Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 10*x^16 - 17*x^15 + 24*x^14 - 28*x^13 + 28*x^12 - 22*x^11 + 14*x^10 - 8*x^9 + 5*x^8 - 3*x^7 + 2*x^6 - 3*x^5 + 2*x^4 + x^3 - 2*x^2 + 1)
gp: K = bnfinit(y^18 - 4*y^17 + 10*y^16 - 17*y^15 + 24*y^14 - 28*y^13 + 28*y^12 - 22*y^11 + 14*y^10 - 8*y^9 + 5*y^8 - 3*y^7 + 2*y^6 - 3*y^5 + 2*y^4 + y^3 - 2*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^17 + 10*x^16 - 17*x^15 + 24*x^14 - 28*x^13 + 28*x^12 - 22*x^11 + 14*x^10 - 8*x^9 + 5*x^8 - 3*x^7 + 2*x^6 - 3*x^5 + 2*x^4 + x^3 - 2*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 10*x^16 - 17*x^15 + 24*x^14 - 28*x^13 + 28*x^12 - 22*x^11 + 14*x^10 - 8*x^9 + 5*x^8 - 3*x^7 + 2*x^6 - 3*x^5 + 2*x^4 + x^3 - 2*x^2 + 1)
\( x^{18} - 4 x^{17} + 10 x^{16} - 17 x^{15} + 24 x^{14} - 28 x^{13} + 28 x^{12} - 22 x^{11} + 14 x^{10} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-923324627983045303\)
\(\medspace = -\,31^{6}\cdot 463\cdot 1499^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.96\) | 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: | $31^{1/2}463^{1/2}1499^{1/2}\approx 4638.442303187569$ | ||
| Ramified primes: |
\(31\), \(463\), \(1499\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-463}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13}a^{17}-\frac{5}{13}a^{16}+\frac{2}{13}a^{15}-\frac{6}{13}a^{14}+\frac{4}{13}a^{13}-\frac{6}{13}a^{12}-\frac{5}{13}a^{11}-\frac{4}{13}a^{10}+\frac{5}{13}a^{9}+\frac{5}{13}a^{7}+\frac{5}{13}a^{6}-\frac{3}{13}a^{5}+\frac{2}{13}a^{3}-\frac{1}{13}a^{2}-\frac{1}{13}a+\frac{1}{13}$
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 $1$
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: | $8$ | 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: |
$\frac{3}{13}a^{17}-\frac{15}{13}a^{16}+\frac{32}{13}a^{15}-\frac{57}{13}a^{14}+\frac{77}{13}a^{13}-\frac{96}{13}a^{12}+\frac{89}{13}a^{11}-\frac{64}{13}a^{10}+\frac{28}{13}a^{9}-\frac{11}{13}a^{7}+\frac{28}{13}a^{6}-\frac{9}{13}a^{5}+\frac{6}{13}a^{3}+\frac{10}{13}a^{2}-\frac{3}{13}a+\frac{3}{13}$, $\frac{10}{13}a^{17}-\frac{37}{13}a^{16}+\frac{72}{13}a^{15}-\frac{99}{13}a^{14}+\frac{105}{13}a^{13}-\frac{99}{13}a^{12}+\frac{54}{13}a^{11}-\frac{1}{13}a^{10}-\frac{41}{13}a^{9}+2a^{8}-\frac{15}{13}a^{7}+\frac{11}{13}a^{6}+\frac{9}{13}a^{5}-2a^{4}+\frac{7}{13}a^{3}+\frac{29}{13}a^{2}+\frac{3}{13}a-\frac{16}{13}$, $\frac{9}{13}a^{17}-\frac{32}{13}a^{16}+\frac{70}{13}a^{15}-\frac{106}{13}a^{14}+\frac{140}{13}a^{13}-\frac{145}{13}a^{12}+\frac{124}{13}a^{11}-\frac{62}{13}a^{10}+\frac{19}{13}a^{9}+a^{8}-\frac{20}{13}a^{7}+\frac{32}{13}a^{6}-\frac{14}{13}a^{5}-\frac{21}{13}a^{3}+\frac{30}{13}a^{2}-\frac{9}{13}a-\frac{4}{13}$, $\frac{14}{13}a^{17}-\frac{44}{13}a^{16}+\frac{80}{13}a^{15}-\frac{97}{13}a^{14}+\frac{95}{13}a^{13}-\frac{71}{13}a^{12}+\frac{8}{13}a^{11}+\frac{61}{13}a^{10}-\frac{99}{13}a^{9}+5a^{8}-\frac{47}{13}a^{7}+\frac{31}{13}a^{6}-\frac{3}{13}a^{5}-2a^{4}-\frac{11}{13}a^{3}+\frac{38}{13}a^{2}-\frac{1}{13}a-\frac{12}{13}$, $\frac{2}{13}a^{17}-\frac{23}{13}a^{16}+\frac{56}{13}a^{15}-\frac{116}{13}a^{14}+\frac{164}{13}a^{13}-\frac{220}{13}a^{12}+\frac{224}{13}a^{11}-\frac{203}{13}a^{10}+\frac{140}{13}a^{9}-7a^{8}+\frac{49}{13}a^{7}-\frac{16}{13}a^{6}+\frac{7}{13}a^{5}-a^{4}+\frac{4}{13}a^{3}+\frac{11}{13}a^{2}-\frac{2}{13}a+\frac{2}{13}$, $\frac{12}{13}a^{17}-\frac{34}{13}a^{16}+\frac{63}{13}a^{15}-\frac{72}{13}a^{14}+\frac{74}{13}a^{13}-\frac{46}{13}a^{12}+\frac{5}{13}a^{11}+\frac{56}{13}a^{10}-\frac{70}{13}a^{9}+4a^{8}-\frac{31}{13}a^{7}+\frac{21}{13}a^{6}+\frac{3}{13}a^{5}-2a^{4}-\frac{15}{13}a^{3}+\frac{27}{13}a^{2}+\frac{1}{13}a-\frac{14}{13}$, $\frac{21}{13}a^{17}-\frac{53}{13}a^{16}+\frac{94}{13}a^{15}-\frac{100}{13}a^{14}+\frac{97}{13}a^{13}-\frac{48}{13}a^{12}-\frac{27}{13}a^{11}+\frac{111}{13}a^{10}-\frac{129}{13}a^{9}+7a^{8}-\frac{77}{13}a^{7}+\frac{53}{13}a^{6}-\frac{11}{13}a^{5}-a^{4}-\frac{36}{13}a^{3}+\frac{31}{13}a^{2}+\frac{18}{13}a-\frac{18}{13}$, $\frac{18}{13}a^{17}-\frac{51}{13}a^{16}+\frac{101}{13}a^{15}-\frac{134}{13}a^{14}+\frac{163}{13}a^{13}-\frac{160}{13}a^{12}+\frac{118}{13}a^{11}-\frac{59}{13}a^{10}+\frac{25}{13}a^{9}-3a^{8}+\frac{25}{13}a^{7}-\frac{14}{13}a^{6}+\frac{24}{13}a^{5}-2a^{4}-\frac{16}{13}a^{3}+\frac{34}{13}a^{2}+\frac{8}{13}a-\frac{21}{13}$
| 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: | \( 19.2198611118 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 19.2198611118 \cdot 1}{2\cdot\sqrt{923324627983045303}}\cr\approx \mathstrut & 0.152637714552 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 10*x^16 - 17*x^15 + 24*x^14 - 28*x^13 + 28*x^12 - 22*x^11 + 14*x^10 - 8*x^9 + 5*x^8 - 3*x^7 + 2*x^6 - 3*x^5 + 2*x^4 + x^3 - 2*x^2 + 1)
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^18 - 4*x^17 + 10*x^16 - 17*x^15 + 24*x^14 - 28*x^13 + 28*x^12 - 22*x^11 + 14*x^10 - 8*x^9 + 5*x^8 - 3*x^7 + 2*x^6 - 3*x^5 + 2*x^4 + x^3 - 2*x^2 + 1, 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^18 - 4*x^17 + 10*x^16 - 17*x^15 + 24*x^14 - 28*x^13 + 28*x^12 - 22*x^11 + 14*x^10 - 8*x^9 + 5*x^8 - 3*x^7 + 2*x^6 - 3*x^5 + 2*x^4 + x^3 - 2*x^2 + 1);
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^18 - 4*x^17 + 10*x^16 - 17*x^15 + 24*x^14 - 28*x^13 + 28*x^12 - 22*x^11 + 14*x^10 - 8*x^9 + 5*x^8 - 3*x^7 + 2*x^6 - 3*x^5 + 2*x^4 + x^3 - 2*x^2 + 1);
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
$C_2\times S_4^3.S_4$ (as 18T912):
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 663552 |
| The 330 conjugacy class representatives for $C_2\times S_4^3.S_4$ are not computed |
| Character table for $C_2\times S_4^3.S_4$ is not computed |
Intermediate fields
| 3.1.31.1, 9.1.44656709.1 |
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 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| 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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | $18$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | $18$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(31\)
| 31.3.0.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 31.3.0.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(463\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
|
\(1499\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |