Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 393*x^10 + 688*x^9 + 62187*x^8 + 97374*x^7 - 3967909*x^6 - 9955650*x^5 + 133133706*x^4 + 364479098*x^3 - 1900721436*x^2 - 1258966476*x + 27667585644)
gp: K = bnfinit(y^12 - 6*y^11 - 393*y^10 + 688*y^9 + 62187*y^8 + 97374*y^7 - 3967909*y^6 - 9955650*y^5 + 133133706*y^4 + 364479098*y^3 - 1900721436*y^2 - 1258966476*y + 27667585644, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 - 393*x^10 + 688*x^9 + 62187*x^8 + 97374*x^7 - 3967909*x^6 - 9955650*x^5 + 133133706*x^4 + 364479098*x^3 - 1900721436*x^2 - 1258966476*x + 27667585644);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 - 393*x^10 + 688*x^9 + 62187*x^8 + 97374*x^7 - 3967909*x^6 - 9955650*x^5 + 133133706*x^4 + 364479098*x^3 - 1900721436*x^2 - 1258966476*x + 27667585644)
\( x^{12} - 6 x^{11} - 393 x^{10} + 688 x^{9} + 62187 x^{8} + 97374 x^{7} - 3967909 x^{6} + \cdots + 27667585644 \)
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: |
\(7844482483973372433572592070830336\)
\(\medspace = 2^{8}\cdot 3^{16}\cdot 23^{6}\cdot 37^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(667.65\) | 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: | $2^{2/3}3^{4/3}23^{1/2}37^{5/6}\approx 667.6471889561029$ | ||
| Ramified primes: |
\(2\), \(3\), \(23\), \(37\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{32}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{10466394}a^{10}-\frac{285284}{5233197}a^{9}+\frac{437041}{3488798}a^{8}-\frac{524949}{3488798}a^{7}+\frac{99626}{1744399}a^{6}-\frac{1005217}{3488798}a^{5}+\frac{30206}{73707}a^{4}-\frac{1184965}{5233197}a^{3}+\frac{5803}{1744399}a^{2}+\frac{789877}{1744399}a+\frac{121525}{1744399}$, $\frac{1}{13\!\cdots\!58}a^{11}-\frac{85\!\cdots\!33}{22\!\cdots\!43}a^{10}+\frac{10\!\cdots\!97}{13\!\cdots\!58}a^{9}-\frac{25\!\cdots\!51}{45\!\cdots\!86}a^{8}+\frac{89\!\cdots\!08}{22\!\cdots\!43}a^{7}+\frac{33\!\cdots\!80}{22\!\cdots\!43}a^{6}-\frac{33\!\cdots\!77}{13\!\cdots\!58}a^{5}-\frac{14\!\cdots\!45}{45\!\cdots\!86}a^{4}-\frac{59\!\cdots\!81}{13\!\cdots\!58}a^{3}+\frac{39\!\cdots\!06}{22\!\cdots\!43}a^{2}-\frac{93\!\cdots\!67}{22\!\cdots\!43}a+\frac{13\!\cdots\!33}{15\!\cdots\!93}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{332955}$, which has order $26969355$ (assuming GRH)
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: |
$\frac{61848826564852}{96\!\cdots\!11}a^{11}+\frac{649169844801682}{96\!\cdots\!11}a^{10}+\frac{22\!\cdots\!76}{96\!\cdots\!11}a^{9}-\frac{57\!\cdots\!18}{32\!\cdots\!37}a^{8}-\frac{10\!\cdots\!54}{32\!\cdots\!37}a^{7}+\frac{45\!\cdots\!09}{32\!\cdots\!37}a^{6}+\frac{19\!\cdots\!17}{96\!\cdots\!11}a^{5}-\frac{63\!\cdots\!13}{96\!\cdots\!11}a^{4}-\frac{55\!\cdots\!95}{96\!\cdots\!11}a^{3}+\frac{48\!\cdots\!68}{32\!\cdots\!37}a^{2}+\frac{13\!\cdots\!74}{32\!\cdots\!37}a-\frac{82\!\cdots\!81}{21\!\cdots\!87}$, $\frac{17\!\cdots\!52}{67\!\cdots\!29}a^{11}-\frac{14\!\cdots\!16}{67\!\cdots\!29}a^{10}-\frac{24\!\cdots\!21}{22\!\cdots\!43}a^{9}+\frac{15\!\cdots\!77}{22\!\cdots\!43}a^{8}+\frac{36\!\cdots\!35}{22\!\cdots\!43}a^{7}-\frac{23\!\cdots\!65}{45\!\cdots\!86}a^{6}-\frac{13\!\cdots\!89}{13\!\cdots\!58}a^{5}+\frac{33\!\cdots\!95}{13\!\cdots\!58}a^{4}+\frac{12\!\cdots\!43}{45\!\cdots\!86}a^{3}-\frac{13\!\cdots\!68}{22\!\cdots\!43}a^{2}-\frac{63\!\cdots\!71}{22\!\cdots\!43}a+\frac{21\!\cdots\!51}{15\!\cdots\!93}$, $\frac{15\!\cdots\!76}{67\!\cdots\!29}a^{11}-\frac{66\!\cdots\!06}{22\!\cdots\!43}a^{10}-\frac{48\!\cdots\!01}{67\!\cdots\!29}a^{9}+\frac{15\!\cdots\!10}{22\!\cdots\!43}a^{8}+\frac{22\!\cdots\!71}{22\!\cdots\!43}a^{7}-\frac{23\!\cdots\!25}{45\!\cdots\!86}a^{6}-\frac{76\!\cdots\!17}{13\!\cdots\!58}a^{5}+\frac{10\!\cdots\!83}{45\!\cdots\!86}a^{4}+\frac{19\!\cdots\!41}{13\!\cdots\!58}a^{3}-\frac{12\!\cdots\!36}{22\!\cdots\!43}a^{2}-\frac{91\!\cdots\!55}{22\!\cdots\!43}a+\frac{12\!\cdots\!83}{15\!\cdots\!93}$, $\frac{24\!\cdots\!63}{22\!\cdots\!43}a^{11}+\frac{53\!\cdots\!83}{45\!\cdots\!86}a^{10}+\frac{16\!\cdots\!43}{45\!\cdots\!86}a^{9}-\frac{12\!\cdots\!57}{45\!\cdots\!86}a^{8}-\frac{11\!\cdots\!78}{22\!\cdots\!43}a^{7}+\frac{84\!\cdots\!75}{45\!\cdots\!86}a^{6}+\frac{12\!\cdots\!57}{45\!\cdots\!86}a^{5}-\frac{37\!\cdots\!79}{45\!\cdots\!86}a^{4}-\frac{16\!\cdots\!85}{22\!\cdots\!43}a^{3}+\frac{49\!\cdots\!17}{22\!\cdots\!43}a^{2}+\frac{95\!\cdots\!66}{22\!\cdots\!43}a-\frac{50\!\cdots\!77}{15\!\cdots\!93}$, $\frac{33\!\cdots\!08}{67\!\cdots\!29}a^{11}+\frac{12\!\cdots\!60}{67\!\cdots\!29}a^{10}+\frac{42\!\cdots\!57}{22\!\cdots\!43}a^{9}+\frac{14\!\cdots\!73}{45\!\cdots\!86}a^{8}-\frac{59\!\cdots\!97}{22\!\cdots\!43}a^{7}-\frac{43\!\cdots\!17}{45\!\cdots\!86}a^{6}+\frac{13\!\cdots\!53}{13\!\cdots\!58}a^{5}+\frac{32\!\cdots\!51}{67\!\cdots\!29}a^{4}-\frac{60\!\cdots\!43}{45\!\cdots\!86}a^{3}-\frac{81\!\cdots\!42}{22\!\cdots\!43}a^{2}+\frac{68\!\cdots\!75}{22\!\cdots\!43}a+\frac{24\!\cdots\!07}{15\!\cdots\!93}$
| 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: | \( 207114.39572456083 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 207114.39572456083 \cdot 26969355}{2\cdot\sqrt{7844482483973372433572592070830336}}\cr\approx \mathstrut & 1.94020630580959 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 393*x^10 + 688*x^9 + 62187*x^8 + 97374*x^7 - 3967909*x^6 - 9955650*x^5 + 133133706*x^4 + 364479098*x^3 - 1900721436*x^2 - 1258966476*x + 27667585644)
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^11 - 393*x^10 + 688*x^9 + 62187*x^8 + 97374*x^7 - 3967909*x^6 - 9955650*x^5 + 133133706*x^4 + 364479098*x^3 - 1900721436*x^2 - 1258966476*x + 27667585644, 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^11 - 393*x^10 + 688*x^9 + 62187*x^8 + 97374*x^7 - 3967909*x^6 - 9955650*x^5 + 133133706*x^4 + 364479098*x^3 - 1900721436*x^2 - 1258966476*x + 27667585644);
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^11 - 393*x^10 + 688*x^9 + 62187*x^8 + 97374*x^7 - 3967909*x^6 - 9955650*x^5 + 133133706*x^4 + 364479098*x^3 - 1900721436*x^2 - 1258966476*x + 27667585644);
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
| \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-851}) \), \(\Q(\sqrt{-23}, \sqrt{37})\), 6.6.7279451230032.1 |
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: | data not computed |
| Degree 18 siblings: | data not computed |
| 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 | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
| 2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
|
\(3\)
| 3.3.4.1 | $x^{3} + 6 x^{2} + 21$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
| 3.3.4.1 | $x^{3} + 6 x^{2} + 21$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
| 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
|
\(23\)
| 23.12.6.1 | $x^{12} + 140 x^{10} + 18 x^{9} + 8092 x^{8} + 20 x^{7} + 244001 x^{6} - 56140 x^{5} + 4059563 x^{4} - 1721304 x^{3} + 36312468 x^{2} - 14694000 x + 141161628$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(37\)
| 37.12.10.1 | $x^{12} + 198 x^{11} + 16347 x^{10} + 720720 x^{9} + 17919555 x^{8} + 239132718 x^{7} + 1363015503 x^{6} + 478272762 x^{5} + 72280395 x^{4} + 32212620 x^{3} + 653616987 x^{2} + 8608429962 x + 47259217318$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |