Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 3*x^10 + 4*x^9 - 16*x^8 - 10*x^7 + 32*x^6 + 61*x^5 + 44*x^4 + 4*x^3 - 10*x^2 - 2*x + 1)
gp: K = bnfinit(y^12 - 3*y^11 + 3*y^10 + 4*y^9 - 16*y^8 - 10*y^7 + 32*y^6 + 61*y^5 + 44*y^4 + 4*y^3 - 10*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 + 3*x^10 + 4*x^9 - 16*x^8 - 10*x^7 + 32*x^6 + 61*x^5 + 44*x^4 + 4*x^3 - 10*x^2 - 2*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 + 3*x^10 + 4*x^9 - 16*x^8 - 10*x^7 + 32*x^6 + 61*x^5 + 44*x^4 + 4*x^3 - 10*x^2 - 2*x + 1)
\( x^{12} - 3 x^{11} + 3 x^{10} + 4 x^{9} - 16 x^{8} - 10 x^{7} + 32 x^{6} + 61 x^{5} + 44 x^{4} + 4 x^{3} + \cdots + 1 \)
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: |
\(21347865140625\)
\(\medspace = 3^{6}\cdot 5^{6}\cdot 37^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.91\) | 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: | $3^{1/2}5^{1/2}37^{2/3}\approx 43.004454742078806$ | ||
| Ramified primes: |
\(3\), \(5\), \(37\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{34997}a^{11}-\frac{8762}{34997}a^{10}-\frac{2060}{34997}a^{9}-\frac{14908}{34997}a^{8}+\frac{5349}{34997}a^{7}+\frac{9082}{34997}a^{6}-\frac{1025}{34997}a^{5}-\frac{16193}{34997}a^{4}-\frac{8310}{34997}a^{3}-\frac{6466}{34997}a^{2}+\frac{10538}{34997}a-\frac{15255}{34997}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: |
\( \frac{1647}{443} a^{11} - \frac{6491}{443} a^{10} + \frac{10749}{443} a^{9} - \frac{2416}{443} a^{8} - \frac{25832}{443} a^{7} + \frac{8133}{443} a^{6} + \frac{49271}{443} a^{5} + \frac{54104}{443} a^{4} + \frac{13205}{443} a^{3} - \frac{18388}{443} a^{2} - \frac{5970}{443} a + \frac{2861}{443} \)
(order $6$)
| 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: |
$\frac{27501}{34997}a^{11}-\frac{114408}{34997}a^{10}+\frac{183068}{34997}a^{9}+\frac{4947}{34997}a^{8}-\frac{584491}{34997}a^{7}+\frac{340463}{34997}a^{6}+\frac{998973}{34997}a^{5}+\frac{643078}{34997}a^{4}-\frac{492858}{34997}a^{3}-\frac{946628}{34997}a^{2}-\frac{214601}{34997}a+\frac{121272}{34997}$, $\frac{67615}{34997}a^{11}-\frac{258393}{34997}a^{10}+\frac{421124}{34997}a^{9}-\frac{90820}{34997}a^{8}-\frac{1001276}{34997}a^{7}+\frac{197053}{34997}a^{6}+\frac{1878523}{34997}a^{5}+\frac{2511234}{34997}a^{4}+\frac{1116089}{34997}a^{3}-\frac{226048}{34997}a^{2}-\frac{152038}{34997}a+\frac{34753}{34997}$, $\frac{39157}{34997}a^{11}-\frac{158031}{34997}a^{10}+\frac{284641}{34997}a^{9}-\frac{142584}{34997}a^{8}-\frac{496210}{34997}a^{7}+\frac{194342}{34997}a^{6}+\frac{950553}{34997}a^{5}+\frac{1301234}{34997}a^{4}+\frac{567388}{34997}a^{3}-\frac{90858}{34997}a^{2}-\frac{48158}{34997}a-\frac{46236}{34997}$, $\frac{57175}{34997}a^{11}-\frac{195277}{34997}a^{10}+\frac{229384}{34997}a^{9}+\frac{232014}{34997}a^{8}-\frac{1199606}{34997}a^{7}+\frac{47858}{34997}a^{6}+\frac{2080423}{34997}a^{5}+\frac{2460650}{34997}a^{4}+\frac{939941}{34997}a^{3}-\frac{825170}{34997}a^{2}-\frac{383169}{34997}a+\frac{165594}{34997}$, $\frac{119118}{34997}a^{11}-\frac{451346}{34997}a^{10}+\frac{715824}{34997}a^{9}-\frac{98361}{34997}a^{8}-\frac{1813044}{34997}a^{7}+\frac{247391}{34997}a^{6}+\frac{3543280}{34997}a^{5}+\frac{4531491}{34997}a^{4}+\frac{1769415}{34997}a^{3}-\frac{947931}{34997}a^{2}-\frac{496870}{34997}a+\frac{144129}{34997}$
| 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: | \( 71.67111211927603 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 71.67111211927603 \cdot 1}{6\cdot\sqrt{21347865140625}}\cr\approx \mathstrut & 0.159072416261730 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 3*x^10 + 4*x^9 - 16*x^8 - 10*x^7 + 32*x^6 + 61*x^5 + 44*x^4 + 4*x^3 - 10*x^2 - 2*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 + 3*x^10 + 4*x^9 - 16*x^8 - 10*x^7 + 32*x^6 + 61*x^5 + 44*x^4 + 4*x^3 - 10*x^2 - 2*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 + 3*x^10 + 4*x^9 - 16*x^8 - 10*x^7 + 32*x^6 + 61*x^5 + 44*x^4 + 4*x^3 - 10*x^2 - 2*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 + 3*x^10 + 4*x^9 - 16*x^8 - 10*x^7 + 32*x^6 + 61*x^5 + 44*x^4 + 4*x^3 - 10*x^2 - 2*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
$C_6\times S_3$ (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
| \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.36963.1 |
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: | 18.6.9372992211440206484080078125.1, 18.0.253070789708885575070162109375.1 |
| 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 | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(5\)
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(37\)
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.6.4.1 | $x^{6} + 99 x^{5} + 3273 x^{4} + 36407 x^{3} + 10209 x^{2} + 120831 x + 1323720$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
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) | $1$ | $1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.555.6t1.a.a | $1$ | $ 3 \cdot 5 \cdot 37 $ | 6.0.6325293375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.185.6t1.b.a | $1$ | $ 5 \cdot 37 $ | 6.6.234270125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.185.6t1.b.b | $1$ | $ 5 \cdot 37 $ | 6.6.234270125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.111.6t1.b.a | $1$ | $ 3 \cdot 37 $ | 6.0.50602347.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.555.6t1.a.b | $1$ | $ 3 \cdot 5 \cdot 37 $ | 6.0.6325293375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.37.3t1.a.a | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.37.3t1.a.b | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.111.6t1.b.b | $1$ | $ 3 \cdot 37 $ | 6.0.50602347.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 2.4107.3t2.a.a | $2$ | $ 3 \cdot 37^{2}$ | 3.1.4107.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.102675.6t3.b.a | $2$ | $ 3 \cdot 5^{2} \cdot 37^{2}$ | 6.2.2108431125.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.111.6t5.a.a | $2$ | $ 3 \cdot 37 $ | 6.0.36963.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.2775.12t18.a.a | $2$ | $ 3 \cdot 5^{2} \cdot 37 $ | 12.0.21347865140625.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.2775.12t18.a.b | $2$ | $ 3 \cdot 5^{2} \cdot 37 $ | 12.0.21347865140625.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.111.6t5.a.b | $2$ | $ 3 \cdot 37 $ | 6.0.36963.1 | $S_3\times C_3$ (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 *.