magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![433, -3165, 11843, -29270, 53800, -77770, 92078, -90906, 76338, -54695, 33819, -17914, 8190, -3164, 1040, -276, 60, -9, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 60*x^16 - 276*x^15 + 1040*x^14 - 3164*x^13 + 8190*x^12 - 17914*x^11 + 33819*x^10 - 54695*x^9 + 76338*x^8 - 90906*x^7 + 92078*x^6 - 77770*x^5 + 53800*x^4 - 29270*x^3 + 11843*x^2 - 3165*x + 433)
gp: K = bnfinit(x^18 - 9*x^17 + 60*x^16 - 276*x^15 + 1040*x^14 - 3164*x^13 + 8190*x^12 - 17914*x^11 + 33819*x^10 - 54695*x^9 + 76338*x^8 - 90906*x^7 + 92078*x^6 - 77770*x^5 + 53800*x^4 - 29270*x^3 + 11843*x^2 - 3165*x + 433, 1)
Normalized defining polynomial
\( x^{18} - 9 x^{17} + 60 x^{16} - 276 x^{15} + 1040 x^{14} - 3164 x^{13} + 8190 x^{12} - 17914 x^{11} + 33819 x^{10} - 54695 x^{9} + 76338 x^{8} - 90906 x^{7} + 92078 x^{6} - 77770 x^{5} + 53800 x^{4} - 29270 x^{3} + 11843 x^{2} - 3165 x + 433 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-216895506996594951140594638751=-\,7^{16}\cdot 13^{8}\cdot 8000831\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $42.64$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $7, 13, 8000831$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{165}$, which has order $165$ (assuming GRH)
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 89545.2892541 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 13824 |
| The 80 conjugacy class representatives for t18n586 are not computed |
| Character table for t18n586 is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 9.9.164648481361.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | R | $18$ | R | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | $18$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | $18$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $7$ | 7.9.8.2 | $x^{9} - 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
| 7.9.8.2 | $x^{9} - 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ | |
| $13$ | 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.3.0.1 | $x^{3} - 2 x + 6$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.6.4.1 | $x^{6} + 39 x^{3} + 676$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 8000831 | Data not computed | ||||||