Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 202*x^10 + 181*x^9 + 13504*x^8 - 9703*x^7 - 357433*x^6 + 184541*x^5 + 3755080*x^4 - 1388646*x^3 - 11454450*x^2 + 4959038*x - 492149)
gp: K = bnfinit(y^12 - y^11 - 202*y^10 + 181*y^9 + 13504*y^8 - 9703*y^7 - 357433*y^6 + 184541*y^5 + 3755080*y^4 - 1388646*y^3 - 11454450*y^2 + 4959038*y - 492149, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 202*x^10 + 181*x^9 + 13504*x^8 - 9703*x^7 - 357433*x^6 + 184541*x^5 + 3755080*x^4 - 1388646*x^3 - 11454450*x^2 + 4959038*x - 492149);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 202*x^10 + 181*x^9 + 13504*x^8 - 9703*x^7 - 357433*x^6 + 184541*x^5 + 3755080*x^4 - 1388646*x^3 - 11454450*x^2 + 4959038*x - 492149)
\( x^{12} - x^{11} - 202 x^{10} + 181 x^{9} + 13504 x^{8} - 9703 x^{7} - 357433 x^{6} + 184541 x^{5} + \cdots - 492149 \)
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: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(65426054573967427251953125\)
\(\medspace = 5^{9}\cdot 7^{10}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(141.68\) | 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: | $5^{3/4}7^{5/6}17^{3/4}\approx 141.681309097028$ | ||
| Ramified primes: |
\(5\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{85}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(595=5\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{595}(256,·)$, $\chi_{595}(1,·)$, $\chi_{595}(132,·)$, $\chi_{595}(293,·)$, $\chi_{595}(38,·)$, $\chi_{595}(424,·)$, $\chi_{595}(169,·)$, $\chi_{595}(47,·)$, $\chi_{595}(208,·)$, $\chi_{595}(86,·)$, $\chi_{595}(472,·)$, $\chi_{595}(254,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{27242209}a^{10}-\frac{9269042}{27242209}a^{9}+\frac{1480181}{27242209}a^{8}+\frac{6669483}{27242209}a^{7}+\frac{5361671}{27242209}a^{6}+\frac{4045628}{27242209}a^{5}-\frac{11051929}{27242209}a^{4}-\frac{7400550}{27242209}a^{3}-\frac{5048011}{27242209}a^{2}-\frac{3324577}{27242209}a-\frac{27665}{163127}$, $\frac{1}{39\!\cdots\!79}a^{11}-\frac{34\!\cdots\!68}{39\!\cdots\!79}a^{10}+\frac{13\!\cdots\!21}{39\!\cdots\!79}a^{9}+\frac{92\!\cdots\!23}{39\!\cdots\!79}a^{8}+\frac{11\!\cdots\!69}{39\!\cdots\!79}a^{7}+\frac{18\!\cdots\!18}{39\!\cdots\!79}a^{6}-\frac{10\!\cdots\!21}{39\!\cdots\!79}a^{5}+\frac{65\!\cdots\!69}{39\!\cdots\!79}a^{4}-\frac{11\!\cdots\!02}{39\!\cdots\!79}a^{3}-\frac{14\!\cdots\!73}{39\!\cdots\!79}a^{2}+\frac{61\!\cdots\!71}{39\!\cdots\!79}a-\frac{44\!\cdots\!20}{23\!\cdots\!37}$
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
$C_{4}$, which has order $4$ (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: | $11$ | 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: |
$\frac{93\!\cdots\!60}{67\!\cdots\!61}a^{11}-\frac{71\!\cdots\!99}{67\!\cdots\!61}a^{10}-\frac{18\!\cdots\!20}{67\!\cdots\!61}a^{9}+\frac{12\!\cdots\!70}{67\!\cdots\!61}a^{8}+\frac{12\!\cdots\!20}{67\!\cdots\!61}a^{7}-\frac{61\!\cdots\!45}{67\!\cdots\!61}a^{6}-\frac{33\!\cdots\!20}{67\!\cdots\!61}a^{5}+\frac{98\!\cdots\!70}{67\!\cdots\!61}a^{4}+\frac{34\!\cdots\!40}{67\!\cdots\!61}a^{3}-\frac{59\!\cdots\!15}{67\!\cdots\!61}a^{2}-\frac{99\!\cdots\!00}{67\!\cdots\!61}a+\frac{14\!\cdots\!14}{40\!\cdots\!83}$, $\frac{68\!\cdots\!10}{67\!\cdots\!61}a^{11}-\frac{41\!\cdots\!71}{67\!\cdots\!61}a^{10}-\frac{13\!\cdots\!20}{67\!\cdots\!61}a^{9}+\frac{69\!\cdots\!60}{67\!\cdots\!61}a^{8}+\frac{92\!\cdots\!80}{67\!\cdots\!61}a^{7}-\frac{30\!\cdots\!45}{67\!\cdots\!61}a^{6}-\frac{14\!\cdots\!13}{40\!\cdots\!83}a^{5}+\frac{34\!\cdots\!30}{67\!\cdots\!61}a^{4}+\frac{25\!\cdots\!65}{67\!\cdots\!61}a^{3}-\frac{42\!\cdots\!35}{67\!\cdots\!61}a^{2}-\frac{81\!\cdots\!45}{67\!\cdots\!61}a+\frac{57\!\cdots\!53}{40\!\cdots\!83}$, $\frac{40\!\cdots\!13}{39\!\cdots\!79}a^{11}-\frac{29\!\cdots\!28}{39\!\cdots\!79}a^{10}-\frac{80\!\cdots\!17}{39\!\cdots\!79}a^{9}+\frac{51\!\cdots\!28}{39\!\cdots\!79}a^{8}+\frac{54\!\cdots\!44}{39\!\cdots\!79}a^{7}-\frac{24\!\cdots\!43}{39\!\cdots\!79}a^{6}-\frac{14\!\cdots\!62}{39\!\cdots\!79}a^{5}+\frac{32\!\cdots\!55}{39\!\cdots\!79}a^{4}+\frac{15\!\cdots\!28}{39\!\cdots\!79}a^{3}-\frac{89\!\cdots\!25}{39\!\cdots\!79}a^{2}-\frac{47\!\cdots\!11}{39\!\cdots\!79}a+\frac{27\!\cdots\!47}{23\!\cdots\!37}$, $\frac{17\!\cdots\!68}{39\!\cdots\!79}a^{11}-\frac{19\!\cdots\!47}{39\!\cdots\!79}a^{10}-\frac{34\!\cdots\!06}{39\!\cdots\!79}a^{9}+\frac{33\!\cdots\!86}{39\!\cdots\!79}a^{8}+\frac{22\!\cdots\!67}{39\!\cdots\!79}a^{7}-\frac{16\!\cdots\!44}{39\!\cdots\!79}a^{6}-\frac{58\!\cdots\!08}{39\!\cdots\!79}a^{5}+\frac{23\!\cdots\!04}{39\!\cdots\!79}a^{4}+\frac{58\!\cdots\!17}{39\!\cdots\!79}a^{3}-\frac{11\!\cdots\!86}{39\!\cdots\!79}a^{2}-\frac{17\!\cdots\!75}{39\!\cdots\!79}a+\frac{28\!\cdots\!67}{23\!\cdots\!37}$, $\frac{10\!\cdots\!74}{14\!\cdots\!31}a^{11}-\frac{87\!\cdots\!26}{14\!\cdots\!31}a^{10}-\frac{21\!\cdots\!54}{14\!\cdots\!31}a^{9}+\frac{15\!\cdots\!34}{14\!\cdots\!31}a^{8}+\frac{14\!\cdots\!55}{14\!\cdots\!31}a^{7}-\frac{73\!\cdots\!94}{14\!\cdots\!31}a^{6}-\frac{36\!\cdots\!03}{14\!\cdots\!31}a^{5}+\frac{10\!\cdots\!40}{14\!\cdots\!31}a^{4}+\frac{38\!\cdots\!58}{14\!\cdots\!31}a^{3}-\frac{42\!\cdots\!54}{14\!\cdots\!31}a^{2}-\frac{11\!\cdots\!97}{14\!\cdots\!31}a+\frac{18\!\cdots\!08}{14\!\cdots\!31}$, $\frac{95\!\cdots\!22}{39\!\cdots\!79}a^{11}-\frac{86\!\cdots\!12}{39\!\cdots\!79}a^{10}-\frac{19\!\cdots\!39}{39\!\cdots\!79}a^{9}+\frac{15\!\cdots\!68}{39\!\cdots\!79}a^{8}+\frac{12\!\cdots\!36}{39\!\cdots\!79}a^{7}-\frac{80\!\cdots\!75}{39\!\cdots\!79}a^{6}-\frac{33\!\cdots\!40}{39\!\cdots\!79}a^{5}+\frac{13\!\cdots\!55}{39\!\cdots\!79}a^{4}+\frac{34\!\cdots\!24}{39\!\cdots\!79}a^{3}-\frac{76\!\cdots\!61}{39\!\cdots\!79}a^{2}-\frac{10\!\cdots\!16}{39\!\cdots\!79}a+\frac{16\!\cdots\!99}{23\!\cdots\!37}$, $\frac{85\!\cdots\!40}{39\!\cdots\!79}a^{11}-\frac{81\!\cdots\!07}{39\!\cdots\!79}a^{10}-\frac{17\!\cdots\!82}{39\!\cdots\!79}a^{9}+\frac{14\!\cdots\!11}{39\!\cdots\!79}a^{8}+\frac{11\!\cdots\!46}{39\!\cdots\!79}a^{7}-\frac{74\!\cdots\!22}{39\!\cdots\!79}a^{6}-\frac{29\!\cdots\!21}{39\!\cdots\!79}a^{5}+\frac{12\!\cdots\!26}{39\!\cdots\!79}a^{4}+\frac{30\!\cdots\!34}{39\!\cdots\!79}a^{3}-\frac{70\!\cdots\!05}{39\!\cdots\!79}a^{2}-\frac{93\!\cdots\!75}{39\!\cdots\!79}a+\frac{14\!\cdots\!65}{23\!\cdots\!37}$, $\frac{19\!\cdots\!92}{39\!\cdots\!79}a^{11}-\frac{24\!\cdots\!54}{39\!\cdots\!79}a^{10}-\frac{39\!\cdots\!66}{39\!\cdots\!79}a^{9}+\frac{44\!\cdots\!23}{39\!\cdots\!79}a^{8}+\frac{26\!\cdots\!78}{39\!\cdots\!79}a^{7}-\frac{22\!\cdots\!59}{39\!\cdots\!79}a^{6}-\frac{67\!\cdots\!55}{39\!\cdots\!79}a^{5}+\frac{38\!\cdots\!20}{39\!\cdots\!79}a^{4}+\frac{67\!\cdots\!15}{39\!\cdots\!79}a^{3}-\frac{20\!\cdots\!31}{39\!\cdots\!79}a^{2}-\frac{20\!\cdots\!61}{39\!\cdots\!79}a+\frac{33\!\cdots\!19}{23\!\cdots\!37}$, $\frac{19\!\cdots\!85}{39\!\cdots\!79}a^{11}+\frac{29\!\cdots\!89}{39\!\cdots\!79}a^{10}-\frac{41\!\cdots\!49}{39\!\cdots\!79}a^{9}-\frac{61\!\cdots\!72}{39\!\cdots\!79}a^{8}+\frac{28\!\cdots\!56}{39\!\cdots\!79}a^{7}+\frac{44\!\cdots\!69}{39\!\cdots\!79}a^{6}-\frac{75\!\cdots\!96}{39\!\cdots\!79}a^{5}-\frac{12\!\cdots\!16}{39\!\cdots\!79}a^{4}+\frac{71\!\cdots\!33}{39\!\cdots\!79}a^{3}+\frac{11\!\cdots\!69}{39\!\cdots\!79}a^{2}-\frac{70\!\cdots\!69}{39\!\cdots\!79}a+\frac{41\!\cdots\!40}{23\!\cdots\!37}$, $\frac{33\!\cdots\!44}{39\!\cdots\!79}a^{11}-\frac{76\!\cdots\!01}{39\!\cdots\!79}a^{10}-\frac{62\!\cdots\!55}{39\!\cdots\!79}a^{9}+\frac{12\!\cdots\!92}{39\!\cdots\!79}a^{8}+\frac{36\!\cdots\!89}{39\!\cdots\!79}a^{7}-\frac{56\!\cdots\!51}{39\!\cdots\!79}a^{6}-\frac{76\!\cdots\!71}{39\!\cdots\!79}a^{5}+\frac{59\!\cdots\!32}{39\!\cdots\!79}a^{4}+\frac{48\!\cdots\!08}{39\!\cdots\!79}a^{3}+\frac{22\!\cdots\!68}{39\!\cdots\!79}a^{2}+\frac{74\!\cdots\!77}{39\!\cdots\!79}a-\frac{89\!\cdots\!17}{23\!\cdots\!37}$, $\frac{11\!\cdots\!97}{39\!\cdots\!79}a^{11}-\frac{71\!\cdots\!48}{39\!\cdots\!79}a^{10}-\frac{22\!\cdots\!53}{39\!\cdots\!79}a^{9}+\frac{13\!\cdots\!94}{39\!\cdots\!79}a^{8}+\frac{14\!\cdots\!34}{39\!\cdots\!79}a^{7}-\frac{77\!\cdots\!24}{39\!\cdots\!79}a^{6}-\frac{34\!\cdots\!64}{39\!\cdots\!79}a^{5}+\frac{15\!\cdots\!06}{39\!\cdots\!79}a^{4}+\frac{25\!\cdots\!91}{39\!\cdots\!79}a^{3}-\frac{94\!\cdots\!26}{39\!\cdots\!79}a^{2}+\frac{32\!\cdots\!85}{39\!\cdots\!79}a-\frac{15\!\cdots\!18}{23\!\cdots\!37}$
| 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: | \( 162126335.029 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 162126335.029 \cdot 4}{2\cdot\sqrt{65426054573967427251953125}}\cr\approx \mathstrut & 0.164198105946 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 202*x^10 + 181*x^9 + 13504*x^8 - 9703*x^7 - 357433*x^6 + 184541*x^5 + 3755080*x^4 - 1388646*x^3 - 11454450*x^2 + 4959038*x - 492149)
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 - x^11 - 202*x^10 + 181*x^9 + 13504*x^8 - 9703*x^7 - 357433*x^6 + 184541*x^5 + 3755080*x^4 - 1388646*x^3 - 11454450*x^2 + 4959038*x - 492149, 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 - x^11 - 202*x^10 + 181*x^9 + 13504*x^8 - 9703*x^7 - 357433*x^6 + 184541*x^5 + 3755080*x^4 - 1388646*x^3 - 11454450*x^2 + 4959038*x - 492149);
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 - x^11 - 202*x^10 + 181*x^9 + 13504*x^8 - 9703*x^7 - 357433*x^6 + 184541*x^5 + 3755080*x^4 - 1388646*x^3 - 11454450*x^2 + 4959038*x - 492149);
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
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 cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{85}) \), \(\Q(\zeta_{7})^+\), 4.4.30092125.1, 6.6.1474514125.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | R | ${\href{/padicField/11.12.0.1}{12} }$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.12.9.3 | $x^{12} + 75 x^{4} - 375$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(7\)
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
|
\(17\)
| 17.12.9.3 | $x^{12} + 289 x^{4} - 68782$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |