magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![211, -104, -640, 597, 65, -103, 75, -126, -79, 218, -222, 169, -75, 9, 21, -25, 14, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 14*x^16 - 25*x^15 + 21*x^14 + 9*x^13 - 75*x^12 + 169*x^11 - 222*x^10 + 218*x^9 - 79*x^8 - 126*x^7 + 75*x^6 - 103*x^5 + 65*x^4 + 597*x^3 - 640*x^2 - 104*x + 211)
gp: K = bnfinit(x^18 - 5*x^17 + 14*x^16 - 25*x^15 + 21*x^14 + 9*x^13 - 75*x^12 + 169*x^11 - 222*x^10 + 218*x^9 - 79*x^8 - 126*x^7 + 75*x^6 - 103*x^5 + 65*x^4 + 597*x^3 - 640*x^2 - 104*x + 211, 1)
Normalized defining polynomial
\( x^{18} - 5 x^{17} + 14 x^{16} - 25 x^{15} + 21 x^{14} + 9 x^{13} - 75 x^{12} + 169 x^{11} - 222 x^{10} + 218 x^{9} - 79 x^{8} - 126 x^{7} + 75 x^{6} - 103 x^{5} + 65 x^{4} + 597 x^{3} - 640 x^{2} - 104 x + 211 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-24704970275997896785381087551=-\,3^{9}\cdot 7^{15}\cdot 43^{2}\cdot 337^{2}\cdot 1259\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.79$ | 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: | $3, 7, 43, 337, 1259$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$ (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: | $12$ | 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: | \( 28055763.1721 \) (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 279936 |
| The 159 conjugacy class representatives for t18n857 are not computed |
| Character table for t18n857 is not computed |
Intermediate fields
| \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{21})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 sibling: | 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 | ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ | R | ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | $18$ | ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $7$ | 7.6.5.5 | $x^{6} + 56$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 7.12.10.1 | $x^{12} - 70 x^{6} + 35721$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |
| $43$ | $\Q_{43}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{43}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{43}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{43}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{43}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{43}$ | $x + 9$ | $1$ | $1$ | $0$ | 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 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.3.2.3 | $x^{3} - 3483$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 337 | Data not computed | ||||||
| 1259 | Data not computed | ||||||