Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + x^16 + 4*x^15 - x^14 + x^13 + 7*x^12 + 3*x^11 + 4*x^10 - 2*x^9 + 10*x^8 + 2*x^6 - 5*x^5 + 4*x^4 + x^3 + 2*x^2 - x + 1)
gp: K = bnfinit(y^18 - y^17 + y^16 + 4*y^15 - y^14 + y^13 + 7*y^12 + 3*y^11 + 4*y^10 - 2*y^9 + 10*y^8 + 2*y^6 - 5*y^5 + 4*y^4 + y^3 + 2*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 + x^16 + 4*x^15 - x^14 + x^13 + 7*x^12 + 3*x^11 + 4*x^10 - 2*x^9 + 10*x^8 + 2*x^6 - 5*x^5 + 4*x^4 + x^3 + 2*x^2 - x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 + x^16 + 4*x^15 - x^14 + x^13 + 7*x^12 + 3*x^11 + 4*x^10 - 2*x^9 + 10*x^8 + 2*x^6 - 5*x^5 + 4*x^4 + x^3 + 2*x^2 - x + 1)
\( x^{18} - x^{17} + x^{16} + 4 x^{15} - x^{14} + x^{13} + 7 x^{12} + 3 x^{11} + 4 x^{10} - 2 x^{9} + \cdots + 1 \)
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: |
\(-129723503874804239427\)
\(\medspace = -\,3^{9}\cdot 433^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.10\) | 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: | $3^{1/2}433^{1/2}\approx 36.0416425818802$ | ||
| Ramified primes: |
\(3\), \(433\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{6}a^{15}-\frac{1}{6}a^{13}+\frac{1}{6}a^{11}+\frac{1}{6}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{6}a^{7}-\frac{1}{2}a^{6}+\frac{1}{6}a^{5}-\frac{1}{3}a^{4}+\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{16}-\frac{1}{6}a^{14}-\frac{1}{6}a^{12}+\frac{1}{6}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{2}a^{8}+\frac{1}{6}a^{7}-\frac{1}{2}a^{6}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}+\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{1986}a^{17}+\frac{11}{662}a^{16}+\frac{65}{993}a^{15}+\frac{121}{1986}a^{14}-\frac{95}{993}a^{13}-\frac{85}{993}a^{12}-\frac{73}{993}a^{11}-\frac{109}{662}a^{10}-\frac{191}{1986}a^{9}-\frac{269}{993}a^{8}-\frac{77}{1986}a^{7}+\frac{346}{993}a^{6}-\frac{151}{993}a^{5}+\frac{325}{993}a^{4}-\frac{245}{662}a^{3}+\frac{167}{1986}a^{2}-\frac{139}{993}a-\frac{185}{1986}$
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: |
\( \frac{113}{662} a^{17} + \frac{44}{331} a^{16} - \frac{142}{993} a^{15} + \frac{815}{993} a^{14} + \frac{1226}{993} a^{13} - \frac{343}{662} a^{12} + \frac{1811}{1986} a^{11} + \frac{5659}{1986} a^{10} + \frac{1451}{1986} a^{9} - \frac{221}{662} a^{8} + \frac{685}{993} a^{7} + \frac{1073}{662} a^{6} + \frac{149}{331} a^{5} - \frac{5723}{1986} a^{4} - \frac{292}{993} a^{3} + \frac{668}{993} a^{2} + \frac{693}{662} a + \frac{175}{1986} \)
(order $6$)
| 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: |
$\frac{32}{331}a^{17}-\frac{142}{993}a^{16}+\frac{45}{662}a^{15}+\frac{362}{993}a^{14}-\frac{401}{1986}a^{13}-\frac{101}{993}a^{12}+\frac{255}{662}a^{11}+\frac{1099}{1986}a^{10}+\frac{200}{993}a^{9}-\frac{1336}{993}a^{8}+\frac{3421}{1986}a^{7}+\frac{265}{662}a^{6}-\frac{461}{662}a^{5}-\frac{53}{331}a^{4}+\frac{293}{662}a^{3}+\frac{427}{662}a^{2}-\frac{1409}{1986}a-\frac{255}{662}$, $\frac{64}{993}a^{17}-\frac{79}{1986}a^{16}-\frac{241}{1986}a^{15}+\frac{1255}{1986}a^{14}-\frac{273}{662}a^{13}-\frac{907}{1986}a^{12}+\frac{1579}{993}a^{11}-\frac{381}{662}a^{10}-\frac{970}{993}a^{9}+\frac{2963}{1986}a^{8}-\frac{98}{331}a^{7}+\frac{265}{993}a^{6}-\frac{3239}{1986}a^{5}+\frac{2105}{1986}a^{4}-\frac{123}{331}a^{3}+\frac{32}{331}a^{2}-\frac{414}{331}a-\frac{179}{1986}$, $\frac{22}{331}a^{17}+\frac{64}{331}a^{16}-\frac{26}{993}a^{15}+\frac{373}{993}a^{14}+\frac{1031}{993}a^{13}+\frac{365}{993}a^{12}+\frac{625}{993}a^{11}+\frac{1919}{993}a^{10}+\frac{432}{331}a^{9}+\frac{411}{331}a^{8}+\frac{292}{331}a^{7}+\frac{656}{993}a^{6}+\frac{259}{993}a^{5}-\frac{130}{993}a^{4}-\frac{1177}{993}a^{3}-\frac{563}{993}a^{2}-\frac{158}{331}a+\frac{233}{331}$, $\frac{80}{993}a^{17}+\frac{105}{662}a^{16}-\frac{64}{331}a^{15}+\frac{385}{662}a^{14}+\frac{688}{993}a^{13}-\frac{389}{1986}a^{12}+\frac{1465}{1986}a^{11}+\frac{982}{993}a^{10}+\frac{313}{331}a^{9}+\frac{545}{662}a^{8}-\frac{3383}{1986}a^{7}+\frac{1269}{662}a^{6}+\frac{1}{331}a^{5}-\frac{1589}{1986}a^{4}-\frac{1135}{662}a^{3}+\frac{1895}{1986}a^{2}+\frac{1529}{1986}a+\frac{142}{331}$, $\frac{44}{331}a^{17}-\frac{887}{1986}a^{16}+\frac{93}{331}a^{15}+\frac{387}{662}a^{14}-\frac{1910}{993}a^{13}+\frac{467}{1986}a^{12}+\frac{2831}{1986}a^{11}-\frac{2782}{993}a^{10}-\frac{718}{993}a^{9}-\frac{1357}{1986}a^{8}+\frac{1849}{1986}a^{7}-\frac{1001}{662}a^{6}-\frac{2461}{993}a^{5}+\frac{1261}{662}a^{4}+\frac{527}{662}a^{3}+\frac{65}{1986}a^{2}-\frac{1565}{1986}a+\frac{736}{993}$, $\frac{667}{1986}a^{17}-\frac{138}{331}a^{16}+\frac{325}{993}a^{15}+\frac{1130}{993}a^{14}-\frac{475}{993}a^{13}+\frac{143}{1986}a^{12}+\frac{2249}{1986}a^{11}+\frac{779}{662}a^{10}+\frac{785}{662}a^{9}-\frac{7655}{1986}a^{8}+\frac{3614}{993}a^{7}+\frac{481}{1986}a^{6}-\frac{362}{331}a^{5}-\frac{4363}{1986}a^{4}+\frac{1307}{993}a^{3}+\frac{84}{331}a^{2}+\frac{1589}{1986}a-\frac{925}{1986}$, $\frac{67}{993}a^{17}+\frac{781}{1986}a^{16}-\frac{186}{331}a^{15}+\frac{1981}{1986}a^{14}+\frac{501}{331}a^{13}-\frac{863}{662}a^{12}+\frac{3937}{1986}a^{11}+\frac{972}{331}a^{10}-\frac{881}{993}a^{9}+\frac{6355}{1986}a^{8}-\frac{719}{1986}a^{7}+\frac{5675}{1986}a^{6}-\frac{1036}{993}a^{5}+\frac{1119}{662}a^{4}-\frac{2831}{1986}a^{3}+\frac{1525}{1986}a^{2}-\frac{511}{1986}a+\frac{1507}{993}$, $\frac{419}{993}a^{17}-\frac{481}{1986}a^{16}+\frac{41}{1986}a^{15}+\frac{1251}{662}a^{14}+\frac{653}{1986}a^{13}-\frac{595}{662}a^{12}+\frac{903}{331}a^{11}+\frac{5669}{1986}a^{10}+\frac{404}{993}a^{9}-\frac{3001}{1986}a^{8}+\frac{3485}{993}a^{7}+\frac{549}{331}a^{6}+\frac{139}{1986}a^{5}-\frac{4429}{1986}a^{4}+\frac{617}{331}a^{3}+\frac{2449}{993}a^{2}+\frac{672}{331}a+\frac{209}{1986}$
| 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: | \( 979.022125559 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 979.022125559 \cdot 1}{6\cdot\sqrt{129723503874804239427}}\cr\approx \mathstrut & 0.218650631681 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + x^16 + 4*x^15 - x^14 + x^13 + 7*x^12 + 3*x^11 + 4*x^10 - 2*x^9 + 10*x^8 + 2*x^6 - 5*x^5 + 4*x^4 + x^3 + 2*x^2 - x + 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 - x^17 + x^16 + 4*x^15 - x^14 + x^13 + 7*x^12 + 3*x^11 + 4*x^10 - 2*x^9 + 10*x^8 + 2*x^6 - 5*x^5 + 4*x^4 + x^3 + 2*x^2 - x + 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 - x^17 + x^16 + 4*x^15 - x^14 + x^13 + 7*x^12 + 3*x^11 + 4*x^10 - 2*x^9 + 10*x^8 + 2*x^6 - 5*x^5 + 4*x^4 + x^3 + 2*x^2 - x + 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 - x^17 + x^16 + 4*x^15 - x^14 + x^13 + 7*x^12 + 3*x^11 + 4*x^10 - 2*x^9 + 10*x^8 + 2*x^6 - 5*x^5 + 4*x^4 + x^3 + 2*x^2 - x + 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
$\SOPlus(4,2)$ (as 18T34):
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 72 |
| The 9 conjugacy class representatives for $\SOPlus(4,2)$ |
| Character table for $\SOPlus(4,2)$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 6.0.11691.1 x2, 9.3.2191933899.1 |
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 6 siblings: | 6.0.11691.1, 6.4.243548211.1 |
| Degree 9 sibling: | 9.3.2191933899.1 |
| Degree 12 siblings: | deg 12, 12.0.59182215273.1, deg 12, deg 12, deg 12, deg 12 |
| Degree 18 siblings: | deg 18, deg 18 |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 6.0.11691.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(433\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ | ||||
| Deg $6$ | $2$ | $3$ | $3$ |