Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 41*x^10 - 6*x^9 + 1878*x^8 - 4406*x^7 + 63987*x^6 - 136784*x^5 + 1352954*x^4 - 3084580*x^3 + 19917769*x^2 - 31791008*x + 115888143)
gp: K = bnfinit(y^12 - 2*y^11 + 41*y^10 - 6*y^9 + 1878*y^8 - 4406*y^7 + 63987*y^6 - 136784*y^5 + 1352954*y^4 - 3084580*y^3 + 19917769*y^2 - 31791008*y + 115888143, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 + 41*x^10 - 6*x^9 + 1878*x^8 - 4406*x^7 + 63987*x^6 - 136784*x^5 + 1352954*x^4 - 3084580*x^3 + 19917769*x^2 - 31791008*x + 115888143);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 + 41*x^10 - 6*x^9 + 1878*x^8 - 4406*x^7 + 63987*x^6 - 136784*x^5 + 1352954*x^4 - 3084580*x^3 + 19917769*x^2 - 31791008*x + 115888143)
\( x^{12} - 2 x^{11} + 41 x^{10} - 6 x^{9} + 1878 x^{8} - 4406 x^{7} + 63987 x^{6} - 136784 x^{5} + \cdots + 115888143 \)
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: |
\(51924135650907635776859701504\)
\(\medspace = 2^{8}\cdot 37^{10}\cdot 59^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(247.14\) | 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: | $2^{2/3}37^{5/6}59^{1/2}\approx 247.14250235137513$ | ||
| Ramified primes: |
\(2\), \(37\), \(59\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{8}-\frac{1}{6}a^{7}+\frac{1}{3}a^{4}-\frac{1}{6}a^{3}-\frac{1}{6}a^{2}+\frac{1}{6}a$, $\frac{1}{18}a^{10}+\frac{1}{18}a^{9}+\frac{1}{6}a^{8}-\frac{1}{9}a^{7}+\frac{1}{6}a^{6}+\frac{5}{18}a^{5}-\frac{1}{6}a^{4}-\frac{1}{18}a^{2}+\frac{5}{18}a-\frac{1}{6}$, $\frac{1}{10\!\cdots\!46}a^{11}+\frac{36\!\cdots\!35}{15\!\cdots\!78}a^{10}-\frac{71\!\cdots\!19}{10\!\cdots\!46}a^{9}+\frac{12\!\cdots\!01}{53\!\cdots\!73}a^{8}+\frac{50\!\cdots\!47}{21\!\cdots\!54}a^{7}-\frac{12\!\cdots\!91}{10\!\cdots\!46}a^{6}+\frac{15\!\cdots\!11}{10\!\cdots\!46}a^{5}-\frac{10\!\cdots\!67}{17\!\cdots\!91}a^{4}+\frac{41\!\cdots\!71}{10\!\cdots\!46}a^{3}+\frac{27\!\cdots\!65}{10\!\cdots\!46}a^{2}-\frac{24\!\cdots\!01}{10\!\cdots\!46}a-\frac{97\!\cdots\!60}{25\!\cdots\!13}$
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
$C_{3}\times C_{18}\times C_{108234}$, which has order $5844636$ (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: |
$\frac{13\!\cdots\!46}{53\!\cdots\!73}a^{11}+\frac{39\!\cdots\!79}{76\!\cdots\!39}a^{10}+\frac{31\!\cdots\!17}{53\!\cdots\!73}a^{9}+\frac{18\!\cdots\!95}{10\!\cdots\!46}a^{8}+\frac{14\!\cdots\!98}{10\!\cdots\!77}a^{7}+\frac{50\!\cdots\!08}{53\!\cdots\!73}a^{6}-\frac{52\!\cdots\!79}{53\!\cdots\!73}a^{5}+\frac{80\!\cdots\!17}{35\!\cdots\!82}a^{4}-\frac{49\!\cdots\!52}{53\!\cdots\!73}a^{3}+\frac{21\!\cdots\!47}{53\!\cdots\!73}a^{2}-\frac{14\!\cdots\!39}{53\!\cdots\!73}a+\frac{17\!\cdots\!35}{50\!\cdots\!26}$, $\frac{77\!\cdots\!21}{53\!\cdots\!73}a^{11}-\frac{26\!\cdots\!23}{15\!\cdots\!78}a^{10}-\frac{84\!\cdots\!65}{10\!\cdots\!46}a^{9}-\frac{38\!\cdots\!35}{53\!\cdots\!73}a^{8}-\frac{40\!\cdots\!54}{10\!\cdots\!77}a^{7}-\frac{24\!\cdots\!71}{10\!\cdots\!46}a^{6}-\frac{60\!\cdots\!27}{10\!\cdots\!46}a^{5}-\frac{10\!\cdots\!27}{17\!\cdots\!91}a^{4}-\frac{14\!\cdots\!08}{53\!\cdots\!73}a^{3}-\frac{83\!\cdots\!79}{10\!\cdots\!46}a^{2}+\frac{78\!\cdots\!57}{10\!\cdots\!46}a-\frac{12\!\cdots\!00}{25\!\cdots\!13}$, $\frac{39\!\cdots\!19}{53\!\cdots\!73}a^{11}+\frac{35\!\cdots\!25}{15\!\cdots\!78}a^{10}+\frac{35\!\cdots\!97}{10\!\cdots\!46}a^{9}+\frac{68\!\cdots\!13}{53\!\cdots\!73}a^{8}+\frac{16\!\cdots\!63}{10\!\cdots\!77}a^{7}+\frac{10\!\cdots\!34}{53\!\cdots\!73}a^{6}+\frac{18\!\cdots\!17}{53\!\cdots\!73}a^{5}+\frac{41\!\cdots\!92}{17\!\cdots\!91}a^{4}+\frac{59\!\cdots\!71}{10\!\cdots\!46}a^{3}-\frac{32\!\cdots\!47}{10\!\cdots\!46}a^{2}+\frac{25\!\cdots\!43}{53\!\cdots\!73}a-\frac{11\!\cdots\!27}{50\!\cdots\!26}$, $\frac{11\!\cdots\!31}{17\!\cdots\!91}a^{11}+\frac{73\!\cdots\!75}{50\!\cdots\!26}a^{10}+\frac{77\!\cdots\!35}{35\!\cdots\!82}a^{9}+\frac{16\!\cdots\!55}{35\!\cdots\!82}a^{8}+\frac{16\!\cdots\!71}{36\!\cdots\!59}a^{7}+\frac{45\!\cdots\!81}{17\!\cdots\!91}a^{6}-\frac{40\!\cdots\!71}{17\!\cdots\!91}a^{5}+\frac{23\!\cdots\!87}{39\!\cdots\!98}a^{4}-\frac{10\!\cdots\!91}{35\!\cdots\!82}a^{3}+\frac{37\!\cdots\!77}{35\!\cdots\!82}a^{2}-\frac{14\!\cdots\!99}{17\!\cdots\!91}a+\frac{79\!\cdots\!26}{84\!\cdots\!71}$, $\frac{18\!\cdots\!56}{76\!\cdots\!39}a^{11}-\frac{64\!\cdots\!52}{10\!\cdots\!77}a^{10}-\frac{20\!\cdots\!15}{76\!\cdots\!39}a^{9}-\frac{34\!\cdots\!21}{15\!\cdots\!78}a^{8}-\frac{11\!\cdots\!35}{10\!\cdots\!77}a^{7}-\frac{14\!\cdots\!35}{15\!\cdots\!78}a^{6}-\frac{31\!\cdots\!63}{15\!\cdots\!78}a^{5}-\frac{10\!\cdots\!29}{50\!\cdots\!26}a^{4}-\frac{39\!\cdots\!17}{15\!\cdots\!78}a^{3}-\frac{24\!\cdots\!82}{76\!\cdots\!39}a^{2}-\frac{70\!\cdots\!55}{15\!\cdots\!78}a-\frac{10\!\cdots\!20}{36\!\cdots\!59}$
| 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: | \( 12302.579850899174 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 12302.579850899174 \cdot 5844636}{2\cdot\sqrt{51924135650907635776859701504}}\cr\approx \mathstrut & 9.70774227249150 \end{aligned}\] (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 + 41*x^10 - 6*x^9 + 1878*x^8 - 4406*x^7 + 63987*x^6 - 136784*x^5 + 1352954*x^4 - 3084580*x^3 + 19917769*x^2 - 31791008*x + 115888143)
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 + 41*x^10 - 6*x^9 + 1878*x^8 - 4406*x^7 + 63987*x^6 - 136784*x^5 + 1352954*x^4 - 3084580*x^3 + 19917769*x^2 - 31791008*x + 115888143, 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 + 41*x^10 - 6*x^9 + 1878*x^8 - 4406*x^7 + 63987*x^6 - 136784*x^5 + 1352954*x^4 - 3084580*x^3 + 19917769*x^2 - 31791008*x + 115888143);
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 + 41*x^10 - 6*x^9 + 1878*x^8 - 4406*x^7 + 63987*x^6 - 136784*x^5 + 1352954*x^4 - 3084580*x^3 + 19917769*x^2 - 31791008*x + 115888143);
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
| \(\Q(\sqrt{-2183}) \), \(\Q(\sqrt{-59}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{37}, \sqrt{-59})\), 6.6.1109503312.1 |
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 | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{6}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | R |
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 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
|
\(37\)
| 37.12.10.1 | $x^{12} + 198 x^{11} + 16347 x^{10} + 720720 x^{9} + 17919555 x^{8} + 239132718 x^{7} + 1363015503 x^{6} + 478272762 x^{5} + 72280395 x^{4} + 32212620 x^{3} + 653616987 x^{2} + 8608429962 x + 47259217318$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
|
\(59\)
| 59.12.6.1 | $x^{12} + 358 x^{10} + 36 x^{9} + 53003 x^{8} + 72 x^{7} + 4118916 x^{6} - 737784 x^{5} + 177619595 x^{4} - 58235760 x^{3} + 4088219302 x^{2} - 1289138436 x + 40329774445$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |