Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 1)
gp: K = bnfinit(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*x + 1)
\( x^{12} - 3 x^{11} + 5 x^{10} - 8 x^{9} + 11 x^{8} - 13 x^{7} + 18 x^{6} - 13 x^{5} + 11 x^{4} - 8 x^{3} + 5 x^{2} - 3 x + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(470596000000\)
\(\medspace = 2^{8}\cdot 5^{6}\cdot 7^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.39\) | 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))
| |
| Ramified primes: |
\(2\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{10}a^{10}-\frac{1}{10}a^{8}-\frac{1}{10}a^{7}-\frac{1}{10}a^{6}-\frac{1}{10}a^{4}+\frac{2}{5}a^{3}-\frac{1}{10}a^{2}-\frac{1}{2}a+\frac{1}{10}$, $\frac{1}{20}a^{11}-\frac{1}{20}a^{9}-\frac{1}{20}a^{8}+\frac{1}{5}a^{7}-\frac{1}{4}a^{6}-\frac{1}{20}a^{5}+\frac{1}{5}a^{4}+\frac{9}{20}a^{3}+\frac{1}{4}a^{2}-\frac{1}{5}a+\frac{1}{4}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
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: | $5$ | 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: |
$\frac{7}{10}a^{11}-\frac{19}{10}a^{10}+\frac{33}{10}a^{9}-\frac{53}{10}a^{8}+\frac{67}{10}a^{7}-\frac{81}{10}a^{6}+\frac{113}{10}a^{5}-\frac{63}{10}a^{4}+\frac{77}{10}a^{3}-\frac{21}{10}a^{2}+\frac{27}{10}a-\frac{19}{10}$, $\frac{2}{5}a^{11}-\frac{3}{10}a^{10}-\frac{2}{5}a^{9}+\frac{2}{5}a^{8}-\frac{11}{10}a^{7}+\frac{9}{5}a^{6}-\frac{2}{5}a^{5}+\frac{59}{10}a^{4}-\frac{3}{5}a^{3}+\frac{9}{5}a^{2}-\frac{1}{10}a+\frac{1}{5}$, $\frac{13}{20}a^{11}-\frac{13}{10}a^{10}+\frac{37}{20}a^{9}-\frac{67}{20}a^{8}+\frac{39}{10}a^{7}-\frac{89}{20}a^{6}+\frac{147}{20}a^{5}-\frac{11}{10}a^{4}+\frac{103}{20}a^{3}-\frac{9}{20}a^{2}+\frac{9}{10}a-\frac{11}{20}$, $\frac{19}{20}a^{11}-\frac{23}{10}a^{10}+\frac{71}{20}a^{9}-\frac{123}{20}a^{8}+\frac{81}{10}a^{7}-\frac{189}{20}a^{6}+\frac{281}{20}a^{5}-\frac{69}{10}a^{4}+\frac{197}{20}a^{3}-\frac{109}{20}a^{2}+\frac{27}{10}a-\frac{31}{20}$, $\frac{17}{20}a^{11}-\frac{12}{5}a^{10}+\frac{73}{20}a^{9}-\frac{119}{20}a^{8}+\frac{83}{10}a^{7}-\frac{187}{20}a^{6}+\frac{263}{20}a^{5}-\frac{41}{5}a^{4}+\frac{131}{20}a^{3}-\frac{117}{20}a^{2}+\frac{21}{10}a-\frac{13}{20}$
| 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: | \( 6.67082664998 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) = \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}} \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 6.67082664998 \cdot 1}{2\cdot\sqrt{470596000000}}\approx 0.299160846811$
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*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))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*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)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*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)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 + 5*x^10 - 8*x^9 + 11*x^8 - 13*x^7 + 18*x^6 - 13*x^5 + 11*x^4 - 8*x^3 + 5*x^2 - 3*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))))
Galois group
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 12 |
| The 6 conjugacy class representatives for $D_6$ |
| Character table for $D_6$ |
Intermediate fields
| \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), 3.1.140.1 x3, \(\Q(\sqrt{5}, \sqrt{-7})\), 6.0.686000.1, 6.0.137200.1 x3, 6.2.98000.1 x3 |
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
| Degree 6 siblings: | 6.0.137200.1, 6.2.98000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |