Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 508*x^10 - 1676*x^9 + 104199*x^8 - 273412*x^7 + 10968102*x^6 - 21538616*x^5 + 623909300*x^4 - 819565924*x^3 + 18410523340*x^2 - 12257477028*x + 222965992201)
gp: K = bnfinit(y^12 - 4*y^11 + 508*y^10 - 1676*y^9 + 104199*y^8 - 273412*y^7 + 10968102*y^6 - 21538616*y^5 + 623909300*y^4 - 819565924*y^3 + 18410523340*y^2 - 12257477028*y + 222965992201, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 508*x^10 - 1676*x^9 + 104199*x^8 - 273412*x^7 + 10968102*x^6 - 21538616*x^5 + 623909300*x^4 - 819565924*x^3 + 18410523340*x^2 - 12257477028*x + 222965992201);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 508*x^10 - 1676*x^9 + 104199*x^8 - 273412*x^7 + 10968102*x^6 - 21538616*x^5 + 623909300*x^4 - 819565924*x^3 + 18410523340*x^2 - 12257477028*x + 222965992201)
\( x^{12} - 4 x^{11} + 508 x^{10} - 1676 x^{9} + 104199 x^{8} - 273412 x^{7} + 10968102 x^{6} + \cdots + 222965992201 \)
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: |
\(71243920337289728000000000\)
\(\medspace = 2^{18}\cdot 5^{9}\cdot 7^{8}\cdot 17^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(142.69\) | 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: | $2^{3/2}5^{3/4}7^{2/3}17^{1/2}\approx 142.69069418206897$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4760=2^{3}\cdot 5\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4760}(1,·)$, $\chi_{4760}(2277,·)$, $\chi_{4760}(1089,·)$, $\chi_{4760}(4453,·)$, $\chi_{4760}(4489,·)$, $\chi_{4760}(3637,·)$, $\chi_{4760}(373,·)$, $\chi_{4760}(681,·)$, $\chi_{4760}(2041,·)$, $\chi_{4760}(3809,·)$, $\chi_{4760}(1597,·)$, $\chi_{4760}(1733,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.2312000.1$^{2}$, 12.0.71243920337289728000000000.1$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{34}a^{6}-\frac{1}{17}a^{5}-\frac{3}{34}a^{4}+\frac{3}{17}a^{3}+\frac{1}{17}a^{2}-\frac{2}{17}a+\frac{1}{34}$, $\frac{1}{34}a^{7}-\frac{7}{34}a^{5}+\frac{7}{17}a^{3}-\frac{7}{34}a+\frac{1}{17}$, $\frac{1}{34}a^{8}-\frac{7}{17}a^{5}-\frac{7}{34}a^{4}+\frac{4}{17}a^{3}+\frac{7}{34}a^{2}+\frac{4}{17}a+\frac{7}{34}$, $\frac{1}{34}a^{9}-\frac{1}{34}a^{5}-\frac{11}{34}a^{3}+\frac{1}{17}a^{2}-\frac{15}{34}a+\frac{7}{17}$, $\frac{1}{13301139930674}a^{10}+\frac{93927102345}{13301139930674}a^{9}-\frac{25133222697}{6650569965337}a^{8}+\frac{20855390959}{6650569965337}a^{7}-\frac{123026019705}{13301139930674}a^{6}-\frac{4017015355575}{13301139930674}a^{5}-\frac{6353606372165}{13301139930674}a^{4}+\frac{1508051930957}{13301139930674}a^{3}+\frac{856442981325}{13301139930674}a^{2}+\frac{4905201814491}{13301139930674}a+\frac{2678224099164}{6650569965337}$, $\frac{1}{11\!\cdots\!74}a^{11}+\frac{146131140508290}{58\!\cdots\!37}a^{10}+\frac{11\!\cdots\!86}{58\!\cdots\!37}a^{9}-\frac{82\!\cdots\!68}{58\!\cdots\!37}a^{8}-\frac{38\!\cdots\!23}{58\!\cdots\!37}a^{7}-\frac{76\!\cdots\!81}{11\!\cdots\!74}a^{6}-\frac{10\!\cdots\!85}{58\!\cdots\!37}a^{5}+\frac{32\!\cdots\!61}{11\!\cdots\!74}a^{4}+\frac{50\!\cdots\!39}{11\!\cdots\!74}a^{3}-\frac{24\!\cdots\!00}{58\!\cdots\!37}a^{2}-\frac{54\!\cdots\!09}{11\!\cdots\!74}a-\frac{33\!\cdots\!65}{11\!\cdots\!74}$
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
$C_{18}\times C_{16866}$, which has order $303588$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $303588$ (assuming GRH)
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: |
$\frac{14}{6650569965337}a^{11}+\frac{19685}{6650569965337}a^{10}-\frac{59095}{6650569965337}a^{9}+\frac{16472945}{13301139930674}a^{8}-\frac{20605280}{6650569965337}a^{7}+\frac{1302319991}{6650569965337}a^{6}-\frac{2475898418}{6650569965337}a^{5}+\frac{190990224595}{13301139930674}a^{4}-\frac{122102252780}{6650569965337}a^{3}+\frac{6719856863321}{13301139930674}a^{2}-\frac{2189974499014}{6650569965337}a+\frac{6736440811337}{782419995922}$, $\frac{12\!\cdots\!20}{34\!\cdots\!61}a^{11}-\frac{26\!\cdots\!10}{34\!\cdots\!61}a^{10}+\frac{58\!\cdots\!20}{34\!\cdots\!61}a^{9}-\frac{12\!\cdots\!15}{34\!\cdots\!61}a^{8}+\frac{10\!\cdots\!40}{34\!\cdots\!61}a^{7}-\frac{23\!\cdots\!90}{34\!\cdots\!61}a^{6}+\frac{91\!\cdots\!04}{34\!\cdots\!61}a^{5}-\frac{20\!\cdots\!25}{34\!\cdots\!61}a^{4}+\frac{38\!\cdots\!80}{34\!\cdots\!61}a^{3}-\frac{86\!\cdots\!40}{34\!\cdots\!61}a^{2}+\frac{62\!\cdots\!20}{34\!\cdots\!61}a-\frac{14\!\cdots\!95}{34\!\cdots\!61}$, $\frac{51\!\cdots\!20}{34\!\cdots\!61}a^{11}+\frac{20\!\cdots\!46}{34\!\cdots\!61}a^{10}+\frac{13\!\cdots\!60}{34\!\cdots\!61}a^{9}+\frac{10\!\cdots\!30}{34\!\cdots\!61}a^{8}+\frac{10\!\cdots\!40}{34\!\cdots\!61}a^{7}+\frac{20\!\cdots\!10}{34\!\cdots\!61}a^{6}-\frac{23\!\cdots\!60}{34\!\cdots\!61}a^{5}+\frac{19\!\cdots\!05}{34\!\cdots\!61}a^{4}-\frac{21\!\cdots\!40}{34\!\cdots\!61}a^{3}+\frac{84\!\cdots\!00}{34\!\cdots\!61}a^{2}-\frac{55\!\cdots\!20}{34\!\cdots\!61}a+\frac{14\!\cdots\!49}{34\!\cdots\!61}$, $\frac{86\!\cdots\!26}{58\!\cdots\!37}a^{11}+\frac{17\!\cdots\!97}{58\!\cdots\!37}a^{10}+\frac{27\!\cdots\!15}{58\!\cdots\!37}a^{9}+\frac{21\!\cdots\!75}{11\!\cdots\!74}a^{8}+\frac{19\!\cdots\!60}{58\!\cdots\!37}a^{7}+\frac{24\!\cdots\!79}{58\!\cdots\!37}a^{6}-\frac{17\!\cdots\!02}{58\!\cdots\!37}a^{5}+\frac{49\!\cdots\!75}{11\!\cdots\!74}a^{4}-\frac{26\!\cdots\!00}{58\!\cdots\!37}a^{3}+\frac{22\!\cdots\!79}{11\!\cdots\!74}a^{2}-\frac{75\!\cdots\!26}{58\!\cdots\!37}a+\frac{23\!\cdots\!61}{68\!\cdots\!22}$, $\frac{30\!\cdots\!66}{58\!\cdots\!37}a^{11}-\frac{28\!\cdots\!73}{58\!\cdots\!37}a^{10}+\frac{12\!\cdots\!55}{58\!\cdots\!37}a^{9}-\frac{21\!\cdots\!35}{11\!\cdots\!74}a^{8}+\frac{19\!\cdots\!40}{58\!\cdots\!37}a^{7}-\frac{15\!\cdots\!51}{58\!\cdots\!37}a^{6}+\frac{13\!\cdots\!66}{58\!\cdots\!37}a^{5}-\frac{20\!\cdots\!75}{11\!\cdots\!74}a^{4}+\frac{39\!\cdots\!60}{58\!\cdots\!37}a^{3}-\frac{65\!\cdots\!81}{11\!\cdots\!74}a^{2}+\frac{29\!\cdots\!14}{58\!\cdots\!37}a-\frac{57\!\cdots\!29}{68\!\cdots\!22}$
| 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: | \( 104.882003477 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 104.882003477 \cdot 303588}{2\cdot\sqrt{71243920337289728000000000}}\cr\approx \mathstrut & 0.116054206267 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 508*x^10 - 1676*x^9 + 104199*x^8 - 273412*x^7 + 10968102*x^6 - 21538616*x^5 + 623909300*x^4 - 819565924*x^3 + 18410523340*x^2 - 12257477028*x + 222965992201)
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 - 4*x^11 + 508*x^10 - 1676*x^9 + 104199*x^8 - 273412*x^7 + 10968102*x^6 - 21538616*x^5 + 623909300*x^4 - 819565924*x^3 + 18410523340*x^2 - 12257477028*x + 222965992201, 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 - 4*x^11 + 508*x^10 - 1676*x^9 + 104199*x^8 - 273412*x^7 + 10968102*x^6 - 21538616*x^5 + 623909300*x^4 - 819565924*x^3 + 18410523340*x^2 - 12257477028*x + 222965992201);
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 - 4*x^11 + 508*x^10 - 1676*x^9 + 104199*x^8 - 273412*x^7 + 10968102*x^6 - 21538616*x^5 + 623909300*x^4 - 819565924*x^3 + 18410523340*x^2 - 12257477028*x + 222965992201);
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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.2312000.1, 6.6.300125.1 |
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 | R | ${\href{/padicField/3.12.0.1}{12} }$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.28 | $x^{12} + 12 x^{11} + 128 x^{10} + 1032 x^{9} + 8068 x^{8} + 54752 x^{7} + 298816 x^{6} + 1148736 x^{5} + 2948656 x^{4} + 4481984 x^{3} + 2851584 x^{2} - 296320 x + 1412800$ | $2$ | $6$ | $18$ | $C_{12}$ | $[3]^{6}$ |
|
\(5\)
| 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(7\)
| 7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(17\)
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |