Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 167*x^10 - 1246*x^9 + 24756*x^8 - 132978*x^7 + 1130569*x^6 - 3335608*x^5 + 25742792*x^4 - 60568948*x^3 + 342253035*x^2 + 20074400*x + 1190281)
gp: K = bnfinit(y^12 - 2*y^11 + 167*y^10 - 1246*y^9 + 24756*y^8 - 132978*y^7 + 1130569*y^6 - 3335608*y^5 + 25742792*y^4 - 60568948*y^3 + 342253035*y^2 + 20074400*y + 1190281, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 + 167*x^10 - 1246*x^9 + 24756*x^8 - 132978*x^7 + 1130569*x^6 - 3335608*x^5 + 25742792*x^4 - 60568948*x^3 + 342253035*x^2 + 20074400*x + 1190281);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 + 167*x^10 - 1246*x^9 + 24756*x^8 - 132978*x^7 + 1130569*x^6 - 3335608*x^5 + 25742792*x^4 - 60568948*x^3 + 342253035*x^2 + 20074400*x + 1190281)
\( x^{12} - 2 x^{11} + 167 x^{10} - 1246 x^{9} + 24756 x^{8} - 132978 x^{7} + 1130569 x^{6} + \cdots + 1190281 \)
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: |
\(279442937001407746455186229208699136\)
\(\medspace = 2^{8}\cdot 3^{6}\cdot 11^{6}\cdot 13^{10}\cdot 19^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(899.20\) | 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: | $2^{2/3}3^{1/2}11^{1/2}13^{5/6}19^{5/6}\approx 899.2030295511601$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\), \(13\), \(19\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{542}a^{10}+\frac{35}{271}a^{9}+\frac{23}{542}a^{8}+\frac{64}{271}a^{7}+\frac{105}{542}a^{6}+\frac{91}{271}a^{5}-\frac{99}{542}a^{4}-\frac{44}{271}a^{3}-\frac{201}{542}a^{2}+\frac{42}{271}a-\frac{107}{542}$, $\frac{1}{12\!\cdots\!66}a^{11}-\frac{10\!\cdots\!77}{12\!\cdots\!66}a^{10}+\frac{11\!\cdots\!16}{60\!\cdots\!83}a^{9}-\frac{11\!\cdots\!74}{60\!\cdots\!83}a^{8}+\frac{11\!\cdots\!78}{60\!\cdots\!83}a^{7}+\frac{11\!\cdots\!21}{12\!\cdots\!66}a^{6}+\frac{80\!\cdots\!11}{12\!\cdots\!66}a^{5}-\frac{31\!\cdots\!17}{12\!\cdots\!66}a^{4}-\frac{25\!\cdots\!12}{60\!\cdots\!83}a^{3}+\frac{22\!\cdots\!49}{60\!\cdots\!83}a^{2}-\frac{22\!\cdots\!61}{60\!\cdots\!83}a+\frac{16\!\cdots\!17}{11\!\cdots\!26}$
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
$C_{3}\times C_{6}\times C_{6}\times C_{178662}$, which has order $19295496$ (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: |
\( \frac{1104275774544408105169912}{22315938126273364211338100369773} a^{11} - \frac{2221226333382221377090028}{22315938126273364211338100369773} a^{10} + \frac{184187854407656776177513118}{22315938126273364211338100369773} a^{9} - \frac{1378160302726217578056273481}{22315938126273364211338100369773} a^{8} + \frac{27316432908721986448249792168}{22315938126273364211338100369773} a^{7} - \frac{146969608742931861558299353276}{22315938126273364211338100369773} a^{6} + \frac{1245217378148184102332848720622}{22315938126273364211338100369773} a^{5} - \frac{3685833742550290806998557842744}{22315938126273364211338100369773} a^{4} + \frac{28357947233214022187898250231264}{22315938126273364211338100369773} a^{3} - \frac{67793372330307910230213077929992}{22315938126273364211338100369773} a^{2} + \frac{376393670947972548439061415236406}{22315938126273364211338100369773} a + \frac{20235439219713677910927419736}{20454572068078244006726031503} \)
(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: |
$\frac{89\!\cdots\!28}{22\!\cdots\!73}a^{11}+\frac{17\!\cdots\!54}{22\!\cdots\!73}a^{10}+\frac{13\!\cdots\!84}{22\!\cdots\!73}a^{9}-\frac{68\!\cdots\!96}{22\!\cdots\!73}a^{8}+\frac{15\!\cdots\!73}{22\!\cdots\!73}a^{7}-\frac{45\!\cdots\!20}{22\!\cdots\!73}a^{6}+\frac{41\!\cdots\!56}{22\!\cdots\!73}a^{5}-\frac{85\!\cdots\!20}{22\!\cdots\!73}a^{4}+\frac{87\!\cdots\!84}{22\!\cdots\!73}a^{3}-\frac{16\!\cdots\!80}{22\!\cdots\!73}a^{2}-\frac{34\!\cdots\!40}{22\!\cdots\!73}a-\frac{53\!\cdots\!08}{20\!\cdots\!03}$, $\frac{13\!\cdots\!65}{60\!\cdots\!83}a^{11}-\frac{70\!\cdots\!72}{60\!\cdots\!83}a^{10}-\frac{16\!\cdots\!19}{60\!\cdots\!83}a^{9}+\frac{52\!\cdots\!93}{12\!\cdots\!66}a^{8}-\frac{13\!\cdots\!42}{60\!\cdots\!83}a^{7}-\frac{62\!\cdots\!28}{60\!\cdots\!83}a^{6}+\frac{41\!\cdots\!09}{60\!\cdots\!83}a^{5}-\frac{10\!\cdots\!33}{12\!\cdots\!66}a^{4}+\frac{11\!\cdots\!59}{60\!\cdots\!83}a^{3}-\frac{42\!\cdots\!46}{60\!\cdots\!83}a^{2}+\frac{91\!\cdots\!22}{60\!\cdots\!83}a-\frac{27\!\cdots\!05}{11\!\cdots\!26}$, $\frac{20\!\cdots\!20}{22\!\cdots\!73}a^{11}+\frac{32\!\cdots\!44}{22\!\cdots\!73}a^{10}-\frac{48\!\cdots\!18}{22\!\cdots\!73}a^{9}+\frac{59\!\cdots\!33}{22\!\cdots\!73}a^{8}-\frac{10\!\cdots\!16}{22\!\cdots\!73}a^{7}+\frac{86\!\cdots\!24}{22\!\cdots\!73}a^{6}-\frac{70\!\cdots\!02}{22\!\cdots\!73}a^{5}+\frac{23\!\cdots\!76}{22\!\cdots\!73}a^{4}-\frac{16\!\cdots\!92}{22\!\cdots\!73}a^{3}+\frac{43\!\cdots\!68}{22\!\cdots\!73}a^{2}-\frac{31\!\cdots\!62}{22\!\cdots\!73}a+\frac{13\!\cdots\!35}{20\!\cdots\!03}$, $\frac{19\!\cdots\!21}{11\!\cdots\!26}a^{11}+\frac{15\!\cdots\!27}{55\!\cdots\!13}a^{10}+\frac{19\!\cdots\!67}{55\!\cdots\!13}a^{9}+\frac{13\!\cdots\!44}{55\!\cdots\!13}a^{8}+\frac{20\!\cdots\!43}{11\!\cdots\!26}a^{7}+\frac{35\!\cdots\!67}{11\!\cdots\!26}a^{6}+\frac{41\!\cdots\!61}{11\!\cdots\!26}a^{5}+\frac{50\!\cdots\!99}{55\!\cdots\!13}a^{4}+\frac{80\!\cdots\!06}{55\!\cdots\!13}a^{3}+\frac{11\!\cdots\!22}{55\!\cdots\!13}a^{2}+\frac{13\!\cdots\!39}{11\!\cdots\!26}a+\frac{78\!\cdots\!89}{11\!\cdots\!26}$, $\frac{50\!\cdots\!31}{60\!\cdots\!83}a^{11}+\frac{20\!\cdots\!11}{12\!\cdots\!66}a^{10}+\frac{73\!\cdots\!92}{60\!\cdots\!83}a^{9}-\frac{38\!\cdots\!46}{60\!\cdots\!83}a^{8}+\frac{17\!\cdots\!35}{12\!\cdots\!66}a^{7}-\frac{24\!\cdots\!62}{60\!\cdots\!83}a^{6}+\frac{22\!\cdots\!48}{60\!\cdots\!83}a^{5}-\frac{87\!\cdots\!65}{12\!\cdots\!66}a^{4}+\frac{47\!\cdots\!39}{60\!\cdots\!83}a^{3}-\frac{89\!\cdots\!14}{60\!\cdots\!83}a^{2}-\frac{10\!\cdots\!33}{12\!\cdots\!66}a-\frac{29\!\cdots\!86}{55\!\cdots\!13}$
| 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: | \( 3173377.1710623284 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3173377.1710623284 \cdot 19295496}{6\cdot\sqrt{279442937001407746455186229208699136}}\cr\approx \mathstrut & 1.18784296828840 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 167*x^10 - 1246*x^9 + 24756*x^8 - 132978*x^7 + 1130569*x^6 - 3335608*x^5 + 25742792*x^4 - 60568948*x^3 + 342253035*x^2 + 20074400*x + 1190281)
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 - 2*x^11 + 167*x^10 - 1246*x^9 + 24756*x^8 - 132978*x^7 + 1130569*x^6 - 3335608*x^5 + 25742792*x^4 - 60568948*x^3 + 342253035*x^2 + 20074400*x + 1190281, 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 - 2*x^11 + 167*x^10 - 1246*x^9 + 24756*x^8 - 132978*x^7 + 1130569*x^6 - 3335608*x^5 + 25742792*x^4 - 60568948*x^3 + 342253035*x^2 + 20074400*x + 1190281);
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 - 2*x^11 + 167*x^10 - 1246*x^9 + 24756*x^8 - 132978*x^7 + 1130569*x^6 - 3335608*x^5 + 25742792*x^4 - 60568948*x^3 + 342253035*x^2 + 20074400*x + 1190281);
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
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2717}) \), \(\Q(\sqrt{-8151}) \), \(\Q(\sqrt{-3}, \sqrt{2717})\), 6.6.19578652781045072.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{6}$ | R | R | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
| 2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(11\)
| 11.12.6.1 | $x^{12} + 72 x^{10} + 8 x^{9} + 2034 x^{8} + 38 x^{7} + 27996 x^{6} - 6312 x^{5} + 196025 x^{4} - 84710 x^{3} + 695581 x^{2} - 235284 x + 1080083$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(13\)
| 13.6.5.1 | $x^{6} + 52$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 13.6.5.1 | $x^{6} + 52$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
|
\(19\)
| 19.6.5.4 | $x^{6} + 76$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 19.6.5.4 | $x^{6} + 76$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |