45}a^{4}-\frac{35\!\cdots\!41}{41\!\cdots\!49}a^{3}+\frac{95\!\cdots\!59}{41\!\cdots\!49}a^{2}+\frac{46\!\cdots\!46}{41\!\cdots\!49}a+\frac{35\!\cdots\!95}{41\!\cdots\!49}$, $\frac{12\!\cdots\!72}{41\!\cdots\!49}a^{15}-\frac{16\!\cdots\!16}{20\!\cdots\!45}a^{14}-\frac{23\!\cdots\!11}{20\!\cdots\!45}a^{13}+\frac{16\!\cdots\!63}{20\!\cdots\!45}a^{12}+\frac{55\!\cdots\!37}{41\!\cdots\!49}a^{11}+\frac{63\!\cdots\!56}{20\!\cdots\!45}a^{10}-\frac{12\!\cdots\!17}{20\!\cdots\!45}a^{9}-\frac{75\!\cdots\!18}{20\!\cdots\!45}a^{8}+\frac{19\!\cdots\!33}{20\!\cdots\!45}a^{7}+\frac{95\!\cdots\!41}{20\!\cdots\!45}a^{6}-\frac{31\!\cdots\!66}{41\!\cdots\!49}a^{5}+\frac{20\!\cdots\!99}{20\!\cdots\!45}a^{4}+\frac{18\!\cdots\!07}{41\!\cdots\!49}a^{3}-\frac{64\!\cdots\!98}{41\!\cdots\!49}a^{2}-\frac{59\!\cdots\!65}{41\!\cdots\!49}a+\frac{68\!\cdots\!86}{41\!\cdots\!49}$, $\frac{41\!\cdots\!08}{20\!\cdots\!45}a^{15}-\frac{16\!\cdots\!01}{20\!\cdots\!45}a^{14}-\frac{15\!\cdots\!08}{20\!\cdots\!45}a^{13}+\frac{34\!\cdots\!42}{20\!\cdots\!45}a^{12}+\frac{20\!\cdots\!67}{20\!\cdots\!45}a^{11}-\frac{45\!\cdots\!20}{41\!\cdots\!49}a^{10}-\frac{13\!\cdots\!59}{20\!\cdots\!45}a^{9}+\frac{44\!\cdots\!74}{20\!\cdots\!45}a^{8}+\frac{39\!\cdots\!14}{20\!\cdots\!45}a^{7}+\frac{50\!\cdots\!69}{20\!\cdots\!45}a^{6}-\frac{48\!\cdots\!52}{20\!\cdots\!45}a^{5}-\frac{14\!\cdots\!14}{20\!\cdots\!45}a^{4}+\frac{47\!\cdots\!28}{41\!\cdots\!49}a^{3}+\frac{13\!\cdots\!36}{41\!\cdots\!49}a^{2}-\frac{58\!\cdots\!98}{41\!\cdots\!49}a-\frac{83\!\cdots\!91}{41\!\cdots\!49}$, $\frac{61\!\cdots\!68}{20\!\cdots\!45}a^{15}-\frac{36\!\cdots\!13}{41\!\cdots\!49}a^{14}-\frac{21\!\cdots\!97}{20\!\cdots\!45}a^{13}+\frac{22\!\cdots\!94}{20\!\cdots\!45}a^{12}+\frac{24\!\cdots\!77}{20\!\cdots\!45}a^{11}-\frac{88\!\cdots\!76}{20\!\cdots\!45}a^{10}-\frac{10\!\cdots\!07}{20\!\cdots\!45}a^{9}-\frac{44\!\cdots\!28}{20\!\cdots\!45}a^{8}+\frac{14\!\cdots\!26}{20\!\cdots\!45}a^{7}+\frac{58\!\cdots\!88}{20\!\cdots\!45}a^{6}-\frac{60\!\cdots\!92}{20\!\cdots\!45}a^{5}-\frac{20\!\cdots\!58}{20\!\cdots\!45}a^{4}-\frac{22\!\cdots\!71}{41\!\cdots\!49}a^{3}+\frac{22\!\cdots\!17}{41\!\cdots\!49}a^{2}+\frac{26\!\cdots\!31}{41\!\cdots\!49}a-\frac{16\!\cdots\!51}{41\!\cdots\!49}$, $\frac{69924376133848}{29\!\cdots\!95}a^{15}-\frac{306130766074584}{29\!\cdots\!95}a^{14}-\frac{24\!\cdots\!81}{29\!\cdots\!95}a^{13}+\frac{67\!\cdots\!91}{29\!\cdots\!95}a^{12}+\frac{32\!\cdots\!17}{29\!\cdots\!95}a^{11}-\frac{50\!\cdots\!62}{29\!\cdots\!95}a^{10}-\frac{41\!\cdots\!34}{583981217200019}a^{9}+\frac{29\!\cdots\!72}{583981217200019}a^{8}+\frac{64\!\cdots\!43}{29\!\cdots\!95}a^{7}-\frac{86\!\cdots\!63}{29\!\cdots\!95}a^{6}-\frac{85\!\cdots\!52}{29\!\cdots\!95}a^{5}-\frac{10\!\cdots\!47}{29\!\cdots\!95}a^{4}+\frac{89\!\cdots\!78}{583981217200019}a^{3}+\frac{19\!\cdots\!09}{583981217200019}a^{2}-\frac{12\!\cdots\!94}{583981217200019}a-\frac{22\!\cdots\!20}{583981217200019}$, $\frac{43\!\cdots\!39}{41\!\cdots\!49}a^{15}-\frac{90\!\cdots\!75}{41\!\cdots\!49}a^{14}-\frac{16\!\cdots\!99}{41\!\cdots\!49}a^{13}+\frac{29\!\cdots\!33}{41\!\cdots\!49}a^{12}+\frac{20\!\cdots\!38}{41\!\cdots\!49}a^{11}+\frac{14\!\cdots\!19}{41\!\cdots\!49}a^{10}-\frac{87\!\cdots\!08}{41\!\cdots\!49}a^{9}-\frac{99\!\cdots\!79}{41\!\cdots\!49}a^{8}+\frac{12\!\cdots\!59}{41\!\cdots\!49}a^{7}+\frac{15\!\cdots\!91}{41\!\cdots\!49}a^{6}-\frac{77\!\cdots\!23}{41\!\cdots\!49}a^{5}-\frac{88\!\cdots\!45}{41\!\cdots\!49}a^{4}+\frac{26\!\cdots\!52}{41\!\cdots\!49}a^{3}+\frac{13\!\cdots\!45}{41\!\cdots\!49}a^{2}-\frac{47\!\cdots\!29}{41\!\cdots\!49}a+\frac{57\!\cdots\!93}{41\!\cdots\!49}$, $\frac{80\!\cdots\!91}{20\!\cdots\!45}a^{15}-\frac{10\!\cdots\!81}{20\!\cdots\!45}a^{14}-\frac{70\!\cdots\!77}{41\!\cdots\!49}a^{13}-\frac{54\!\cdots\!29}{20\!\cdots\!45}a^{12}+\frac{40\!\cdots\!99}{20\!\cdots\!45}a^{11}+\frac{96\!\cdots\!44}{20\!\cdots\!45}a^{10}-\frac{13\!\cdots\!56}{20\!\cdots\!45}a^{9}-\frac{49\!\cdots\!54}{20\!\cdots\!45}a^{8}-\frac{75\!\cdots\!97}{41\!\cdots\!49}a^{7}+\frac{73\!\cdots\!72}{20\!\cdots\!45}a^{6}+\frac{29\!\cdots\!56}{20\!\cdots\!45}a^{5}-\frac{39\!\cdots\!77}{20\!\cdots\!45}a^{4}-\frac{35\!\cdots\!41}{41\!\cdots\!49}a^{3}+\frac{95\!\cdots\!59}{41\!\cdots\!49}a^{2}+\frac{46\!\cdots\!46}{41\!\cdots\!49}a+\frac{76\!\cdots\!44}{41\!\cdots\!49}$, $\frac{555703951954023}{41\!\cdots\!49}a^{15}+\frac{46\!\cdots\!56}{20\!\cdots\!45}a^{14}-\frac{12\!\cdots\!04}{20\!\cdots\!45}a^{13}-\frac{41\!\cdots\!28}{20\!\cdots\!45}a^{12}+\frac{23\!\cdots\!58}{41\!\cdots\!49}a^{11}+\frac{61\!\cdots\!94}{20\!\cdots\!45}a^{10}+\frac{23\!\cdots\!37}{20\!\cdots\!45}a^{9}-\frac{29\!\cdots\!82}{20\!\cdots\!45}a^{8}-\frac{27\!\cdots\!13}{20\!\cdots\!45}a^{7}+\frac{42\!\cdots\!19}{20\!\cdots\!45}a^{6}+\frac{10\!\cdots\!68}{41\!\cdots\!49}a^{5}-\frac{27\!\cdots\!59}{20\!\cdots\!45}a^{4}-\frac{67\!\cdots\!24}{41\!\cdots\!49}a^{3}+\frac{20\!\cdots\!43}{41\!\cdots\!49}a^{2}+\frac{10\!\cdots\!51}{41\!\cdots\!49}a-\frac{34\!\cdots\!36}{41\!\cdots\!49}$, $\frac{498675632913381}{20\!\cdots\!45}a^{15}+\frac{15\!\cdots\!89}{41\!\cdots\!49}a^{14}-\frac{42\!\cdots\!39}{20\!\cdots\!45}a^{13}-\frac{32\!\cdots\!32}{20\!\cdots\!45}a^{12}+\frac{46\!\cdots\!89}{20\!\cdots\!45}a^{11}+\frac{40\!\cdots\!13}{20\!\cdots\!45}a^{10}-\frac{14\!\cdots\!04}{20\!\cdots\!45}a^{9}-\frac{18\!\cdots\!96}{20\!\cdots\!45}a^{8}-\frac{94\!\cdots\!43}{20\!\cdots\!45}a^{7}+\frac{29\!\cdots\!46}{20\!\cdots\!45}a^{6}+\frac{15\!\cdots\!16}{20\!\cdots\!45}a^{5}-\frac{19\!\cdots\!16}{20\!\cdots\!45}a^{4}-\frac{91\!\cdots\!81}{41\!\cdots\!49}a^{3}+\frac{11\!\cdots\!69}{41\!\cdots\!49}a^{2}-\frac{29\!\cdots\!43}{41\!\cdots\!49}a+\frac{27\!\cdots\!05}{41\!\cdots\!49}$, $\frac{63\!\cdots\!81}{20\!\cdots\!45}a^{15}-\frac{12\!\cdots\!21}{20\!\cdots\!45}a^{14}-\frac{50\!\cdots\!48}{41\!\cdots\!49}a^{13}+\frac{39\!\cdots\!51}{20\!\cdots\!45}a^{12}+\frac{30\!\cdots\!09}{20\!\cdots\!45}a^{11}+\frac{20\!\cdots\!09}{20\!\cdots\!45}a^{10}-\frac{13\!\cdots\!46}{20\!\cdots\!45}a^{9}-\frac{13\!\cdots\!39}{20\!\cdots\!45}a^{8}+\frac{44\!\cdots\!05}{41\!\cdots\!49}a^{7}+\frac{21\!\cdots\!72}{20\!\cdots\!45}a^{6}-\frac{23\!\cdots\!74}{20\!\cdots\!45}a^{5}-\frac{17\!\cdots\!07}{20\!\cdots\!45}a^{4}+\frac{21\!\cdots\!95}{41\!\cdots\!49}a^{3}+\frac{12\!\cdots\!68}{41\!\cdots\!49}a^{2}-\frac{26\!\cdots\!79}{41\!\cdots\!49}a-\frac{54\!\cdots\!51}{41\!\cdots\!49}$, $\frac{32\!\cdots\!96}{20\!\cdots\!45}a^{15}-\frac{53\!\cdots\!66}{20\!\cdots\!45}a^{14}-\frac{25\!\cdots\!31}{41\!\cdots\!49}a^{13}-\frac{32\!\cdots\!24}{20\!\cdots\!45}a^{12}+\frac{15\!\cdots\!59}{20\!\cdots\!45}a^{11}+\frac{17\!\cdots\!69}{20\!\cdots\!45}a^{10}-\frac{64\!\cdots\!96}{20\!\cdots\!45}a^{9}-\frac{10\!\cdots\!34}{20\!\cdots\!45}a^{8}+\frac{14\!\cdots\!23}{41\!\cdots\!49}a^{7}+\frac{18\!\cdots\!52}{20\!\cdots\!45}a^{6}-\frac{15\!\cdots\!89}{20\!\cdots\!45}a^{5}-\frac{12\!\cdots\!57}{20\!\cdots\!45}a^{4}-\frac{22\!\cdots\!15}{41\!\cdots\!49}a^{3}+\frac{69\!\cdots\!43}{41\!\cdots\!49}a^{2}+\frac{57\!\cdots\!26}{41\!\cdots\!49}a-\frac{64\!\cdots\!56}{41\!\cdots\!49}$, $\frac{31\!\cdots\!68}{20\!\cdots\!45}a^{15}-\frac{10\!\cdots\!84}{20\!\cdots\!45}a^{14}-\frac{10\!\cdots\!61}{20\!\cdots\!45}a^{13}+\frac{14\!\cdots\!01}{20\!\cdots\!45}a^{12}+\frac{10\!\cdots\!82}{20\!\cdots\!45}a^{11}-\frac{55\!\cdots\!07}{20\!\cdots\!45}a^{10}-\frac{62\!\cdots\!66}{41\!\cdots\!49}a^{9}+\frac{42\!\cdots\!90}{41\!\cdots\!49}a^{8}-\frac{55\!\cdots\!02}{20\!\cdots\!45}a^{7}-\frac{13\!\cdots\!58}{20\!\cdots\!45}a^{6}+\frac{26\!\cdots\!23}{20\!\cdots\!45}a^{5}+\frac{18\!\cdots\!53}{20\!\cdots\!45}a^{4}+\frac{55\!\cdots\!03}{41\!\cdots\!49}a^{3}-\frac{15\!\cdots\!06}{41\!\cdots\!49}a^{2}-\frac{70\!\cdots\!23}{41\!\cdots\!49}a+\frac{18\!\cdots\!55}{41\!\cdots\!49}$, $\frac{10\!\cdots\!17}{20\!\cdots\!45}a^{15}-\frac{24\!\cdots\!21}{20\!\cdots\!45}a^{14}-\frac{45\!\cdots\!64}{20\!\cdots\!45}a^{13}-\frac{69\!\cdots\!56}{20\!\cdots\!45}a^{12}+\frac{50\!\cdots\!08}{20\!\cdots\!45}a^{11}+\frac{12\!\cdots\!57}{20\!\cdots\!45}a^{10}-\frac{29\!\cdots\!05}{41\!\cdots\!49}a^{9}-\frac{12\!\cdots\!06}{41\!\cdots\!49}a^{8}-\frac{16\!\cdots\!48}{20\!\cdots\!45}a^{7}+\frac{91\!\cdots\!98}{20\!\cdots\!45}a^{6}+\frac{58\!\cdots\!42}{20\!\cdots\!45}a^{5}-\frac{44\!\cdots\!63}{20\!\cdots\!45}a^{4}-\frac{73\!\cdots\!06}{41\!\cdots\!49}a^{3}+\frac{10\!\cdots\!18}{41\!\cdots\!49}a^{2}+\frac{10\!\cdots\!93}{41\!\cdots\!49}a-\frac{45\!\cdots\!64}{41\!\cdots\!49}$, $\frac{56\!\cdots\!07}{20\!\cdots\!45}a^{15}-\frac{83\!\cdots\!52}{20\!\cdots\!45}a^{14}-\frac{45\!\cdots\!00}{41\!\cdots\!49}a^{13}-\frac{90\!\cdots\!78}{20\!\cdots\!45}a^{12}+\frac{26\!\cdots\!93}{20\!\cdots\!45}a^{11}+\frac{33\!\cdots\!28}{20\!\cdots\!45}a^{10}-\frac{10\!\cdots\!52}{20\!\cdots\!45}a^{9}-\frac{19\!\cdots\!23}{20\!\cdots\!45}a^{8}+\frac{19\!\cdots\!02}{41\!\cdots\!49}a^{7}+\frac{30\!\cdots\!69}{20\!\cdots\!45}a^{6}+\frac{11\!\cdots\!87}{20\!\cdots\!45}a^{5}-\frac{16\!\cdots\!84}{20\!\cdots\!45}a^{4}-\frac{52\!\cdots\!65}{41\!\cdots\!49}a^{3}+\frac{48\!\cdots\!39}{41\!\cdots\!49}a^{2}+\frac{39\!\cdots\!21}{41\!\cdots\!49}a-\frac{76\!\cdots\!00}{41\!\cdots\!49}$
| 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: | \( 16333161.075 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 16333161.075 \cdot 1}{2\cdot\sqrt{2643693128974804931640625}}\cr\approx \mathstrut & 0.32916576417 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 40*x^14 + 5*x^13 + 496*x^12 + 365*x^11 - 2354*x^10 - 2765*x^9 + 4166*x^8 + 5870*x^7 - 3235*x^6 - 5108*x^5 + 1076*x^4 + 1825*x^3 - 100*x^2 - 200*x + 5)
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 - 2*x^15 - 40*x^14 + 5*x^13 + 496*x^12 + 365*x^11 - 2354*x^10 - 2765*x^9 + 4166*x^8 + 5870*x^7 - 3235*x^6 - 5108*x^5 + 1076*x^4 + 1825*x^3 - 100*x^2 - 200*x + 5, 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 - 2*x^15 - 40*x^14 + 5*x^13 + 496*x^12 + 365*x^11 - 2354*x^10 - 2765*x^9 + 4166*x^8 + 5870*x^7 - 3235*x^6 - 5108*x^5 + 1076*x^4 + 1825*x^3 - 100*x^2 - 200*x + 5);
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 - 2*x^15 - 40*x^14 + 5*x^13 + 496*x^12 + 365*x^11 - 2354*x^10 - 2765*x^9 + 4166*x^8 + 5870*x^7 - 3235*x^6 - 5108*x^5 + 1076*x^4 + 1825*x^3 - 100*x^2 - 200*x + 5);
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
$\SD_{16}$ (as 16T12):
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 16 |
| The 7 conjugacy class representatives for $QD_{16}$ |
| Character table for $QD_{16}$ |
Intermediate fields
| \(\Q(\sqrt{505}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}, \sqrt{101})\), 4.4.2525.1 x2, 4.4.51005.1 x2, 8.8.65037750625.1, 8.8.16098453125.1 x4 |
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 8 sibling: | 8.8.16098453125.1 |
| Minimal sibling: | 8.8.16098453125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(101\)
| 101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |