Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 12*x^14 + 12*x^13 + 19*x^12 - 318*x^11 + 2100*x^10 - 5213*x^9 + 14622*x^8 - 4630*x^7 + 927*x^6 + 121358*x^5 + 476036*x^4 + 432383*x^3 + 2307490*x^2 + 427097*x + 1543193)
gp: K = bnfinit(y^16 - 6*y^15 + 12*y^14 + 12*y^13 + 19*y^12 - 318*y^11 + 2100*y^10 - 5213*y^9 + 14622*y^8 - 4630*y^7 + 927*y^6 + 121358*y^5 + 476036*y^4 + 432383*y^3 + 2307490*y^2 + 427097*y + 1543193, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 12*x^14 + 12*x^13 + 19*x^12 - 318*x^11 + 2100*x^10 - 5213*x^9 + 14622*x^8 - 4630*x^7 + 927*x^6 + 121358*x^5 + 476036*x^4 + 432383*x^3 + 2307490*x^2 + 427097*x + 1543193);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 12*x^14 + 12*x^13 + 19*x^12 - 318*x^11 + 2100*x^10 - 5213*x^9 + 14622*x^8 - 4630*x^7 + 927*x^6 + 121358*x^5 + 476036*x^4 + 432383*x^3 + 2307490*x^2 + 427097*x + 1543193)
\( x^{16} - 6 x^{15} + 12 x^{14} + 12 x^{13} + 19 x^{12} - 318 x^{11} + 2100 x^{10} - 5213 x^{9} + \cdots + 1543193 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1968033759133442969054057291329\)
\(\medspace = 17^{14}\cdot 43^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(78.23\) | 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: | $17^{7/8}43^{1/2}\approx 78.23066716385114$ | ||
| Ramified primes: |
\(17\), \(43\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2314}a^{14}+\frac{75}{2314}a^{13}-\frac{40}{1157}a^{12}+\frac{749}{2314}a^{11}-\frac{155}{2314}a^{10}+\frac{336}{1157}a^{9}-\frac{1115}{2314}a^{8}+\frac{653}{2314}a^{7}-\frac{297}{1157}a^{6}-\frac{213}{2314}a^{5}-\frac{1149}{2314}a^{4}-\frac{10}{89}a^{3}-\frac{467}{2314}a^{2}-\frac{163}{2314}a+\frac{86}{1157}$, $\frac{1}{23\!\cdots\!34}a^{15}-\frac{13\!\cdots\!23}{23\!\cdots\!34}a^{14}-\frac{34\!\cdots\!57}{11\!\cdots\!67}a^{13}+\frac{25\!\cdots\!08}{11\!\cdots\!67}a^{12}-\frac{63\!\cdots\!87}{23\!\cdots\!34}a^{11}-\frac{36\!\cdots\!02}{11\!\cdots\!67}a^{10}+\frac{19\!\cdots\!48}{11\!\cdots\!67}a^{9}-\frac{71\!\cdots\!57}{23\!\cdots\!34}a^{8}-\frac{57\!\cdots\!26}{11\!\cdots\!67}a^{7}+\frac{11\!\cdots\!22}{11\!\cdots\!67}a^{6}+\frac{11\!\cdots\!41}{23\!\cdots\!34}a^{5}-\frac{35\!\cdots\!85}{11\!\cdots\!67}a^{4}-\frac{42\!\cdots\!23}{11\!\cdots\!67}a^{3}+\frac{19\!\cdots\!53}{23\!\cdots\!34}a^{2}+\frac{16\!\cdots\!78}{88\!\cdots\!59}a+\frac{93\!\cdots\!79}{23\!\cdots\!34}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{178}$, which has order $356$ (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: | $7$ | 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: |
$\frac{15\!\cdots\!88}{80\!\cdots\!91}a^{15}-\frac{72\!\cdots\!11}{80\!\cdots\!91}a^{14}+\frac{38\!\cdots\!23}{80\!\cdots\!91}a^{13}-\frac{37\!\cdots\!93}{61\!\cdots\!07}a^{12}-\frac{17\!\cdots\!69}{80\!\cdots\!91}a^{11}-\frac{14\!\cdots\!44}{61\!\cdots\!07}a^{10}+\frac{16\!\cdots\!86}{61\!\cdots\!07}a^{9}-\frac{13\!\cdots\!01}{80\!\cdots\!91}a^{8}+\frac{31\!\cdots\!03}{80\!\cdots\!91}a^{7}-\frac{67\!\cdots\!56}{80\!\cdots\!91}a^{6}-\frac{66\!\cdots\!84}{80\!\cdots\!91}a^{5}+\frac{19\!\cdots\!34}{80\!\cdots\!91}a^{4}-\frac{91\!\cdots\!45}{89\!\cdots\!19}a^{3}-\frac{43\!\cdots\!93}{80\!\cdots\!91}a^{2}-\frac{10\!\cdots\!84}{80\!\cdots\!91}a-\frac{20\!\cdots\!64}{80\!\cdots\!91}$, $\frac{34\!\cdots\!16}{80\!\cdots\!91}a^{15}+\frac{24\!\cdots\!68}{80\!\cdots\!91}a^{14}-\frac{10\!\cdots\!18}{80\!\cdots\!91}a^{13}+\frac{15\!\cdots\!34}{61\!\cdots\!07}a^{12}-\frac{84\!\cdots\!92}{80\!\cdots\!91}a^{11}+\frac{12\!\cdots\!08}{61\!\cdots\!07}a^{10}+\frac{89\!\cdots\!88}{61\!\cdots\!07}a^{9}+\frac{37\!\cdots\!95}{80\!\cdots\!91}a^{8}-\frac{13\!\cdots\!90}{80\!\cdots\!91}a^{7}+\frac{41\!\cdots\!96}{80\!\cdots\!91}a^{6}+\frac{34\!\cdots\!64}{80\!\cdots\!91}a^{5}-\frac{76\!\cdots\!16}{80\!\cdots\!91}a^{4}+\frac{51\!\cdots\!06}{89\!\cdots\!19}a^{3}+\frac{19\!\cdots\!67}{80\!\cdots\!91}a^{2}+\frac{55\!\cdots\!36}{80\!\cdots\!91}a+\frac{43\!\cdots\!56}{80\!\cdots\!91}$, $\frac{11\!\cdots\!72}{80\!\cdots\!91}a^{15}-\frac{97\!\cdots\!79}{80\!\cdots\!91}a^{14}+\frac{48\!\cdots\!41}{80\!\cdots\!91}a^{13}-\frac{52\!\cdots\!27}{61\!\cdots\!07}a^{12}-\frac{16\!\cdots\!77}{80\!\cdots\!91}a^{11}-\frac{13\!\cdots\!52}{61\!\cdots\!07}a^{10}+\frac{16\!\cdots\!98}{61\!\cdots\!07}a^{9}-\frac{17\!\cdots\!96}{80\!\cdots\!91}a^{8}+\frac{45\!\cdots\!93}{80\!\cdots\!91}a^{7}-\frac{10\!\cdots\!52}{80\!\cdots\!91}a^{6}-\frac{10\!\cdots\!48}{80\!\cdots\!91}a^{5}+\frac{27\!\cdots\!50}{80\!\cdots\!91}a^{4}-\frac{14\!\cdots\!51}{89\!\cdots\!19}a^{3}-\frac{62\!\cdots\!60}{80\!\cdots\!91}a^{2}-\frac{16\!\cdots\!20}{80\!\cdots\!91}a-\frac{32\!\cdots\!11}{80\!\cdots\!91}$, $\frac{15\!\cdots\!89}{23\!\cdots\!34}a^{15}-\frac{44\!\cdots\!44}{11\!\cdots\!67}a^{14}+\frac{12\!\cdots\!81}{23\!\cdots\!34}a^{13}+\frac{15\!\cdots\!68}{88\!\cdots\!59}a^{12}+\frac{20\!\cdots\!10}{11\!\cdots\!67}a^{11}-\frac{52\!\cdots\!87}{17\!\cdots\!18}a^{10}+\frac{12\!\cdots\!00}{88\!\cdots\!59}a^{9}-\frac{27\!\cdots\!47}{11\!\cdots\!67}a^{8}+\frac{13\!\cdots\!87}{23\!\cdots\!34}a^{7}+\frac{16\!\cdots\!62}{11\!\cdots\!67}a^{6}+\frac{59\!\cdots\!45}{11\!\cdots\!67}a^{5}+\frac{10\!\cdots\!69}{23\!\cdots\!34}a^{4}+\frac{51\!\cdots\!56}{12\!\cdots\!03}a^{3}+\frac{47\!\cdots\!69}{11\!\cdots\!67}a^{2}+\frac{31\!\cdots\!07}{23\!\cdots\!34}a+\frac{21\!\cdots\!81}{23\!\cdots\!34}$, $\frac{42\!\cdots\!69}{23\!\cdots\!34}a^{15}-\frac{16\!\cdots\!93}{11\!\cdots\!67}a^{14}+\frac{98\!\cdots\!01}{23\!\cdots\!34}a^{13}-\frac{28\!\cdots\!60}{11\!\cdots\!67}a^{12}-\frac{10\!\cdots\!45}{11\!\cdots\!67}a^{11}-\frac{13\!\cdots\!47}{23\!\cdots\!34}a^{10}+\frac{63\!\cdots\!14}{11\!\cdots\!67}a^{9}-\frac{19\!\cdots\!05}{11\!\cdots\!67}a^{8}+\frac{88\!\cdots\!55}{23\!\cdots\!34}a^{7}-\frac{27\!\cdots\!46}{88\!\cdots\!59}a^{6}-\frac{38\!\cdots\!56}{11\!\cdots\!67}a^{5}+\frac{62\!\cdots\!19}{23\!\cdots\!34}a^{4}+\frac{86\!\cdots\!30}{12\!\cdots\!03}a^{3}-\frac{17\!\cdots\!04}{11\!\cdots\!67}a^{2}+\frac{75\!\cdots\!95}{23\!\cdots\!34}a-\frac{23\!\cdots\!59}{23\!\cdots\!34}$, $\frac{48\!\cdots\!58}{11\!\cdots\!67}a^{15}-\frac{31\!\cdots\!04}{11\!\cdots\!67}a^{14}-\frac{89\!\cdots\!28}{11\!\cdots\!67}a^{13}+\frac{50\!\cdots\!66}{88\!\cdots\!59}a^{12}+\frac{21\!\cdots\!32}{11\!\cdots\!67}a^{11}-\frac{24\!\cdots\!45}{88\!\cdots\!59}a^{10}-\frac{79\!\cdots\!34}{88\!\cdots\!59}a^{9}+\frac{40\!\cdots\!03}{11\!\cdots\!67}a^{8}+\frac{10\!\cdots\!84}{11\!\cdots\!67}a^{7}-\frac{36\!\cdots\!46}{11\!\cdots\!67}a^{6}+\frac{52\!\cdots\!06}{11\!\cdots\!67}a^{5}+\frac{10\!\cdots\!88}{11\!\cdots\!67}a^{4}+\frac{30\!\cdots\!14}{11\!\cdots\!67}a^{3}+\frac{11\!\cdots\!23}{11\!\cdots\!67}a^{2}+\frac{31\!\cdots\!84}{11\!\cdots\!67}a+\frac{82\!\cdots\!95}{11\!\cdots\!67}$, $\frac{12\!\cdots\!73}{17\!\cdots\!18}a^{15}-\frac{37\!\cdots\!89}{17\!\cdots\!18}a^{14}-\frac{86\!\cdots\!39}{17\!\cdots\!18}a^{13}-\frac{90\!\cdots\!30}{88\!\cdots\!59}a^{12}+\frac{49\!\cdots\!59}{17\!\cdots\!18}a^{11}-\frac{88\!\cdots\!87}{17\!\cdots\!18}a^{10}-\frac{12\!\cdots\!01}{88\!\cdots\!59}a^{9}+\frac{59\!\cdots\!35}{17\!\cdots\!18}a^{8}+\frac{43\!\cdots\!19}{17\!\cdots\!18}a^{7}-\frac{71\!\cdots\!74}{88\!\cdots\!59}a^{6}+\frac{54\!\cdots\!79}{17\!\cdots\!18}a^{5}+\frac{49\!\cdots\!13}{17\!\cdots\!18}a^{4}+\frac{14\!\cdots\!99}{88\!\cdots\!59}a^{3}-\frac{14\!\cdots\!49}{17\!\cdots\!18}a^{2}+\frac{86\!\cdots\!29}{17\!\cdots\!18}a-\frac{11\!\cdots\!35}{17\!\cdots\!18}$
| 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: | \( 535402.98004 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 535402.98004 \cdot 356}{2\cdot\sqrt{1968033759133442969054057291329}}\cr\approx \mathstrut & 0.16501501429 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 12*x^14 + 12*x^13 + 19*x^12 - 318*x^11 + 2100*x^10 - 5213*x^9 + 14622*x^8 - 4630*x^7 + 927*x^6 + 121358*x^5 + 476036*x^4 + 432383*x^3 + 2307490*x^2 + 427097*x + 1543193)
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 - 6*x^15 + 12*x^14 + 12*x^13 + 19*x^12 - 318*x^11 + 2100*x^10 - 5213*x^9 + 14622*x^8 - 4630*x^7 + 927*x^6 + 121358*x^5 + 476036*x^4 + 432383*x^3 + 2307490*x^2 + 427097*x + 1543193, 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 - 6*x^15 + 12*x^14 + 12*x^13 + 19*x^12 - 318*x^11 + 2100*x^10 - 5213*x^9 + 14622*x^8 - 4630*x^7 + 927*x^6 + 121358*x^5 + 476036*x^4 + 432383*x^3 + 2307490*x^2 + 427097*x + 1543193);
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 - 6*x^15 + 12*x^14 + 12*x^13 + 19*x^12 - 318*x^11 + 2100*x^10 - 5213*x^9 + 14622*x^8 - 4630*x^7 + 927*x^6 + 121358*x^5 + 476036*x^4 + 432383*x^3 + 2307490*x^2 + 427097*x + 1543193);
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
$C_2^2:C_8$ (as 16T24):
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 $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.2.12427.1, 4.4.4913.1, 4.2.211259.1, 8.0.1402866265591073.2, 8.4.758716206377.1, 8.4.44630365081.1 |
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
| Galois closure: | deg 32 |
| Degree 16 sibling: | 16.8.575650281819106455466129.1 |
| Minimal sibling: | 16.8.575650281819106455466129.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.16.14.1 | $x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |
|
\(43\)
| 43.4.2.2 | $x^{4} - 1806 x^{2} + 5547$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 43.4.2.2 | $x^{4} - 1806 x^{2} + 5547$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 43.8.4.1 | $x^{8} + 182 x^{6} + 84 x^{5} + 11555 x^{4} - 6804 x^{3} + 301934 x^{2} - 447636 x + 2755621$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |