sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*x + 1)
gp: K = bnfinit(y^6 - y^5 - 22*y^4 + 5*y^3 + 73*y^2 - 58*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*x + 1)
x 6 − x 5 − 22 x 4 + 5 x 3 + 73 x 2 − 58 x + 1 x^{6} - x^{5} - 22x^{4} + 5x^{3} + 73x^{2} - 58x + 1 x 6 − x 5 − 2 2 x 4 + 5 x 3 + 7 3 x 2 − 5 8 x + 1
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree : 6 6 6
sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
Signature : [ 6 , 0 ] [6, 0] [ 6 , 0 ]
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant :
286315237 286315237 2 8 6 3 1 5 2 3 7
= 1 3 3 ⋅ 1 9 4 \medspace = 13^{3}\cdot 19^{4} = 1 3 3 ⋅ 1 9 4
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : 25.67 25.67 2 5 . 6 7
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 : 1 3 1 / 2 1 9 2 / 3 ≈ 25.67284961266124 13^{1/2}19^{2/3}\approx 25.67284961266124 1 3 1 / 2 1 9 2 / 3 ≈ 2 5 . 6 7 2 8 4 9 6 1 2 6 6 1 2 4
Ramified primes :
13 13 1 3 , 19 19 1 9
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
Discriminant root field : Q ( 13 ) \Q(\sqrt{13}) Q ( 1 3 )
Aut ( K / Q ) \Aut(K/\Q) A u t ( K / Q )
= = =
Gal ( K / Q ) \Gal(K/\Q) G a l ( K / Q ) :
C 6 C_6 C 6
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is Galois and abelian over Q \Q Q .
Conductor : 247 = 13 ⋅ 19 247=13\cdot 19 2 4 7 = 1 3 ⋅ 1 9
Dirichlet character group :
{ \lbrace { χ 247 ( 64 , ⋅ ) \chi_{247}(64,·) χ 2 4 7 ( 6 4 , ⋅ ) , χ 247 ( 1 , ⋅ ) \chi_{247}(1,·) χ 2 4 7 ( 1 , ⋅ ) , χ 247 ( 144 , ⋅ ) \chi_{247}(144,·) χ 2 4 7 ( 1 4 4 , ⋅ ) , χ 247 ( 235 , ⋅ ) \chi_{247}(235,·) χ 2 4 7 ( 2 3 5 , ⋅ ) , χ 247 ( 220 , ⋅ ) \chi_{247}(220,·) χ 2 4 7 ( 2 2 0 , ⋅ ) , χ 247 ( 77 , ⋅ ) \chi_{247}(77,·) χ 2 4 7 ( 7 7 , ⋅ ) } \rbrace }
This is not a CM field .
This field has no CM subfields.
1 1 1 , a a a , a 2 a^{2} a 2 , a 3 a^{3} a 3 , a 4 a^{4} a 4 , 1 107 a 5 + 19 107 a 4 + 37 107 a 3 − 4 107 a 2 − 7 107 a + 16 107 \frac{1}{107}a^{5}+\frac{19}{107}a^{4}+\frac{37}{107}a^{3}-\frac{4}{107}a^{2}-\frac{7}{107}a+\frac{16}{107} 1 0 7 1 a 5 + 1 0 7 1 9 a 4 + 1 0 7 3 7 a 3 − 1 0 7 4 a 2 − 1 0 7 7 a + 1 0 7 1 6
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
Ideal class group : Trivial group, which has order 1 1 1
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Narrow class group : Trivial group, which has order 1 1 1
sage: K.narrow_class_group().invariants()
gp: bnfnarrow(K)
magma: NarrowClassGroup(K);
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
Rank : 5 5 5
sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
Torsion generator :
− 1 -1 − 1
(order 2 2 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 :
22 107 a 5 − 10 107 a 4 − 470 107 a 3 − 195 107 a 2 + 1237 107 a − 397 107 \frac{22}{107}a^{5}-\frac{10}{107}a^{4}-\frac{470}{107}a^{3}-\frac{195}{107}a^{2}+\frac{1237}{107}a-\frac{397}{107} 1 0 7 2 2 a 5 − 1 0 7 1 0 a 4 − 1 0 7 4 7 0 a 3 − 1 0 7 1 9 5 a 2 + 1 0 7 1 2 3 7 a − 1 0 7 3 9 7 , 62 107 a 5 + 1 107 a 4 − 1344 107 a 3 − 1104 107 a 2 + 3097 107 a − 78 107 \frac{62}{107}a^{5}+\frac{1}{107}a^{4}-\frac{1344}{107}a^{3}-\frac{1104}{107}a^{2}+\frac{3097}{107}a-\frac{78}{107} 1 0 7 6 2 a 5 + 1 0 7 1 a 4 − 1 0 7 1 3 4 4 a 3 − 1 0 7 1 1 0 4 a 2 + 1 0 7 3 0 9 7 a − 1 0 7 7 8 , 40 107 a 5 + 11 107 a 4 − 874 107 a 3 − 909 107 a 2 + 1967 107 a + 319 107 \frac{40}{107}a^{5}+\frac{11}{107}a^{4}-\frac{874}{107}a^{3}-\frac{909}{107}a^{2}+\frac{1967}{107}a+\frac{319}{107} 1 0 7 4 0 a 5 + 1 0 7 1 1 a 4 − 1 0 7 8 7 4 a 3 − 1 0 7 9 0 9 a 2 + 1 0 7 1 9 6 7 a + 1 0 7 3 1 9 , 22 107 a 5 − 10 107 a 4 − 470 107 a 3 − 195 107 a 2 + 1130 107 a − 397 107 \frac{22}{107}a^{5}-\frac{10}{107}a^{4}-\frac{470}{107}a^{3}-\frac{195}{107}a^{2}+\frac{1130}{107}a-\frac{397}{107} 1 0 7 2 2 a 5 − 1 0 7 1 0 a 4 − 1 0 7 4 7 0 a 3 − 1 0 7 1 9 5 a 2 + 1 0 7 1 1 3 0 a − 1 0 7 3 9 7 , 62 107 a 5 + 1 107 a 4 − 1344 107 a 3 − 1104 107 a 2 + 3204 107 a − 78 107 \frac{62}{107}a^{5}+\frac{1}{107}a^{4}-\frac{1344}{107}a^{3}-\frac{1104}{107}a^{2}+\frac{3204}{107}a-\frac{78}{107} 1 0 7 6 2 a 5 + 1 0 7 1 a 4 − 1 0 7 1 3 4 4 a 3 − 1 0 7 1 1 0 4 a 2 + 1 0 7 3 2 0 4 a − 1 0 7 7 8
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 : 81.4434827116 81.4434827116 8 1 . 4 4 3 4 8 2 7 1 1 6
sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
lim s → 1 ( s − 1 ) ζ K ( s ) = ( 2 r 1 ⋅ ( 2 π ) r 2 ⋅ R ⋅ h w ⋅ ∣ D ∣ ≈ ( 2 6 ⋅ ( 2 π ) 0 ⋅ 81.4434827116 ⋅ 1 2 ⋅ 286315237 ≈ ( 0.154022470373
\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)^{0}\cdot 81.4434827116 \cdot 1}{2\cdot\sqrt{286315237}}\cr\approx \mathstrut & 0.154022470373
\end{aligned} s → 1 lim ( s − 1 ) ζ K ( s ) = ( ≈ ( ≈ ( w ⋅ ∣ D ∣ 2 r 1 ⋅ ( 2 π ) r 2 ⋅ R ⋅ h 2 ⋅ 2 8 6 3 1 5 2 3 7 2 6 ⋅ ( 2 π ) 0 ⋅ 8 1 . 4 4 3 4 8 2 7 1 1 6 ⋅ 1 0 . 1 5 4 0 2 2 4 7 0 3 7 3
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*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))))
gp: \\ self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*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)))]
magma: /* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*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)));
oscar: # self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^6 - x^5 - 22*x^4 + 5*x^3 + 73*x^2 - 58*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))))
C 6 C_6 C 6 (as 6T1 ):
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)
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]
p p p
2 2 2
3 3 3
5 5 5
7 7 7
11 11 1 1
13 13 1 3
17 17 1 7
19 19 1 9
23 23 2 3
29 29 2 9
31 31 3 1
37 37 3 7
41 41 4 1
43 43 4 3
47 47 4 7
53 53 5 3
59 59 5 9
Cycle type
6 {\href{/padicField/2.6.0.1}{6} } 6
3 2 {\href{/padicField/3.3.0.1}{3} }^{2} 3 2
6 {\href{/padicField/5.6.0.1}{6} } 6
2 3 {\href{/padicField/7.2.0.1}{2} }^{3} 2 3
2 3 {\href{/padicField/11.2.0.1}{2} }^{3} 2 3
R
3 2 {\href{/padicField/17.3.0.1}{3} }^{2} 3 2
R
3 2 {\href{/padicField/23.3.0.1}{3} }^{2} 3 2
3 2 {\href{/padicField/29.3.0.1}{3} }^{2} 3 2
2 3 {\href{/padicField/31.2.0.1}{2} }^{3} 2 3
2 3 {\href{/padicField/37.2.0.1}{2} }^{3} 2 3
6 {\href{/padicField/41.6.0.1}{6} } 6
3 2 {\href{/padicField/43.3.0.1}{3} }^{2} 3 2
6 {\href{/padicField/47.6.0.1}{6} } 6
3 2 {\href{/padicField/53.3.0.1}{3} }^{2} 3 2
6 {\href{/padicField/59.6.0.1}{6} } 6
In the table, R denotes a ramified prime.
Cycle lengths which are repeated in a cycle type are indicated by
exponents.
sage: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: \\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
magma: // to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
oscar: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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]
p p p Label Polynomial
e e e
f f f
c c c
Galois group
Slope content
13 13 1 3
13.3.2.3a1.2 x 6 + 4 x 4 + 22 x 3 + 4 x 2 + 44 x + 134 x^{6} + 4 x^{4} + 22 x^{3} + 4 x^{2} + 44 x + 134 x 6 + 4 x 4 + 2 2 x 3 + 4 x 2 + 4 4 x + 1 3 4 2 2 2 3 3 3 3 3 3 C 6 C_6 C 6 [ ] 2 3 [\ ]_{2}^{3} [ ] 2 3
19 19 1 9
19.2.3.4a1.2 x 6 + 54 x 5 + 978 x 4 + 6048 x 3 + 1956 x 2 + 216 x + 27 x^{6} + 54 x^{5} + 978 x^{4} + 6048 x^{3} + 1956 x^{2} + 216 x + 27 x 6 + 5 4 x 5 + 9 7 8 x 4 + 6 0 4 8 x 3 + 1 9 5 6 x 2 + 2 1 6 x + 2 7 3 3 3 2 2 2 4 4 4 C 6 C_6 C 6 [ ] 3 2 [\ ]_{3}^{2} [ ] 3 2
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)