Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 30*x^13 - 39*x^12 - 110*x^11 + 1336*x^10 - 5341*x^9 + 18122*x^8 - 41394*x^7 + 94489*x^6 - 151578*x^5 + 256578*x^4 - 246727*x^3 + 388080*x^2 - 78291*x + 242827)
gp: K = bnfinit(y^16 - 2*y^15 - 8*y^14 + 30*y^13 - 39*y^12 - 110*y^11 + 1336*y^10 - 5341*y^9 + 18122*y^8 - 41394*y^7 + 94489*y^6 - 151578*y^5 + 256578*y^4 - 246727*y^3 + 388080*y^2 - 78291*y + 242827, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 8*x^14 + 30*x^13 - 39*x^12 - 110*x^11 + 1336*x^10 - 5341*x^9 + 18122*x^8 - 41394*x^7 + 94489*x^6 - 151578*x^5 + 256578*x^4 - 246727*x^3 + 388080*x^2 - 78291*x + 242827);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 8*x^14 + 30*x^13 - 39*x^12 - 110*x^11 + 1336*x^10 - 5341*x^9 + 18122*x^8 - 41394*x^7 + 94489*x^6 - 151578*x^5 + 256578*x^4 - 246727*x^3 + 388080*x^2 - 78291*x + 242827)
\( x^{16} - 2 x^{15} - 8 x^{14} + 30 x^{13} - 39 x^{12} - 110 x^{11} + 1336 x^{10} - 5341 x^{9} + \cdots + 242827 \)
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: |
\(2859655432078149808865465089\)
\(\medspace = 17^{14}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.00\) | 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: | $17^{7/8}19^{1/2}\approx 52.00194728311084$ | ||
| Ramified primes: |
\(17\), \(19\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{202}a^{14}-\frac{1}{101}a^{13}-\frac{22}{101}a^{12}+\frac{1}{202}a^{11}+\frac{15}{101}a^{10}+\frac{28}{101}a^{9}-\frac{47}{202}a^{8}+\frac{8}{101}a^{7}+\frac{9}{101}a^{6}-\frac{55}{202}a^{5}+\frac{6}{101}a^{4}+\frac{42}{101}a^{3}-\frac{91}{202}a^{2}+\frac{11}{101}a+\frac{41}{101}$, $\frac{1}{30\!\cdots\!98}a^{15}+\frac{32\!\cdots\!02}{15\!\cdots\!99}a^{14}+\frac{73\!\cdots\!81}{15\!\cdots\!99}a^{13}+\frac{24\!\cdots\!11}{30\!\cdots\!98}a^{12}+\frac{23\!\cdots\!77}{15\!\cdots\!99}a^{11}+\frac{32\!\cdots\!06}{15\!\cdots\!99}a^{10}-\frac{19\!\cdots\!95}{30\!\cdots\!98}a^{9}-\frac{31\!\cdots\!99}{15\!\cdots\!99}a^{8}+\frac{61\!\cdots\!65}{15\!\cdots\!99}a^{7}-\frac{13\!\cdots\!65}{30\!\cdots\!98}a^{6}-\frac{44\!\cdots\!16}{15\!\cdots\!99}a^{5}+\frac{30\!\cdots\!06}{15\!\cdots\!99}a^{4}+\frac{28\!\cdots\!83}{30\!\cdots\!98}a^{3}+\frac{72\!\cdots\!07}{15\!\cdots\!99}a^{2}-\frac{22\!\cdots\!74}{15\!\cdots\!99}a+\frac{68\!\cdots\!36}{15\!\cdots\!99}$
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
$C_{10}\times C_{10}$, which has order $100$ (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: |
$\frac{22\!\cdots\!19}{98\!\cdots\!03}a^{15}-\frac{43\!\cdots\!93}{98\!\cdots\!03}a^{14}-\frac{49\!\cdots\!43}{19\!\cdots\!06}a^{13}+\frac{17\!\cdots\!75}{19\!\cdots\!06}a^{12}-\frac{19\!\cdots\!83}{98\!\cdots\!03}a^{11}-\frac{11\!\cdots\!19}{19\!\cdots\!06}a^{10}+\frac{65\!\cdots\!79}{19\!\cdots\!06}a^{9}-\frac{10\!\cdots\!98}{98\!\cdots\!03}a^{8}+\frac{54\!\cdots\!85}{19\!\cdots\!06}a^{7}-\frac{89\!\cdots\!05}{19\!\cdots\!06}a^{6}+\frac{78\!\cdots\!42}{98\!\cdots\!03}a^{5}-\frac{14\!\cdots\!27}{19\!\cdots\!06}a^{4}+\frac{17\!\cdots\!87}{19\!\cdots\!06}a^{3}+\frac{90\!\cdots\!73}{98\!\cdots\!03}a^{2}+\frac{20\!\cdots\!85}{19\!\cdots\!06}a+\frac{30\!\cdots\!35}{19\!\cdots\!06}$, $\frac{13\!\cdots\!48}{98\!\cdots\!03}a^{15}-\frac{28\!\cdots\!12}{98\!\cdots\!03}a^{14}-\frac{15\!\cdots\!97}{98\!\cdots\!03}a^{13}+\frac{51\!\cdots\!95}{98\!\cdots\!03}a^{12}-\frac{16\!\cdots\!80}{98\!\cdots\!03}a^{11}-\frac{29\!\cdots\!77}{98\!\cdots\!03}a^{10}+\frac{20\!\cdots\!99}{98\!\cdots\!03}a^{9}-\frac{68\!\cdots\!21}{98\!\cdots\!03}a^{8}+\frac{19\!\cdots\!63}{98\!\cdots\!03}a^{7}-\frac{36\!\cdots\!45}{98\!\cdots\!03}a^{6}+\frac{65\!\cdots\!44}{98\!\cdots\!03}a^{5}-\frac{82\!\cdots\!89}{98\!\cdots\!03}a^{4}+\frac{96\!\cdots\!31}{98\!\cdots\!03}a^{3}+\frac{69\!\cdots\!81}{98\!\cdots\!03}a^{2}+\frac{84\!\cdots\!19}{98\!\cdots\!03}a+\frac{90\!\cdots\!61}{98\!\cdots\!03}$, $\frac{29\!\cdots\!71}{98\!\cdots\!03}a^{15}+\frac{23\!\cdots\!09}{98\!\cdots\!03}a^{14}-\frac{34\!\cdots\!81}{19\!\cdots\!06}a^{13}-\frac{71\!\cdots\!17}{19\!\cdots\!06}a^{12}-\frac{95\!\cdots\!83}{98\!\cdots\!03}a^{11}-\frac{31\!\cdots\!69}{19\!\cdots\!06}a^{10}+\frac{49\!\cdots\!75}{19\!\cdots\!06}a^{9}-\frac{84\!\cdots\!44}{98\!\cdots\!03}a^{8}+\frac{82\!\cdots\!23}{19\!\cdots\!06}a^{7}-\frac{13\!\cdots\!29}{19\!\cdots\!06}a^{6}+\frac{23\!\cdots\!72}{98\!\cdots\!03}a^{5}-\frac{32\!\cdots\!83}{19\!\cdots\!06}a^{4}+\frac{11\!\cdots\!03}{19\!\cdots\!06}a^{3}+\frac{91\!\cdots\!27}{98\!\cdots\!03}a^{2}+\frac{87\!\cdots\!41}{19\!\cdots\!06}a+\frac{97\!\cdots\!69}{19\!\cdots\!06}$, $\frac{99\!\cdots\!98}{15\!\cdots\!99}a^{15}+\frac{14\!\cdots\!62}{15\!\cdots\!99}a^{14}-\frac{33\!\cdots\!35}{30\!\cdots\!98}a^{13}-\frac{15\!\cdots\!17}{30\!\cdots\!98}a^{12}+\frac{10\!\cdots\!04}{15\!\cdots\!99}a^{11}-\frac{32\!\cdots\!91}{30\!\cdots\!98}a^{10}+\frac{95\!\cdots\!21}{30\!\cdots\!98}a^{9}-\frac{35\!\cdots\!05}{15\!\cdots\!99}a^{8}+\frac{11\!\cdots\!57}{30\!\cdots\!98}a^{7}+\frac{26\!\cdots\!81}{30\!\cdots\!98}a^{6}-\frac{21\!\cdots\!76}{15\!\cdots\!99}a^{5}+\frac{12\!\cdots\!39}{30\!\cdots\!98}a^{4}-\frac{14\!\cdots\!69}{30\!\cdots\!98}a^{3}+\frac{20\!\cdots\!34}{15\!\cdots\!99}a^{2}+\frac{23\!\cdots\!03}{30\!\cdots\!98}a+\frac{25\!\cdots\!67}{30\!\cdots\!98}$, $\frac{34\!\cdots\!61}{30\!\cdots\!98}a^{15}-\frac{57\!\cdots\!61}{15\!\cdots\!99}a^{14}-\frac{31\!\cdots\!19}{30\!\cdots\!98}a^{13}+\frac{19\!\cdots\!47}{30\!\cdots\!98}a^{12}-\frac{99\!\cdots\!85}{15\!\cdots\!99}a^{11}-\frac{94\!\cdots\!33}{30\!\cdots\!98}a^{10}+\frac{65\!\cdots\!13}{30\!\cdots\!98}a^{9}-\frac{11\!\cdots\!61}{15\!\cdots\!99}a^{8}+\frac{61\!\cdots\!39}{30\!\cdots\!98}a^{7}-\frac{11\!\cdots\!51}{30\!\cdots\!98}a^{6}+\frac{93\!\cdots\!72}{15\!\cdots\!99}a^{5}-\frac{22\!\cdots\!23}{30\!\cdots\!98}a^{4}+\frac{22\!\cdots\!43}{30\!\cdots\!98}a^{3}+\frac{15\!\cdots\!27}{15\!\cdots\!99}a^{2}-\frac{10\!\cdots\!25}{30\!\cdots\!98}a-\frac{47\!\cdots\!21}{15\!\cdots\!99}$, $\frac{10\!\cdots\!25}{15\!\cdots\!99}a^{15}+\frac{45\!\cdots\!05}{30\!\cdots\!98}a^{14}-\frac{33\!\cdots\!63}{30\!\cdots\!98}a^{13}-\frac{17\!\cdots\!95}{15\!\cdots\!99}a^{12}+\frac{17\!\cdots\!37}{30\!\cdots\!98}a^{11}-\frac{29\!\cdots\!65}{30\!\cdots\!98}a^{10}+\frac{88\!\cdots\!92}{15\!\cdots\!99}a^{9}+\frac{89\!\cdots\!91}{30\!\cdots\!98}a^{8}-\frac{77\!\cdots\!63}{30\!\cdots\!98}a^{7}+\frac{19\!\cdots\!30}{15\!\cdots\!99}a^{6}-\frac{79\!\cdots\!73}{30\!\cdots\!98}a^{5}+\frac{23\!\cdots\!79}{30\!\cdots\!98}a^{4}-\frac{11\!\cdots\!55}{15\!\cdots\!99}a^{3}+\frac{51\!\cdots\!25}{30\!\cdots\!98}a^{2}-\frac{13\!\cdots\!83}{30\!\cdots\!98}a+\frac{13\!\cdots\!10}{15\!\cdots\!99}$, $\frac{13\!\cdots\!00}{15\!\cdots\!99}a^{15}-\frac{39\!\cdots\!70}{15\!\cdots\!99}a^{14}-\frac{30\!\cdots\!87}{30\!\cdots\!98}a^{13}+\frac{13\!\cdots\!75}{30\!\cdots\!98}a^{12}-\frac{36\!\cdots\!64}{15\!\cdots\!99}a^{11}-\frac{77\!\cdots\!81}{30\!\cdots\!98}a^{10}+\frac{46\!\cdots\!89}{30\!\cdots\!98}a^{9}-\frac{75\!\cdots\!99}{15\!\cdots\!99}a^{8}+\frac{40\!\cdots\!53}{30\!\cdots\!98}a^{7}-\frac{81\!\cdots\!31}{30\!\cdots\!98}a^{6}+\frac{69\!\cdots\!95}{15\!\cdots\!99}a^{5}-\frac{19\!\cdots\!81}{30\!\cdots\!98}a^{4}+\frac{19\!\cdots\!23}{30\!\cdots\!98}a^{3}-\frac{60\!\cdots\!68}{15\!\cdots\!99}a^{2}+\frac{12\!\cdots\!75}{30\!\cdots\!98}a-\frac{33\!\cdots\!01}{30\!\cdots\!98}$
| 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: | \( 149969.889475 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 149969.889475 \cdot 100}{2\cdot\sqrt{2859655432078149808865465089}}\cr\approx \mathstrut & 0.340609039130 \end{aligned}\] (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 - 8*x^14 + 30*x^13 - 39*x^12 - 110*x^11 + 1336*x^10 - 5341*x^9 + 18122*x^8 - 41394*x^7 + 94489*x^6 - 151578*x^5 + 256578*x^4 - 246727*x^3 + 388080*x^2 - 78291*x + 242827)
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 - 8*x^14 + 30*x^13 - 39*x^12 - 110*x^11 + 1336*x^10 - 5341*x^9 + 18122*x^8 - 41394*x^7 + 94489*x^6 - 151578*x^5 + 256578*x^4 - 246727*x^3 + 388080*x^2 - 78291*x + 242827, 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 - 8*x^14 + 30*x^13 - 39*x^12 - 110*x^11 + 1336*x^10 - 5341*x^9 + 18122*x^8 - 41394*x^7 + 94489*x^6 - 151578*x^5 + 256578*x^4 - 246727*x^3 + 388080*x^2 - 78291*x + 242827);
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 - 8*x^14 + 30*x^13 - 39*x^12 - 110*x^11 + 1336*x^10 - 5341*x^9 + 18122*x^8 - 41394*x^7 + 94489*x^6 - 151578*x^5 + 256578*x^4 - 246727*x^3 + 388080*x^2 - 78291*x + 242827);
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
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.5491.1, 4.2.93347.1, 8.0.53475746204033.2, 8.4.148132260953.1, 8.4.8713662409.1 |
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: | 16.8.21943166735047688468209.1 |
| Minimal sibling: | 16.8.21943166735047688468209.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$ | R | R | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 | $x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |
|
\(19\)
| 19.4.2.2 | $x^{4} - 2888 x^{3} - 767106 x^{2} - 76532 x + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 19.4.2.2 | $x^{4} - 2888 x^{3} - 767106 x^{2} - 76532 x + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 19.8.4.1 | $x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |