Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 54*x^14 - 42*x^12 + 603*x^10 + 1539*x^8 + 1368*x^6 + 450*x^4 + 27*x^2 - 1)
gp: K = bnfinit(y^18 - 54*y^14 - 42*y^12 + 603*y^10 + 1539*y^8 + 1368*y^6 + 450*y^4 + 27*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 54*x^14 - 42*x^12 + 603*x^10 + 1539*x^8 + 1368*x^6 + 450*x^4 + 27*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 54*x^14 - 42*x^12 + 603*x^10 + 1539*x^8 + 1368*x^6 + 450*x^4 + 27*x^2 - 1)
\( x^{18} - 54x^{14} - 42x^{12} + 603x^{10} + 1539x^{8} + 1368x^{6} + 450x^{4} + 27x^{2} - 1 \)
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: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(258151783382020583032356864\)
\(\medspace = 2^{18}\cdot 3^{44}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.33\) | 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: | $2^{3/2}3^{22/9}\approx 41.48025274304532$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{1970958043}a^{16}-\frac{680001050}{1970958043}a^{14}-\frac{698369646}{1970958043}a^{12}-\frac{881856185}{1970958043}a^{10}+\frac{591077472}{1970958043}a^{8}-\frac{400531092}{1970958043}a^{6}-\frac{411694146}{1970958043}a^{4}-\frac{648557274}{1970958043}a^{2}+\frac{939046168}{1970958043}$, $\frac{1}{1970958043}a^{17}-\frac{680001050}{1970958043}a^{15}-\frac{698369646}{1970958043}a^{13}-\frac{881856185}{1970958043}a^{11}+\frac{591077472}{1970958043}a^{9}-\frac{400531092}{1970958043}a^{7}-\frac{411694146}{1970958043}a^{5}-\frac{648557274}{1970958043}a^{3}+\frac{939046168}{1970958043}a$
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 $1$ (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: | $11$ | 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: |
$\frac{359402940}{1970958043}a^{17}-\frac{99977449}{1970958043}a^{15}-\frac{19308685533}{1970958043}a^{13}-\frac{9782623899}{1970958043}a^{11}+\frac{215644500610}{1970958043}a^{9}+\frac{493236135597}{1970958043}a^{7}+\frac{397121550498}{1970958043}a^{5}+\frac{126959471407}{1970958043}a^{3}+\frac{13457256054}{1970958043}a$, $\frac{181178343}{1970958043}a^{17}-\frac{177828305}{1970958043}a^{15}-\frac{9599158803}{1970958043}a^{13}+\frac{1782765666}{1970958043}a^{11}+\frac{107009520446}{1970958043}a^{9}+\frac{174905319288}{1970958043}a^{7}+\frac{81181492281}{1970958043}a^{5}+\frac{1002580072}{1970958043}a^{3}-\frac{5967056259}{1970958043}a$, $\frac{96713475}{1970958043}a^{16}-\frac{58913042}{1970958043}a^{14}-\frac{5225088909}{1970958043}a^{12}-\frac{826575155}{1970958043}a^{10}+\frac{60809995670}{1970958043}a^{8}+\frac{110706599775}{1970958043}a^{6}+\frac{43840581472}{1970958043}a^{4}-\frac{13508117432}{1970958043}a^{2}-\frac{3522507659}{1970958043}$, $\frac{33042823}{1970958043}a^{16}-\frac{71715076}{1970958043}a^{14}-\frac{1692591906}{1970958043}a^{12}+\frac{2328287958}{1970958043}a^{10}+\frac{18273838782}{1970958043}a^{8}+\frac{11607043877}{1970958043}a^{6}-\frac{17889914889}{1970958043}a^{4}-\frac{19188843308}{1970958043}a^{2}-\frac{2937029575}{1970958043}$, $\frac{248896287}{1970958043}a^{17}+\frac{56324523}{1970958043}a^{15}-\frac{13415523127}{1970958043}a^{13}-\frac{13598799745}{1970958043}a^{11}+\frac{146577127564}{1970958043}a^{9}+\frac{421229700015}{1970958043}a^{7}+\frac{436119778574}{1970958043}a^{5}+\frac{173789266522}{1970958043}a^{3}+\frac{16689938415}{1970958043}a$, $\frac{332073948}{1970958043}a^{17}-\frac{417919690}{1970958043}a^{15}-\frac{17572254427}{1970958043}a^{13}+\frac{8279699854}{1970958043}a^{11}+\frac{198708574840}{1970958043}a^{9}+\frac{261988696687}{1970958043}a^{7}+\frac{24237908305}{1970958043}a^{5}-\frac{69305071141}{1970958043}a^{3}-\frac{9029226480}{1970958043}a$, $\frac{84464868}{1970958043}a^{16}-\frac{118915263}{1970958043}a^{14}-\frac{4374069894}{1970958043}a^{12}+\frac{2609340821}{1970958043}a^{10}+\frac{46199524776}{1970958043}a^{8}+\frac{64198719513}{1970958043}a^{6}+\frac{37340910809}{1970958043}a^{4}+\frac{14510697504}{1970958043}a^{2}+\frac{1497367486}{1970958043}$, $\frac{102342905}{1970958043}a^{16}-\frac{7203426}{1970958043}a^{14}-\frac{5555991621}{1970958043}a^{12}-\frac{3832167584}{1970958043}a^{10}+\frac{63508178826}{1970958043}a^{8}+\frac{150585008774}{1970958043}a^{6}+\frac{113678524908}{1970958043}a^{4}+\frac{26727796937}{1970958043}a^{2}+\frac{347550625}{1970958043}$, $\frac{1189442691}{1970958043}a^{17}+\frac{126011631}{1970958043}a^{16}-\frac{28740605}{1970958043}a^{15}+\frac{88576425}{1970958043}a^{14}-\frac{64182713808}{1970958043}a^{13}-\frac{6887941397}{1970958043}a^{12}-\frac{48430136529}{1970958043}a^{11}-\frac{9990128754}{1970958043}a^{10}+\frac{715880225988}{1970958043}a^{9}+\frac{76662281628}{1970958043}a^{8}+\frac{1812554110662}{1970958043}a^{7}+\frac{246337040671}{1970958043}a^{6}+\frac{1612674829395}{1970958043}a^{5}+\frac{260218513041}{1970958043}a^{4}+\frac{554648093918}{1970958043}a^{3}+\frac{100110536955}{1970958043}a^{2}+\frac{50582128215}{1970958043}a+\frac{7949269956}{1970958043}$, $\frac{8153867264}{1970958043}a^{17}+\frac{1181278943}{1970958043}a^{16}+\frac{331646917}{1970958043}a^{15}+\frac{120357941}{1970958043}a^{14}-\frac{440446844586}{1970958043}a^{13}-\frac{63845543023}{1970958043}a^{12}-\frac{360207864819}{1970958043}a^{11}-\frac{56078479959}{1970958043}a^{10}+\frac{4910095004685}{1970958043}a^{9}+\frac{710321031768}{1970958043}a^{8}+\frac{12745906209213}{1970958043}a^{7}+\frac{1891132683854}{1970958043}a^{6}+\frac{11585435281849}{1970958043}a^{5}+\frac{1765351582379}{1970958043}a^{4}+\frac{4007507846954}{1970958043}a^{3}+\frac{628205685970}{1970958043}a^{2}+\frac{325243598034}{1970958043}a+\frac{52191403158}{1970958043}$, $\frac{412770155}{1970958043}a^{17}-\frac{213035077}{1970958043}a^{16}+\frac{36883336}{1970958043}a^{15}+\frac{137800090}{1970958043}a^{14}-\frac{22320580365}{1970958043}a^{13}+\frac{11360553696}{1970958043}a^{12}-\frac{19289670027}{1970958043}a^{11}+\frac{1721217538}{1970958043}a^{10}+\frac{249100635181}{1970958043}a^{9}-\frac{126542525363}{1970958043}a^{8}+\frac{657113011514}{1970958043}a^{7}-\frac{249175421310}{1970958043}a^{6}+\frac{599121831802}{1970958043}a^{5}-\frac{168795135415}{1970958043}a^{4}+\frac{197040089509}{1970958043}a^{3}-\frac{38434520884}{1970958043}a^{2}+\frac{9366498823}{1970958043}a-\frac{952362815}{1970958043}$
| 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: | \( 2459714.37439 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 2459714.37439 \cdot 1}{2\cdot\sqrt{258151783382020583032356864}}\cr\approx \mathstrut & 0.301422940880 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 54*x^14 - 42*x^12 + 603*x^10 + 1539*x^8 + 1368*x^6 + 450*x^4 + 27*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 - 54*x^14 - 42*x^12 + 603*x^10 + 1539*x^8 + 1368*x^6 + 450*x^4 + 27*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 - 54*x^14 - 42*x^12 + 603*x^10 + 1539*x^8 + 1368*x^6 + 450*x^4 + 27*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 - 54*x^14 - 42*x^12 + 603*x^10 + 1539*x^8 + 1368*x^6 + 450*x^4 + 27*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
$C_2^2:C_9$ (as 18T7):
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 $C_2^2 : C_9$ |
| Character table for $C_2^2 : C_9$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 6.2.419904.1, \(\Q(\zeta_{27})^+\) |
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 |
| 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 | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.18.89 | $x^{18} + 18 x^{17} + 66 x^{16} - 272 x^{15} - 1608 x^{14} + 7008 x^{13} + 83536 x^{12} + 346688 x^{11} + 922880 x^{10} + 2307136 x^{9} + 7066496 x^{8} + 20902656 x^{7} + 47520384 x^{6} + 81117696 x^{5} + 108969728 x^{4} + 117408768 x^{3} + 95319808 x^{2} + 50121216 x + 12416512$ | $2$ | $9$ | $18$ | $C_2^2 : C_9$ | $[2, 2]^{9}$ |
|
\(3\)
| 3.9.22.8 | $x^{9} + 24 x^{6} + 18 x^{5} + 9 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.8 | $x^{9} + 24 x^{6} + 18 x^{5} + 9 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |