Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 16*x^16 - 20*x^15 - 2*x^14 + 51*x^13 - 87*x^12 + 64*x^11 + 16*x^10 - 108*x^9 + 181*x^8 - 232*x^7 + 253*x^6 - 224*x^5 + 154*x^4 - 79*x^3 + 29*x^2 - 7*x + 1)
gp: K = bnfinit(y^18 - 6*y^17 + 16*y^16 - 20*y^15 - 2*y^14 + 51*y^13 - 87*y^12 + 64*y^11 + 16*y^10 - 108*y^9 + 181*y^8 - 232*y^7 + 253*y^6 - 224*y^5 + 154*y^4 - 79*y^3 + 29*y^2 - 7*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 16*x^16 - 20*x^15 - 2*x^14 + 51*x^13 - 87*x^12 + 64*x^11 + 16*x^10 - 108*x^9 + 181*x^8 - 232*x^7 + 253*x^6 - 224*x^5 + 154*x^4 - 79*x^3 + 29*x^2 - 7*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 16*x^16 - 20*x^15 - 2*x^14 + 51*x^13 - 87*x^12 + 64*x^11 + 16*x^10 - 108*x^9 + 181*x^8 - 232*x^7 + 253*x^6 - 224*x^5 + 154*x^4 - 79*x^3 + 29*x^2 - 7*x + 1)
\( x^{18} - 6 x^{17} + 16 x^{16} - 20 x^{15} - 2 x^{14} + 51 x^{13} - 87 x^{12} + 64 x^{11} + 16 x^{10} + \cdots + 1 \)
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: |
\(-5132738882980786176\)
\(\medspace = -\,2^{12}\cdot 3^{12}\cdot 11^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.95\) | 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: | $2^{2/3}3^{4/3}11^{1/2}\approx 22.779525808351707$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $9$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{2733}a^{17}+\frac{233}{2733}a^{16}+\frac{44}{911}a^{15}-\frac{119}{911}a^{14}+\frac{103}{911}a^{13}+\frac{37}{911}a^{12}+\frac{934}{2733}a^{11}-\frac{272}{911}a^{10}+\frac{857}{2733}a^{9}-\frac{1171}{2733}a^{8}+\frac{300}{911}a^{7}-\frac{128}{2733}a^{6}-\frac{1187}{2733}a^{5}+\frac{1226}{2733}a^{4}-\frac{58}{911}a^{3}-\frac{670}{2733}a^{2}-\frac{676}{2733}a+\frac{587}{2733}$
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: |
$\frac{5708}{2733}a^{17}-\frac{28337}{2733}a^{16}+\frac{57452}{2733}a^{15}-\frac{29912}{2733}a^{14}-\frac{33378}{911}a^{13}+\frac{239125}{2733}a^{12}-\frac{193943}{2733}a^{11}-\frac{17541}{911}a^{10}+\frac{98587}{911}a^{9}-\frac{392702}{2733}a^{8}+\frac{143658}{911}a^{7}-\frac{470989}{2733}a^{6}+\frac{138983}{911}a^{5}-\frac{211646}{2733}a^{4}+\frac{7828}{911}a^{3}+\frac{48301}{2733}a^{2}-\frac{32408}{2733}a+\frac{3016}{911}$, $\frac{1423}{911}a^{17}-\frac{23821}{2733}a^{16}+\frac{56081}{2733}a^{15}-\frac{16982}{911}a^{14}-\frac{17615}{911}a^{13}+\frac{70497}{911}a^{12}-\frac{253459}{2733}a^{11}+\frac{26776}{911}a^{10}+\frac{191267}{2733}a^{9}-\frac{397538}{2733}a^{8}+\frac{174746}{911}a^{7}-\frac{615668}{2733}a^{6}+\frac{204868}{911}a^{5}-\frac{455401}{2733}a^{4}+\frac{233786}{2733}a^{3}-\frac{74392}{2733}a^{2}+\frac{6581}{2733}a+\frac{1564}{2733}$, $\frac{4097}{2733}a^{17}-\frac{7634}{911}a^{16}+\frac{56152}{2733}a^{15}-\frac{56956}{2733}a^{14}-\frac{41312}{2733}a^{13}+\frac{70510}{911}a^{12}-\frac{93093}{911}a^{11}+\frac{113182}{2733}a^{10}+\frac{190534}{2733}a^{9}-\frac{424787}{2733}a^{8}+\frac{548905}{2733}a^{7}-\frac{636469}{2733}a^{6}+\frac{655699}{2733}a^{5}-\frac{503204}{2733}a^{4}+\frac{253693}{2733}a^{3}-\frac{66650}{2733}a^{2}+\frac{6245}{2733}a+\frac{1721}{2733}$, $\frac{383}{911}a^{17}-\frac{1028}{2733}a^{16}-\frac{11401}{2733}a^{15}+\frac{41663}{2733}a^{14}-\frac{51265}{2733}a^{13}-\frac{5770}{911}a^{12}+\frac{47982}{911}a^{11}-\frac{190564}{2733}a^{10}+\frac{79159}{2733}a^{9}+\frac{104833}{2733}a^{8}-\frac{235834}{2733}a^{7}+\frac{321182}{2733}a^{6}-\frac{398203}{2733}a^{5}+\frac{138865}{911}a^{4}-\frac{308335}{2733}a^{3}+\frac{157568}{2733}a^{2}-\frac{46102}{2733}a+\frac{2537}{911}$, $\frac{4192}{2733}a^{17}-\frac{6329}{911}a^{16}+\frac{37718}{2733}a^{15}-\frac{7819}{911}a^{14}-\frac{16436}{911}a^{13}+\frac{139174}{2733}a^{12}-\frac{144088}{2733}a^{11}+\frac{35662}{2733}a^{10}+\frac{128011}{2733}a^{9}-\frac{261821}{2733}a^{8}+\frac{357461}{2733}a^{7}-\frac{408125}{2733}a^{6}+\frac{394441}{2733}a^{5}-\frac{96419}{911}a^{4}+\frac{53850}{911}a^{3}-\frac{16407}{911}a^{2}+\frac{9439}{2733}a+\frac{1915}{2733}$, $a$, $\frac{2608}{2733}a^{17}-\frac{11816}{2733}a^{16}+\frac{7254}{911}a^{15}-\frac{2434}{911}a^{14}-\frac{43180}{2733}a^{13}+\frac{89068}{2733}a^{12}-\frac{63001}{2733}a^{11}-\frac{27362}{2733}a^{10}+\frac{36868}{911}a^{9}-\frac{145145}{2733}a^{8}+\frac{54511}{911}a^{7}-\frac{176221}{2733}a^{6}+\frac{151108}{2733}a^{5}-\frac{24968}{911}a^{4}+\frac{9907}{2733}a^{3}+\frac{5749}{911}a^{2}-\frac{10244}{2733}a+\frac{1327}{2733}$, $\frac{6830}{2733}a^{17}-\frac{12189}{911}a^{16}+\frac{83482}{2733}a^{15}-\frac{70621}{2733}a^{14}-\frac{87773}{2733}a^{13}+\frac{106039}{911}a^{12}-\frac{121334}{911}a^{11}+\frac{96784}{2733}a^{10}+\frac{302587}{2733}a^{9}-\frac{591500}{2733}a^{8}+\frac{764812}{2733}a^{7}-\frac{887905}{2733}a^{6}+\frac{879805}{2733}a^{5}-\frac{639854}{2733}a^{4}+\frac{313819}{2733}a^{3}-\frac{85781}{2733}a^{2}+\frac{6245}{2733}a+\frac{4454}{2733}$
| 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: | \( 51.3669704577 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 51.3669704577 \cdot 1}{2\cdot\sqrt{5132738882980786176}}\cr\approx \mathstrut & 0.173020748521 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 16*x^16 - 20*x^15 - 2*x^14 + 51*x^13 - 87*x^12 + 64*x^11 + 16*x^10 - 108*x^9 + 181*x^8 - 232*x^7 + 253*x^6 - 224*x^5 + 154*x^4 - 79*x^3 + 29*x^2 - 7*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 - 6*x^17 + 16*x^16 - 20*x^15 - 2*x^14 + 51*x^13 - 87*x^12 + 64*x^11 + 16*x^10 - 108*x^9 + 181*x^8 - 232*x^7 + 253*x^6 - 224*x^5 + 154*x^4 - 79*x^3 + 29*x^2 - 7*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 - 6*x^17 + 16*x^16 - 20*x^15 - 2*x^14 + 51*x^13 - 87*x^12 + 64*x^11 + 16*x^10 - 108*x^9 + 181*x^8 - 232*x^7 + 253*x^6 - 224*x^5 + 154*x^4 - 79*x^3 + 29*x^2 - 7*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 - 6*x^17 + 16*x^16 - 20*x^15 - 2*x^14 + 51*x^13 - 87*x^12 + 64*x^11 + 16*x^10 - 108*x^9 + 181*x^8 - 232*x^7 + 253*x^6 - 224*x^5 + 154*x^4 - 79*x^3 + 29*x^2 - 7*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
$C_3^2:C_6$ (as 18T23):
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 54 |
| The 18 conjugacy class representatives for $C_3^2:C_6$ |
| Character table for $C_3^2:C_6$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \), 3.1.44.1 x3, 6.0.107811.1, 6.0.1724976.2, 6.0.1724976.1, 6.0.21296.1 |
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 27 sibling: | 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 | R | R | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{9}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.9.12.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$ | $3$ | $3$ | $12$ | $C_3^2$ | $[2]^{3}$ | |
|
\(11\)
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |