Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 99*x^10 - 140*x^9 + 3366*x^8 + 8820*x^7 - 40049*x^6 - 160020*x^5 + 39531*x^4 + 826140*x^3 + 1229700*x^2 + 647850*x + 91405)
gp: K = bnfinit(y^12 - 99*y^10 - 140*y^9 + 3366*y^8 + 8820*y^7 - 40049*y^6 - 160020*y^5 + 39531*y^4 + 826140*y^3 + 1229700*y^2 + 647850*y + 91405, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 99*x^10 - 140*x^9 + 3366*x^8 + 8820*x^7 - 40049*x^6 - 160020*x^5 + 39531*x^4 + 826140*x^3 + 1229700*x^2 + 647850*x + 91405);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 99*x^10 - 140*x^9 + 3366*x^8 + 8820*x^7 - 40049*x^6 - 160020*x^5 + 39531*x^4 + 826140*x^3 + 1229700*x^2 + 647850*x + 91405)
\( x^{12} - 99 x^{10} - 140 x^{9} + 3366 x^{8} + 8820 x^{7} - 40049 x^{6} - 160020 x^{5} + 39531 x^{4} + \cdots + 91405 \)
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: |
\(1985246242140168000000000\)
\(\medspace = 2^{12}\cdot 3^{16}\cdot 5^{9}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(105.88\) | 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: | $2\cdot 3^{4/3}5^{3/4}7^{2/3}\approx 105.88095876054969$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| 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: | \(1260=2^{2}\cdot 3^{2}\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1260}(1,·)$, $\chi_{1260}(67,·)$, $\chi_{1260}(709,·)$, $\chi_{1260}(961,·)$, $\chi_{1260}(583,·)$, $\chi_{1260}(1009,·)$, $\chi_{1260}(1201,·)$, $\chi_{1260}(883,·)$, $\chi_{1260}(949,·)$, $\chi_{1260}(823,·)$, $\chi_{1260}(127,·)$, $\chi_{1260}(1087,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{44230159}a^{10}+\frac{2287418}{44230159}a^{9}-\frac{21018866}{44230159}a^{8}+\frac{6133293}{44230159}a^{7}+\frac{21070967}{44230159}a^{6}-\frac{18937369}{44230159}a^{5}-\frac{14633798}{44230159}a^{4}+\frac{14964032}{44230159}a^{3}-\frac{594802}{44230159}a^{2}+\frac{4302793}{44230159}a-\frac{7875186}{44230159}$, $\frac{1}{19\!\cdots\!79}a^{11}+\frac{27293812}{19\!\cdots\!79}a^{10}-\frac{11\!\cdots\!87}{19\!\cdots\!79}a^{9}-\frac{90\!\cdots\!98}{19\!\cdots\!79}a^{8}+\frac{63\!\cdots\!76}{19\!\cdots\!79}a^{7}-\frac{719241744546782}{19\!\cdots\!79}a^{6}-\frac{91\!\cdots\!67}{19\!\cdots\!79}a^{5}-\frac{45\!\cdots\!57}{19\!\cdots\!79}a^{4}-\frac{50\!\cdots\!32}{19\!\cdots\!79}a^{3}+\frac{80\!\cdots\!39}{19\!\cdots\!79}a^{2}-\frac{61\!\cdots\!10}{19\!\cdots\!79}a+\frac{14\!\cdots\!82}{19\!\cdots\!79}$
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
$C_{3}$, which has order $3$ (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: |
$\frac{3780}{44230159}a^{11}-\frac{1416}{44230159}a^{10}-\frac{384300}{44230159}a^{9}-\frac{329745}{44230159}a^{8}+\frac{13652100}{44230159}a^{7}+\frac{25237142}{44230159}a^{6}-\frac{184258410}{44230159}a^{5}-\frac{491661375}{44230159}a^{4}+\frac{625505650}{44230159}a^{3}+\frac{2792973969}{44230159}a^{2}+\frac{2334175830}{44230159}a+\frac{412134598}{44230159}$, $\frac{2070310267032}{19\!\cdots\!79}a^{11}-\frac{5755061670654}{19\!\cdots\!79}a^{10}-\frac{184858437048075}{19\!\cdots\!79}a^{9}+\frac{204816973731675}{19\!\cdots\!79}a^{8}+\frac{60\!\cdots\!68}{19\!\cdots\!79}a^{7}+\frac{23\!\cdots\!23}{19\!\cdots\!79}a^{6}-\frac{80\!\cdots\!57}{19\!\cdots\!79}a^{5}-\frac{12\!\cdots\!70}{19\!\cdots\!79}a^{4}+\frac{30\!\cdots\!29}{19\!\cdots\!79}a^{3}+\frac{85\!\cdots\!56}{19\!\cdots\!79}a^{2}+\frac{61\!\cdots\!98}{19\!\cdots\!79}a+\frac{10\!\cdots\!17}{19\!\cdots\!79}$, $\frac{2287049839560}{19\!\cdots\!79}a^{11}-\frac{1507783464906}{19\!\cdots\!79}a^{10}-\frac{228633749407340}{19\!\cdots\!79}a^{9}-\frac{153019107662145}{19\!\cdots\!79}a^{8}+\frac{80\!\cdots\!20}{19\!\cdots\!79}a^{7}+\frac{13\!\cdots\!52}{19\!\cdots\!79}a^{6}-\frac{10\!\cdots\!64}{19\!\cdots\!79}a^{5}-\frac{28\!\cdots\!50}{19\!\cdots\!79}a^{4}+\frac{36\!\cdots\!20}{19\!\cdots\!79}a^{3}+\frac{16\!\cdots\!14}{19\!\cdots\!79}a^{2}+\frac{13\!\cdots\!39}{19\!\cdots\!79}a+\frac{28\!\cdots\!82}{19\!\cdots\!79}$, $\frac{660526495380}{19\!\cdots\!79}a^{11}-\frac{898482656610}{19\!\cdots\!79}a^{10}-\frac{63270542749040}{19\!\cdots\!79}a^{9}-\frac{11130763713300}{19\!\cdots\!79}a^{8}+\frac{21\!\cdots\!20}{19\!\cdots\!79}a^{7}+\frac{31\!\cdots\!50}{19\!\cdots\!79}a^{6}-\frac{28\!\cdots\!54}{19\!\cdots\!79}a^{5}-\frac{69\!\cdots\!75}{19\!\cdots\!79}a^{4}+\frac{98\!\cdots\!70}{19\!\cdots\!79}a^{3}+\frac{40\!\cdots\!25}{19\!\cdots\!79}a^{2}+\frac{31\!\cdots\!30}{19\!\cdots\!79}a+\frac{53\!\cdots\!07}{19\!\cdots\!79}$, $\frac{4744541648160}{19\!\cdots\!79}a^{11}-\frac{6649426708676}{19\!\cdots\!79}a^{10}-\frac{454590183436180}{19\!\cdots\!79}a^{9}-\frac{61609446514725}{19\!\cdots\!79}a^{8}+\frac{15\!\cdots\!40}{19\!\cdots\!79}a^{7}+\frac{21\!\cdots\!70}{19\!\cdots\!79}a^{6}-\frac{20\!\cdots\!68}{19\!\cdots\!79}a^{5}-\frac{49\!\cdots\!05}{19\!\cdots\!79}a^{4}+\frac{74\!\cdots\!40}{19\!\cdots\!79}a^{3}+\frac{29\!\cdots\!00}{19\!\cdots\!79}a^{2}+\frac{23\!\cdots\!60}{19\!\cdots\!79}a+\frac{23\!\cdots\!41}{19\!\cdots\!79}$, $\frac{18654139582236}{19\!\cdots\!79}a^{11}-\frac{6156065569832}{19\!\cdots\!79}a^{10}-\frac{18\!\cdots\!90}{19\!\cdots\!79}a^{9}-\frac{18\!\cdots\!97}{19\!\cdots\!79}a^{8}+\frac{66\!\cdots\!62}{19\!\cdots\!79}a^{7}+\frac{13\!\cdots\!19}{19\!\cdots\!79}a^{6}-\frac{88\!\cdots\!80}{19\!\cdots\!79}a^{5}-\frac{25\!\cdots\!62}{19\!\cdots\!79}a^{4}+\frac{27\!\cdots\!30}{19\!\cdots\!79}a^{3}+\frac{14\!\cdots\!55}{19\!\cdots\!79}a^{2}+\frac{12\!\cdots\!50}{19\!\cdots\!79}a+\frac{21\!\cdots\!62}{19\!\cdots\!79}$, $\frac{17300284444140}{19\!\cdots\!79}a^{11}-\frac{21827582622343}{19\!\cdots\!79}a^{10}-\frac{16\!\cdots\!90}{19\!\cdots\!79}a^{9}-\frac{293164999697984}{19\!\cdots\!79}a^{8}+\frac{58\!\cdots\!90}{19\!\cdots\!79}a^{7}+\frac{78\!\cdots\!16}{19\!\cdots\!79}a^{6}-\frac{78\!\cdots\!50}{19\!\cdots\!79}a^{5}-\frac{17\!\cdots\!59}{19\!\cdots\!79}a^{4}+\frac{28\!\cdots\!40}{19\!\cdots\!79}a^{3}+\frac{10\!\cdots\!36}{19\!\cdots\!79}a^{2}+\frac{78\!\cdots\!70}{19\!\cdots\!79}a+\frac{12\!\cdots\!17}{19\!\cdots\!79}$, $\frac{17028182593636}{19\!\cdots\!79}a^{11}-\frac{56924767798053}{19\!\cdots\!79}a^{10}-\frac{14\!\cdots\!93}{19\!\cdots\!79}a^{9}+\frac{22\!\cdots\!66}{19\!\cdots\!79}a^{8}+\frac{46\!\cdots\!86}{19\!\cdots\!79}a^{7}+\frac{60\!\cdots\!70}{19\!\cdots\!79}a^{6}-\frac{60\!\cdots\!40}{19\!\cdots\!79}a^{5}-\frac{85\!\cdots\!55}{19\!\cdots\!79}a^{4}+\frac{22\!\cdots\!94}{19\!\cdots\!79}a^{3}+\frac{64\!\cdots\!51}{19\!\cdots\!79}a^{2}+\frac{50\!\cdots\!27}{19\!\cdots\!79}a+\frac{11\!\cdots\!17}{19\!\cdots\!79}$, $\frac{3354848502495}{19\!\cdots\!79}a^{11}-\frac{21906128641523}{19\!\cdots\!79}a^{10}-\frac{247847019307716}{19\!\cdots\!79}a^{9}+\frac{13\!\cdots\!06}{19\!\cdots\!79}a^{8}+\frac{72\!\cdots\!61}{19\!\cdots\!79}a^{7}-\frac{27\!\cdots\!87}{19\!\cdots\!79}a^{6}-\frac{98\!\cdots\!95}{19\!\cdots\!79}a^{5}+\frac{18\!\cdots\!51}{19\!\cdots\!79}a^{4}+\frac{54\!\cdots\!18}{19\!\cdots\!79}a^{3}-\frac{31\!\cdots\!34}{19\!\cdots\!79}a^{2}-\frac{38\!\cdots\!30}{19\!\cdots\!79}a-\frac{80\!\cdots\!79}{19\!\cdots\!79}$, $\frac{25962783726079}{19\!\cdots\!79}a^{11}-\frac{74409633649635}{19\!\cdots\!79}a^{10}-\frac{23\!\cdots\!08}{19\!\cdots\!79}a^{9}+\frac{27\!\cdots\!31}{19\!\cdots\!79}a^{8}+\frac{75\!\cdots\!74}{19\!\cdots\!79}a^{7}+\frac{27\!\cdots\!29}{19\!\cdots\!79}a^{6}-\frac{99\!\cdots\!98}{19\!\cdots\!79}a^{5}-\frac{15\!\cdots\!87}{19\!\cdots\!79}a^{4}+\frac{37\!\cdots\!01}{19\!\cdots\!79}a^{3}+\frac{10\!\cdots\!59}{19\!\cdots\!79}a^{2}+\frac{80\!\cdots\!15}{19\!\cdots\!79}a+\frac{12\!\cdots\!49}{19\!\cdots\!79}$, $\frac{8456486598536}{19\!\cdots\!79}a^{11}-\frac{33063567950087}{19\!\cdots\!79}a^{10}-\frac{703582960938740}{19\!\cdots\!79}a^{9}+\frac{15\!\cdots\!58}{19\!\cdots\!79}a^{8}+\frac{22\!\cdots\!06}{19\!\cdots\!79}a^{7}-\frac{11\!\cdots\!15}{19\!\cdots\!79}a^{6}-\frac{28\!\cdots\!66}{19\!\cdots\!79}a^{5}-\frac{23\!\cdots\!30}{19\!\cdots\!79}a^{4}+\frac{11\!\cdots\!41}{19\!\cdots\!79}a^{3}+\frac{23\!\cdots\!96}{19\!\cdots\!79}a^{2}+\frac{14\!\cdots\!95}{19\!\cdots\!79}a+\frac{22\!\cdots\!29}{19\!\cdots\!79}$
| 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: | \( 58663402.6391 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 58663402.6391 \cdot 3}{2\cdot\sqrt{1985246242140168000000000}}\cr\approx \mathstrut & 0.255806317248 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 99*x^10 - 140*x^9 + 3366*x^8 + 8820*x^7 - 40049*x^6 - 160020*x^5 + 39531*x^4 + 826140*x^3 + 1229700*x^2 + 647850*x + 91405)
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 - 99*x^10 - 140*x^9 + 3366*x^8 + 8820*x^7 - 40049*x^6 - 160020*x^5 + 39531*x^4 + 826140*x^3 + 1229700*x^2 + 647850*x + 91405, 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 - 99*x^10 - 140*x^9 + 3366*x^8 + 8820*x^7 - 40049*x^6 - 160020*x^5 + 39531*x^4 + 826140*x^3 + 1229700*x^2 + 647850*x + 91405);
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 - 99*x^10 - 140*x^9 + 3366*x^8 + 8820*x^7 - 40049*x^6 - 160020*x^5 + 39531*x^4 + 826140*x^3 + 1229700*x^2 + 647850*x + 91405);
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
| \(\Q(\sqrt{5}) \), 3.3.3969.2, \(\Q(\zeta_{20})^+\), 6.6.1969120125.2 |
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 | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.12.0.1}{12} }$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.25 | $x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ |
|
\(3\)
| 3.12.16.25 | $x^{12} + 24 x^{11} + 216 x^{10} + 804 x^{9} + 216 x^{8} - 6480 x^{7} - 11610 x^{6} + 16200 x^{5} + 48600 x^{4} + 33156 x^{3} + 198936 x^{2} + 190593$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ |
|
\(5\)
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(7\)
| 7.12.8.2 | $x^{12} - 70 x^{9} + 1519 x^{6} - 4802 x^{3} + 21609$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |