Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 6*x^10 - 3*x^9 + 3*x^8 - 9*x^7 - 9*x^6 + 9*x^5 + 3*x^4 + 3*x^3 + 6*x^2 + 1)
gp: K = bnfinit(y^12 + 6*y^10 - 3*y^9 + 3*y^8 - 9*y^7 - 9*y^6 + 9*y^5 + 3*y^4 + 3*y^3 + 6*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 6*x^10 - 3*x^9 + 3*x^8 - 9*x^7 - 9*x^6 + 9*x^5 + 3*x^4 + 3*x^3 + 6*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 6*x^10 - 3*x^9 + 3*x^8 - 9*x^7 - 9*x^6 + 9*x^5 + 3*x^4 + 3*x^3 + 6*x^2 + 1)
\( x^{12} + 6x^{10} - 3x^{9} + 3x^{8} - 9x^{7} - 9x^{6} + 9x^{5} + 3x^{4} + 3x^{3} + 6x^{2} + 1 \)
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: |
\(12060860765625\)
\(\medspace = 3^{8}\cdot 5^{6}\cdot 7^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.31\) | 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: | $3^{4/3}5^{1/2}7^{1/2}\approx 25.597390575239302$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{27}a^{10}-\frac{4}{27}a^{9}-\frac{4}{27}a^{8}+\frac{1}{3}a^{7}-\frac{10}{27}a^{6}+\frac{13}{27}a^{5}+\frac{10}{27}a^{4}+\frac{1}{3}a^{3}+\frac{4}{27}a^{2}-\frac{4}{27}a-\frac{1}{27}$, $\frac{1}{351}a^{11}+\frac{5}{351}a^{10}+\frac{5}{351}a^{9}+\frac{1}{39}a^{8}+\frac{152}{351}a^{7}+\frac{49}{351}a^{6}+\frac{145}{351}a^{5}+\frac{5}{39}a^{4}+\frac{85}{351}a^{3}+\frac{77}{351}a^{2}-\frac{64}{351}a-\frac{11}{39}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{88}{351}a^{11}+\frac{206}{351}a^{10}+\frac{557}{351}a^{9}+\frac{101}{39}a^{8}-\frac{196}{351}a^{7}-\frac{602}{351}a^{6}-\frac{2450}{351}a^{5}-\frac{145}{39}a^{4}+\frac{2332}{351}a^{3}+\frac{1745}{351}a^{2}+\frac{452}{351}a+\frac{24}{13}$, $\frac{19}{39}a^{11}-\frac{40}{117}a^{10}+\frac{337}{117}a^{9}-\frac{410}{117}a^{8}+\frac{31}{13}a^{7}-\frac{626}{117}a^{6}-\frac{55}{117}a^{5}+\frac{797}{117}a^{4}-\frac{10}{39}a^{3}-\frac{187}{117}a^{2}+\frac{187}{117}a-\frac{92}{117}$, $a$, $\frac{112}{117}a^{11}-\frac{64}{117}a^{10}+\frac{638}{117}a^{9}-\frac{236}{39}a^{8}+\frac{332}{117}a^{7}-\frac{1025}{117}a^{6}-\frac{530}{117}a^{5}+\frac{640}{39}a^{4}-\frac{35}{117}a^{3}-\frac{307}{117}a^{2}+\frac{437}{117}a-\frac{36}{13}$, $\frac{206}{351}a^{11}+\frac{29}{351}a^{10}+\frac{391}{117}a^{9}-\frac{460}{351}a^{8}+\frac{190}{351}a^{7}-\frac{1658}{351}a^{6}-\frac{233}{39}a^{5}+\frac{2068}{351}a^{4}+\frac{1481}{351}a^{3}-\frac{76}{351}a^{2}+\frac{37}{13}a+\frac{263}{351}$
| 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: | \( 42.05225160280841 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 42.05225160280841 \cdot 1}{2\cdot\sqrt{12060860765625}}\cr\approx \mathstrut & 0.372519761928199 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 + 6*x^10 - 3*x^9 + 3*x^8 - 9*x^7 - 9*x^6 + 9*x^5 + 3*x^4 + 3*x^3 + 6*x^2 + 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 + 6*x^10 - 3*x^9 + 3*x^8 - 9*x^7 - 9*x^6 + 9*x^5 + 3*x^4 + 3*x^3 + 6*x^2 + 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 + 6*x^10 - 3*x^9 + 3*x^8 - 9*x^7 - 9*x^6 + 9*x^5 + 3*x^4 + 3*x^3 + 6*x^2 + 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 + 6*x^10 - 3*x^9 + 3*x^8 - 9*x^7 - 9*x^6 + 9*x^5 + 3*x^4 + 3*x^3 + 6*x^2 + 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
$C_6\times S_3$ (as 12T18):
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 36 |
| The 18 conjugacy class representatives for $C_6\times S_3$ |
| Character table for $C_6\times S_3$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 6.0.3472875.1 |
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
| Galois closure: | deg 36 |
| Degree 18 siblings: | deg 18, 18.0.178078914380413077609375.1 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | R | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.6.8.3 | $x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ | $3$ | $2$ | $8$ | $C_6$ | $[2]^{2}$ | |
|
\(5\)
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(7\)
| 7.12.6.1 | $x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.315.6t1.h.a | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.281302875.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.315.6t1.h.b | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.281302875.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.63.6t1.c.a | $1$ | $ 3^{2} \cdot 7 $ | 6.0.2250423.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.45.6t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.45.6t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.63.6t1.c.b | $1$ | $ 3^{2} \cdot 7 $ | 6.0.2250423.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 2.2835.3t2.a.a | $2$ | $ 3^{4} \cdot 5 \cdot 7 $ | 3.1.2835.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.2835.6t3.c.a | $2$ | $ 3^{4} \cdot 5 \cdot 7 $ | 6.2.40186125.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.315.12t18.a.a | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.0.12060860765625.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.315.6t5.a.a | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.3472875.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.315.12t18.a.b | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 12.0.12060860765625.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.315.6t5.a.b | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.3472875.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
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 *.