Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 369*x^10 + 568*x^9 + 58059*x^8 + 114606*x^7 - 3548525*x^6 - 9052842*x^5 + 135909666*x^4 + 393289626*x^3 - 1893538260*x^2 + 975185460*x + 49479090012)
gp: K = bnfinit(y^12 - 6*y^11 - 369*y^10 + 568*y^9 + 58059*y^8 + 114606*y^7 - 3548525*y^6 - 9052842*y^5 + 135909666*y^4 + 393289626*y^3 - 1893538260*y^2 + 975185460*y + 49479090012, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 - 369*x^10 + 568*x^9 + 58059*x^8 + 114606*x^7 - 3548525*x^6 - 9052842*x^5 + 135909666*x^4 + 393289626*x^3 - 1893538260*x^2 + 975185460*x + 49479090012);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 - 369*x^10 + 568*x^9 + 58059*x^8 + 114606*x^7 - 3548525*x^6 - 9052842*x^5 + 135909666*x^4 + 393289626*x^3 - 1893538260*x^2 + 975185460*x + 49479090012)
\( x^{12} - 6 x^{11} - 369 x^{10} + 568 x^{9} + 58059 x^{8} + 114606 x^{7} - 3548525 x^{6} + \cdots + 49479090012 \)
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: |
\(2301971300255880980351195824191744\)
\(\medspace = 2^{8}\cdot 3^{18}\cdot 13^{6}\cdot 37^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(602.80\) | 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: | $2^{2/3}3^{3/2}13^{1/2}37^{5/6}\approx 602.802426811825$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\), \(37\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}$, $\frac{1}{6}a^{7}+\frac{1}{3}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{8}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{345391002}a^{10}-\frac{14240476}{172695501}a^{9}-\frac{216628}{172695501}a^{8}+\frac{12964222}{172695501}a^{7}-\frac{9032378}{172695501}a^{6}-\frac{85642123}{345391002}a^{5}+\frac{4036259}{38376778}a^{4}+\frac{21916457}{115130334}a^{3}+\frac{2393390}{57565167}a^{2}+\frac{5669256}{19188389}a-\frac{98665}{1744399}$, $\frac{1}{19\!\cdots\!38}a^{11}+\frac{23\!\cdots\!61}{19\!\cdots\!38}a^{10}-\frac{81\!\cdots\!76}{97\!\cdots\!19}a^{9}+\frac{71\!\cdots\!64}{97\!\cdots\!19}a^{8}+\frac{25\!\cdots\!73}{97\!\cdots\!19}a^{7}-\frac{10\!\cdots\!21}{19\!\cdots\!38}a^{6}-\frac{22\!\cdots\!87}{10\!\cdots\!91}a^{5}-\frac{33\!\cdots\!70}{32\!\cdots\!73}a^{4}-\frac{11\!\cdots\!51}{65\!\cdots\!46}a^{3}+\frac{21\!\cdots\!70}{10\!\cdots\!91}a^{2}+\frac{51\!\cdots\!33}{10\!\cdots\!91}a-\frac{16\!\cdots\!17}{98\!\cdots\!81}$
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
$C_{3}\times C_{12}\times C_{283476}$, which has order $10205136$ (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: |
$\frac{17\!\cdots\!80}{45\!\cdots\!67}a^{11}-\frac{20\!\cdots\!50}{45\!\cdots\!67}a^{10}-\frac{56\!\cdots\!48}{45\!\cdots\!67}a^{9}+\frac{49\!\cdots\!30}{45\!\cdots\!67}a^{8}+\frac{77\!\cdots\!90}{45\!\cdots\!67}a^{7}-\frac{36\!\cdots\!79}{45\!\cdots\!67}a^{6}-\frac{44\!\cdots\!97}{45\!\cdots\!67}a^{5}+\frac{18\!\cdots\!45}{45\!\cdots\!67}a^{4}+\frac{13\!\cdots\!51}{45\!\cdots\!67}a^{3}-\frac{42\!\cdots\!24}{45\!\cdots\!67}a^{2}-\frac{68\!\cdots\!94}{45\!\cdots\!67}a+\frac{10\!\cdots\!17}{45\!\cdots\!67}$, $\frac{14\!\cdots\!34}{32\!\cdots\!73}a^{11}+\frac{22\!\cdots\!31}{10\!\cdots\!91}a^{10}+\frac{48\!\cdots\!30}{32\!\cdots\!73}a^{9}+\frac{55\!\cdots\!88}{10\!\cdots\!91}a^{8}-\frac{58\!\cdots\!93}{29\!\cdots\!43}a^{7}-\frac{59\!\cdots\!93}{65\!\cdots\!46}a^{6}+\frac{35\!\cdots\!77}{65\!\cdots\!46}a^{5}+\frac{18\!\cdots\!83}{59\!\cdots\!86}a^{4}-\frac{16\!\cdots\!05}{21\!\cdots\!82}a^{3}-\frac{16\!\cdots\!82}{10\!\cdots\!91}a^{2}+\frac{29\!\cdots\!71}{10\!\cdots\!91}a+\frac{13\!\cdots\!07}{98\!\cdots\!81}$, $\frac{22\!\cdots\!56}{10\!\cdots\!91}a^{11}+\frac{71\!\cdots\!04}{32\!\cdots\!73}a^{10}+\frac{22\!\cdots\!91}{32\!\cdots\!73}a^{9}-\frac{17\!\cdots\!31}{32\!\cdots\!73}a^{8}-\frac{31\!\cdots\!77}{32\!\cdots\!73}a^{7}+\frac{24\!\cdots\!51}{65\!\cdots\!46}a^{6}+\frac{36\!\cdots\!53}{65\!\cdots\!46}a^{5}-\frac{12\!\cdots\!39}{65\!\cdots\!46}a^{4}-\frac{39\!\cdots\!35}{21\!\cdots\!82}a^{3}+\frac{44\!\cdots\!58}{10\!\cdots\!91}a^{2}+\frac{16\!\cdots\!57}{98\!\cdots\!81}a-\frac{15\!\cdots\!39}{98\!\cdots\!81}$, $\frac{14\!\cdots\!45}{32\!\cdots\!73}a^{11}-\frac{33\!\cdots\!01}{65\!\cdots\!46}a^{10}-\frac{93\!\cdots\!79}{65\!\cdots\!46}a^{9}+\frac{12\!\cdots\!06}{10\!\cdots\!91}a^{8}+\frac{65\!\cdots\!87}{32\!\cdots\!73}a^{7}-\frac{27\!\cdots\!08}{32\!\cdots\!73}a^{6}-\frac{38\!\cdots\!81}{32\!\cdots\!73}a^{5}+\frac{27\!\cdots\!05}{65\!\cdots\!46}a^{4}+\frac{81\!\cdots\!95}{21\!\cdots\!82}a^{3}-\frac{95\!\cdots\!78}{10\!\cdots\!91}a^{2}-\frac{32\!\cdots\!26}{98\!\cdots\!81}a+\frac{30\!\cdots\!45}{98\!\cdots\!81}$, $\frac{68\!\cdots\!38}{10\!\cdots\!91}a^{11}-\frac{74\!\cdots\!35}{10\!\cdots\!91}a^{10}-\frac{23\!\cdots\!41}{10\!\cdots\!91}a^{9}+\frac{11\!\cdots\!77}{65\!\cdots\!46}a^{8}+\frac{92\!\cdots\!98}{29\!\cdots\!43}a^{7}-\frac{13\!\cdots\!62}{10\!\cdots\!91}a^{6}-\frac{65\!\cdots\!11}{32\!\cdots\!73}a^{5}+\frac{34\!\cdots\!57}{59\!\cdots\!86}a^{4}+\frac{75\!\cdots\!02}{10\!\cdots\!91}a^{3}-\frac{10\!\cdots\!60}{10\!\cdots\!91}a^{2}-\frac{10\!\cdots\!73}{10\!\cdots\!91}a+\frac{28\!\cdots\!87}{98\!\cdots\!81}$
| 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
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 207114.39572456083 \cdot 10205136}{2\cdot\sqrt{2301971300255880980351195824191744}}\cr\approx \mathstrut & 1.35527786391201 \end{aligned}\] (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 - 369*x^10 + 568*x^9 + 58059*x^8 + 114606*x^7 - 3548525*x^6 - 9052842*x^5 + 135909666*x^4 + 393289626*x^3 - 1893538260*x^2 + 975185460*x + 49479090012)
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 - 369*x^10 + 568*x^9 + 58059*x^8 + 114606*x^7 - 3548525*x^6 - 9052842*x^5 + 135909666*x^4 + 393289626*x^3 - 1893538260*x^2 + 975185460*x + 49479090012, 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 - 369*x^10 + 568*x^9 + 58059*x^8 + 114606*x^7 - 3548525*x^6 - 9052842*x^5 + 135909666*x^4 + 393289626*x^3 - 1893538260*x^2 + 975185460*x + 49479090012);
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 - 369*x^10 + 568*x^9 + 58059*x^8 + 114606*x^7 - 3548525*x^6 - 9052842*x^5 + 135909666*x^4 + 393289626*x^3 - 1893538260*x^2 + 975185460*x + 49479090012);
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
| \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-1443}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{37}, \sqrt{-39})\), 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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{6}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | 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 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
|
\(3\)
| 3.6.9.9 | $x^{6} + 6 x^{4} + 21$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
|
\(13\)
| 13.12.6.1 | $x^{12} + 780 x^{11} + 253578 x^{10} + 43990720 x^{9} + 4297346257 x^{8} + 224493831662 x^{7} + 4938918346310 x^{6} + 2961720498866 x^{5} + 3005850529646 x^{4} + 51307643736852 x^{3} + 70292613843513 x^{2} + 65587287977710 x + 15475747398037$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(37\)
| 37.12.10.1 | $x^{12} + 198 x^{11} + 16347 x^{10} + 720720 x^{9} + 17919555 x^{8} + 239132718 x^{7} + 1363015503 x^{6} + 478272762 x^{5} + 72280395 x^{4} + 32212620 x^{3} + 653616987 x^{2} + 8608429962 x + 47259217318$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |