magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![463, -3436, 12258, -29154, 52138, -73836, 86276, -84430, 70611, -50541, 31325, -16674, 7680, -2996, 996, -268, 59, -9, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 59*x^16 - 268*x^15 + 996*x^14 - 2996*x^13 + 7680*x^12 - 16674*x^11 + 31325*x^10 - 50541*x^9 + 70611*x^8 - 84430*x^7 + 86276*x^6 - 73836*x^5 + 52138*x^4 - 29154*x^3 + 12258*x^2 - 3436*x + 463)
gp: K = bnfinit(x^18 - 9*x^17 + 59*x^16 - 268*x^15 + 996*x^14 - 2996*x^13 + 7680*x^12 - 16674*x^11 + 31325*x^10 - 50541*x^9 + 70611*x^8 - 84430*x^7 + 86276*x^6 - 73836*x^5 + 52138*x^4 - 29154*x^3 + 12258*x^2 - 3436*x + 463, 1)
Normalized defining polynomial
\( x^{18} - 9 x^{17} + 59 x^{16} - 268 x^{15} + 996 x^{14} - 2996 x^{13} + 7680 x^{12} - 16674 x^{11} + 31325 x^{10} - 50541 x^{9} + 70611 x^{8} - 84430 x^{7} + 86276 x^{6} - 73836 x^{5} + 52138 x^{4} - 29154 x^{3} + 12258 x^{2} - 3436 x + 463 \)
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: | \(-2152830165694741754368725851=-\,7^{12}\cdot 13^{4}\cdot 41\cdot 43^{4}\cdot 38851\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.00$ | 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: | $7, 13, 41, 43, 38851$ | 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
$C_{32}$, which has order $32$ (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: | \( 35085.5946511 \) (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 41472 |
| The 192 conjugacy class representatives for t18n696 are not computed |
| Character table for t18n696 is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 9.9.36763077169.1 |
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 | $18$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | R | $18$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | $18$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ | $18$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ | R | R | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| $13$ | 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 41 | Data not computed | ||||||
| $43$ | $\Q_{43}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{43}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.6.4.1 | $x^{6} + 344 x^{3} + 49923$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 43.6.0.1 | $x^{6} - x + 26$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 38851 | Data not computed | ||||||