Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 5*x^16 - x^15 - 4*x^13 + 6*x^12 + 15*x^11 - x^10 + 60*x^7 + 25*x^6 + 64*x^3 + 64*x^2 + 20*x + 1)
gp: K = bnfinit(y^18 - 4*y^17 + 5*y^16 - y^15 - 4*y^13 + 6*y^12 + 15*y^11 - y^10 + 60*y^7 + 25*y^6 + 64*y^3 + 64*y^2 + 20*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^17 + 5*x^16 - x^15 - 4*x^13 + 6*x^12 + 15*x^11 - x^10 + 60*x^7 + 25*x^6 + 64*x^3 + 64*x^2 + 20*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 5*x^16 - x^15 - 4*x^13 + 6*x^12 + 15*x^11 - x^10 + 60*x^7 + 25*x^6 + 64*x^3 + 64*x^2 + 20*x + 1)
\( x^{18} - 4 x^{17} + 5 x^{16} - x^{15} - 4 x^{13} + 6 x^{12} + 15 x^{11} - x^{10} + 60 x^{7} + 25 x^{6} + \cdots + 1 \)
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: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2389861611358642578125\)
\(\medspace = 5^{15}\cdot 23^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.41\) | 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: | $5^{5/6}23^{1/2}\approx 18.337449110889978$ | ||
| Ramified primes: |
\(5\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7}a^{15}+\frac{1}{7}a^{14}+\frac{3}{7}a^{13}+\frac{1}{7}a^{12}-\frac{1}{7}a^{11}+\frac{1}{7}a^{10}-\frac{2}{7}a^{9}-\frac{3}{7}a^{8}-\frac{1}{7}a^{7}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}-\frac{3}{7}a^{2}+\frac{1}{7}a-\frac{1}{7}$, $\frac{1}{133}a^{16}-\frac{2}{133}a^{15}-\frac{2}{19}a^{14}+\frac{20}{133}a^{13}+\frac{3}{133}a^{12}+\frac{25}{133}a^{11}+\frac{30}{133}a^{10}-\frac{53}{133}a^{9}-\frac{6}{133}a^{8}-\frac{53}{133}a^{7}+\frac{60}{133}a^{6}+\frac{25}{133}a^{5}-\frac{27}{133}a^{4}+\frac{46}{133}a^{3}+\frac{3}{133}a^{2}+\frac{45}{133}a+\frac{52}{133}$, $\frac{1}{3325036900717}a^{17}+\frac{11778030749}{3325036900717}a^{16}+\frac{32287422905}{3325036900717}a^{15}+\frac{974745058759}{3325036900717}a^{14}-\frac{200134429162}{3325036900717}a^{13}-\frac{225102781499}{3325036900717}a^{12}+\frac{150412542461}{475005271531}a^{11}+\frac{1010422044717}{3325036900717}a^{10}+\frac{50033342028}{3325036900717}a^{9}-\frac{1585202501755}{3325036900717}a^{8}-\frac{257057708056}{3325036900717}a^{7}+\frac{429332549701}{3325036900717}a^{6}+\frac{1074834199997}{3325036900717}a^{5}+\frac{1224340117828}{3325036900717}a^{4}+\frac{1075622607015}{3325036900717}a^{3}-\frac{675000409416}{3325036900717}a^{2}-\frac{1487841397450}{3325036900717}a-\frac{991283206661}{3325036900717}$
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: | $9$ | 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: |
$\frac{4200995717}{175001942143}a^{17}-\frac{8694363912}{175001942143}a^{16}-\frac{16228704213}{175001942143}a^{15}+\frac{58441337450}{175001942143}a^{14}-\frac{48447001665}{175001942143}a^{13}+\frac{21919069667}{175001942143}a^{12}-\frac{5837135176}{25000277449}a^{11}+\frac{157278268457}{175001942143}a^{10}+\frac{64103005690}{175001942143}a^{9}-\frac{69856326690}{175001942143}a^{8}+\frac{45380890355}{175001942143}a^{7}+\frac{185479124955}{175001942143}a^{6}+\frac{565051484168}{175001942143}a^{5}-\frac{95442126720}{175001942143}a^{4}+\frac{38831972455}{175001942143}a^{3}+\frac{165614790858}{175001942143}a^{2}+\frac{776849238532}{175001942143}a+\frac{179099724708}{175001942143}$, $\frac{2671836861}{13247158967}a^{17}-\frac{11084633425}{13247158967}a^{16}+\frac{14620301832}{13247158967}a^{15}-\frac{2922079759}{13247158967}a^{14}-\frac{3192783100}{13247158967}a^{13}-\frac{7296606773}{13247158967}a^{12}+\frac{339123824}{270350183}a^{11}+\frac{36893424030}{13247158967}a^{10}-\frac{8495652846}{13247158967}a^{9}-\frac{3324206051}{13247158967}a^{8}+\frac{4219964803}{13247158967}a^{7}+\frac{157760044609}{13247158967}a^{6}+\frac{43447152873}{13247158967}a^{5}-\frac{26440565296}{13247158967}a^{4}+\frac{11121039004}{13247158967}a^{3}+\frac{168468662932}{13247158967}a^{2}+\frac{145284072141}{13247158967}a+\frac{16460607444}{13247158967}$, $\frac{144333178837}{3325036900717}a^{17}-\frac{667477562008}{3325036900717}a^{16}+\frac{1214020296758}{3325036900717}a^{15}-\frac{1210991895537}{3325036900717}a^{14}+\frac{54509790431}{175001942143}a^{13}-\frac{701845657927}{3325036900717}a^{12}+\frac{25561034900}{475005271531}a^{11}+\frac{2481114834171}{3325036900717}a^{10}-\frac{1220139644614}{3325036900717}a^{9}+\frac{1769412674759}{3325036900717}a^{8}-\frac{1495346069835}{3325036900717}a^{7}+\frac{8242701235313}{3325036900717}a^{6}+\frac{542955146221}{3325036900717}a^{5}+\frac{3438145366812}{3325036900717}a^{4}-\frac{166090859104}{3325036900717}a^{3}+\frac{5439582986099}{3325036900717}a^{2}+\frac{7280326006434}{3325036900717}a+\frac{4113815734519}{3325036900717}$, $\frac{84058693693}{3325036900717}a^{17}-\frac{530705873855}{3325036900717}a^{16}+\frac{1325298361148}{3325036900717}a^{15}-\frac{1610623733433}{3325036900717}a^{14}+\frac{1086396395330}{3325036900717}a^{13}-\frac{959872205385}{3325036900717}a^{12}+\frac{216113416825}{475005271531}a^{11}-\frac{130644298846}{3325036900717}a^{10}-\frac{2193400825963}{3325036900717}a^{9}+\frac{1329601097969}{3325036900717}a^{8}+\frac{34670772328}{3325036900717}a^{7}+\frac{5513315320980}{3325036900717}a^{6}-\frac{9689144303318}{3325036900717}a^{5}+\frac{2372018069691}{3325036900717}a^{4}+\frac{461449299692}{3325036900717}a^{3}+\frac{7058305252816}{3325036900717}a^{2}-\frac{5530238022075}{3325036900717}a-\frac{155086537537}{175001942143}$, $\frac{442626009491}{3325036900717}a^{17}-\frac{1822243495952}{3325036900717}a^{16}+\frac{2442758153965}{3325036900717}a^{15}-\frac{715556491185}{3325036900717}a^{14}-\frac{146931988262}{3325036900717}a^{13}-\frac{1342087292498}{3325036900717}a^{12}+\frac{51007219007}{67857895933}a^{11}+\frac{351992896930}{175001942143}a^{10}-\frac{1417105616845}{3325036900717}a^{9}+\frac{366062923515}{3325036900717}a^{8}+\frac{1490138286880}{3325036900717}a^{7}+\frac{25838663932514}{3325036900717}a^{6}+\frac{9089473119683}{3325036900717}a^{5}-\frac{1119617445511}{3325036900717}a^{4}+\frac{3690410363064}{3325036900717}a^{3}+\frac{30373432520539}{3325036900717}a^{2}+\frac{27562534563254}{3325036900717}a+\frac{5826533552631}{3325036900717}$, $\frac{149586578290}{3325036900717}a^{17}-\frac{623591476518}{3325036900717}a^{16}+\frac{865935034684}{3325036900717}a^{15}-\frac{309292783815}{3325036900717}a^{14}-\frac{39046220457}{3325036900717}a^{13}-\frac{546924252296}{3325036900717}a^{12}+\frac{217948412008}{475005271531}a^{11}+\frac{1207078030944}{3325036900717}a^{10}-\frac{290856487982}{3325036900717}a^{9}+\frac{582978628446}{3325036900717}a^{8}+\frac{255389545134}{3325036900717}a^{7}+\frac{9521225082458}{3325036900717}a^{6}-\frac{1258479808555}{3325036900717}a^{5}+\frac{1515519670956}{3325036900717}a^{4}+\frac{1614407257872}{3325036900717}a^{3}+\frac{10787449222311}{3325036900717}a^{2}+\frac{2582453440409}{3325036900717}a-\frac{1133252093153}{3325036900717}$, $\frac{814624378678}{3325036900717}a^{17}-\frac{3513583003753}{3325036900717}a^{16}+\frac{5018056041797}{3325036900717}a^{15}-\frac{1687088360817}{3325036900717}a^{14}-\frac{541875854731}{3325036900717}a^{13}-\frac{2610833171826}{3325036900717}a^{12}+\frac{120508452450}{67857895933}a^{11}+\frac{10244319552836}{3325036900717}a^{10}-\frac{4609458982516}{3325036900717}a^{9}-\frac{148310048205}{3325036900717}a^{8}+\frac{1242423918614}{3325036900717}a^{7}+\frac{47946096312655}{3325036900717}a^{6}+\frac{6702107062039}{3325036900717}a^{5}-\frac{9267570014273}{3325036900717}a^{4}+\frac{4713106737854}{3325036900717}a^{3}+\frac{51967774583803}{3325036900717}a^{2}+\frac{34915013116918}{3325036900717}a+\frac{1018402978542}{3325036900717}$, $\frac{8224894852}{3325036900717}a^{17}+\frac{17684599973}{3325036900717}a^{16}-\frac{288179365827}{3325036900717}a^{15}+\frac{842471042634}{3325036900717}a^{14}-\frac{1218794995051}{3325036900717}a^{13}+\frac{1181492319638}{3325036900717}a^{12}-\frac{109439073026}{475005271531}a^{11}+\frac{290067518236}{3325036900717}a^{10}+\frac{47578815528}{175001942143}a^{9}-\frac{2315591628261}{3325036900717}a^{8}+\frac{1245861646906}{3325036900717}a^{7}-\frac{1741999191169}{3325036900717}a^{6}+\frac{2361226240410}{3325036900717}a^{5}-\frac{3909176804038}{3325036900717}a^{4}+\frac{645193698179}{3325036900717}a^{3}-\frac{1075972151817}{3325036900717}a^{2}+\frac{1938127571473}{3325036900717}a-\frac{1955245362512}{3325036900717}$, $\frac{335534044864}{3325036900717}a^{17}-\frac{1333111456816}{3325036900717}a^{16}+\frac{1455299950634}{3325036900717}a^{15}+\frac{598892920008}{3325036900717}a^{14}-\frac{1632797061894}{3325036900717}a^{13}-\frac{150290440262}{3325036900717}a^{12}+\frac{312661897826}{475005271531}a^{11}+\frac{4175874167840}{3325036900717}a^{10}-\frac{591129037906}{3325036900717}a^{9}-\frac{73915561465}{175001942143}a^{8}+\frac{749040245589}{3325036900717}a^{7}+\frac{18541410162180}{3325036900717}a^{6}+\frac{7805579563928}{3325036900717}a^{5}-\frac{8777007313857}{3325036900717}a^{4}+\frac{3461380596652}{3325036900717}a^{3}+\frac{20749314867965}{3325036900717}a^{2}+\frac{14956331631926}{3325036900717}a+\frac{2544798828709}{3325036900717}$
| 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: | \( 2645.14483524 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 2645.14483524 \cdot 1}{2\cdot\sqrt{2389861611358642578125}}\cr\approx \mathstrut & 0.262864551121 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 5*x^16 - x^15 - 4*x^13 + 6*x^12 + 15*x^11 - x^10 + 60*x^7 + 25*x^6 + 64*x^3 + 64*x^2 + 20*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 - 4*x^17 + 5*x^16 - x^15 - 4*x^13 + 6*x^12 + 15*x^11 - x^10 + 60*x^7 + 25*x^6 + 64*x^3 + 64*x^2 + 20*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 - 4*x^17 + 5*x^16 - x^15 - 4*x^13 + 6*x^12 + 15*x^11 - x^10 + 60*x^7 + 25*x^6 + 64*x^3 + 64*x^2 + 20*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 - 4*x^17 + 5*x^16 - x^15 - 4*x^13 + 6*x^12 + 15*x^11 - x^10 + 60*x^7 + 25*x^6 + 64*x^3 + 64*x^2 + 20*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
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 36 |
| The 12 conjugacy class representatives for $D_{18}$ |
| Character table for $D_{18}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.1.23.1, 6.2.66125.1, 9.1.4372515625.1 |
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: | data not computed |
| Degree 18 sibling: | deg 18 |
| 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 | $18$ | $18$ | R | ${\href{/padicField/7.2.0.1}{2} }^{9}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $18$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | $18$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.6.5.1 | $x^{6} + 5$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
| 5.12.10.1 | $x^{12} + 20 x^{7} + 10 x^{6} + 50 x^{2} + 100 x + 25$ | $6$ | $2$ | $10$ | $D_6$ | $[\ ]_{6}^{2}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |