Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 99*x^10 + 108*x^9 + 3615*x^8 + 1617*x^7 - 55041*x^6 - 87213*x^5 + 268551*x^4 + 688433*x^3 + 130815*x^2 - 469788*x - 108289)
gp: K = bnfinit(y^12 - 3*y^11 - 99*y^10 + 108*y^9 + 3615*y^8 + 1617*y^7 - 55041*y^6 - 87213*y^5 + 268551*y^4 + 688433*y^3 + 130815*y^2 - 469788*y - 108289, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 - 99*x^10 + 108*x^9 + 3615*x^8 + 1617*x^7 - 55041*x^6 - 87213*x^5 + 268551*x^4 + 688433*x^3 + 130815*x^2 - 469788*x - 108289);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 3*x^11 - 99*x^10 + 108*x^9 + 3615*x^8 + 1617*x^7 - 55041*x^6 - 87213*x^5 + 268551*x^4 + 688433*x^3 + 130815*x^2 - 469788*x - 108289)
\( x^{12} - 3 x^{11} - 99 x^{10} + 108 x^{9} + 3615 x^{8} + 1617 x^{7} - 55041 x^{6} - 87213 x^{5} + \cdots - 108289 \)
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: |
\(29428267022381449968353937\)
\(\medspace = 3^{16}\cdot 7^{8}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(132.56\) | 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: | $3^{4/3}7^{2/3}17^{3/4}\approx 132.55528790956586$ | ||
| Ramified primes: |
\(3\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1071=3^{2}\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1071}(256,·)$, $\chi_{1071}(1,·)$, $\chi_{1071}(67,·)$, $\chi_{1071}(4,·)$, $\chi_{1071}(1024,·)$, $\chi_{1071}(64,·)$, $\chi_{1071}(268,·)$, $\chi_{1071}(205,·)$, $\chi_{1071}(16,·)$, $\chi_{1071}(883,·)$, $\chi_{1071}(820,·)$, $\chi_{1071}(319,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{34\!\cdots\!86}a^{11}+\frac{55\!\cdots\!81}{34\!\cdots\!86}a^{10}+\frac{29\!\cdots\!93}{34\!\cdots\!86}a^{9}-\frac{16\!\cdots\!95}{34\!\cdots\!86}a^{8}-\frac{16\!\cdots\!77}{34\!\cdots\!86}a^{7}-\frac{54\!\cdots\!50}{17\!\cdots\!43}a^{6}+\frac{10\!\cdots\!97}{34\!\cdots\!86}a^{5}+\frac{15\!\cdots\!73}{17\!\cdots\!43}a^{4}+\frac{23\!\cdots\!00}{17\!\cdots\!43}a^{3}+\frac{10\!\cdots\!03}{34\!\cdots\!86}a^{2}-\frac{15\!\cdots\!57}{34\!\cdots\!86}a+\frac{25\!\cdots\!16}{17\!\cdots\!43}$
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: |
$\frac{34\!\cdots\!21}{34\!\cdots\!86}a^{11}-\frac{99\!\cdots\!00}{17\!\cdots\!43}a^{10}-\frac{29\!\cdots\!51}{34\!\cdots\!86}a^{9}+\frac{57\!\cdots\!35}{17\!\cdots\!43}a^{8}+\frac{95\!\cdots\!27}{34\!\cdots\!86}a^{7}-\frac{19\!\cdots\!03}{34\!\cdots\!86}a^{6}-\frac{70\!\cdots\!45}{17\!\cdots\!43}a^{5}+\frac{32\!\cdots\!27}{17\!\cdots\!43}a^{4}+\frac{38\!\cdots\!31}{17\!\cdots\!43}a^{3}+\frac{38\!\cdots\!35}{34\!\cdots\!86}a^{2}-\frac{28\!\cdots\!39}{17\!\cdots\!43}a-\frac{63\!\cdots\!71}{17\!\cdots\!43}$, $\frac{71\!\cdots\!51}{17\!\cdots\!43}a^{11}-\frac{39\!\cdots\!28}{17\!\cdots\!43}a^{10}-\frac{60\!\cdots\!81}{17\!\cdots\!43}a^{9}+\frac{23\!\cdots\!95}{17\!\cdots\!43}a^{8}+\frac{19\!\cdots\!17}{17\!\cdots\!43}a^{7}-\frac{40\!\cdots\!07}{17\!\cdots\!43}a^{6}-\frac{28\!\cdots\!88}{17\!\cdots\!43}a^{5}+\frac{15\!\cdots\!39}{17\!\cdots\!43}a^{4}+\frac{15\!\cdots\!52}{17\!\cdots\!43}a^{3}+\frac{61\!\cdots\!42}{17\!\cdots\!43}a^{2}-\frac{10\!\cdots\!10}{17\!\cdots\!43}a+\frac{23\!\cdots\!94}{17\!\cdots\!43}$, $\frac{63\!\cdots\!99}{34\!\cdots\!86}a^{11}-\frac{17\!\cdots\!06}{17\!\cdots\!43}a^{10}-\frac{54\!\cdots\!09}{34\!\cdots\!86}a^{9}+\frac{10\!\cdots\!15}{17\!\cdots\!43}a^{8}+\frac{17\!\cdots\!13}{34\!\cdots\!86}a^{7}-\frac{33\!\cdots\!73}{34\!\cdots\!86}a^{6}-\frac{13\!\cdots\!03}{17\!\cdots\!43}a^{5}+\frac{47\!\cdots\!48}{17\!\cdots\!43}a^{4}+\frac{72\!\cdots\!89}{17\!\cdots\!43}a^{3}+\frac{81\!\cdots\!43}{34\!\cdots\!86}a^{2}-\frac{55\!\cdots\!98}{17\!\cdots\!43}a-\frac{12\!\cdots\!92}{17\!\cdots\!43}$, $\frac{58\!\cdots\!16}{17\!\cdots\!43}a^{11}-\frac{33\!\cdots\!70}{17\!\cdots\!43}a^{10}-\frac{49\!\cdots\!50}{17\!\cdots\!43}a^{9}+\frac{19\!\cdots\!91}{17\!\cdots\!43}a^{8}+\frac{16\!\cdots\!90}{17\!\cdots\!43}a^{7}-\frac{34\!\cdots\!06}{17\!\cdots\!43}a^{6}-\frac{23\!\cdots\!86}{17\!\cdots\!43}a^{5}+\frac{13\!\cdots\!72}{17\!\cdots\!43}a^{4}+\frac{12\!\cdots\!84}{17\!\cdots\!43}a^{3}+\frac{51\!\cdots\!49}{17\!\cdots\!43}a^{2}-\frac{10\!\cdots\!84}{17\!\cdots\!43}a-\frac{12\!\cdots\!74}{17\!\cdots\!43}$, $\frac{51\!\cdots\!65}{17\!\cdots\!43}a^{11}-\frac{29\!\cdots\!42}{17\!\cdots\!43}a^{10}-\frac{43\!\cdots\!69}{17\!\cdots\!43}a^{9}+\frac{17\!\cdots\!96}{17\!\cdots\!43}a^{8}+\frac{14\!\cdots\!73}{17\!\cdots\!43}a^{7}-\frac{30\!\cdots\!99}{17\!\cdots\!43}a^{6}-\frac{20\!\cdots\!98}{17\!\cdots\!43}a^{5}+\frac{12\!\cdots\!33}{17\!\cdots\!43}a^{4}+\frac{11\!\cdots\!32}{17\!\cdots\!43}a^{3}+\frac{44\!\cdots\!07}{17\!\cdots\!43}a^{2}-\frac{90\!\cdots\!31}{17\!\cdots\!43}a-\frac{21\!\cdots\!40}{17\!\cdots\!43}$, $\frac{60\!\cdots\!11}{34\!\cdots\!86}a^{11}-\frac{25\!\cdots\!29}{34\!\cdots\!86}a^{10}-\frac{57\!\cdots\!45}{34\!\cdots\!86}a^{9}+\frac{13\!\cdots\!71}{34\!\cdots\!86}a^{8}+\frac{20\!\cdots\!19}{34\!\cdots\!86}a^{7}-\frac{90\!\cdots\!39}{17\!\cdots\!43}a^{6}-\frac{32\!\cdots\!13}{34\!\cdots\!86}a^{5}-\frac{43\!\cdots\!39}{17\!\cdots\!43}a^{4}+\frac{92\!\cdots\!88}{17\!\cdots\!43}a^{3}+\frac{16\!\cdots\!97}{34\!\cdots\!86}a^{2}-\frac{16\!\cdots\!23}{34\!\cdots\!86}a-\frac{21\!\cdots\!36}{17\!\cdots\!43}$, $\frac{57\!\cdots\!15}{34\!\cdots\!86}a^{11}-\frac{23\!\cdots\!51}{34\!\cdots\!86}a^{10}-\frac{26\!\cdots\!08}{17\!\cdots\!43}a^{9}+\frac{10\!\cdots\!23}{34\!\cdots\!86}a^{8}+\frac{93\!\cdots\!78}{17\!\cdots\!43}a^{7}-\frac{20\!\cdots\!62}{17\!\cdots\!43}a^{6}-\frac{14\!\cdots\!21}{17\!\cdots\!43}a^{5}-\frac{30\!\cdots\!55}{34\!\cdots\!86}a^{4}+\frac{72\!\cdots\!07}{17\!\cdots\!43}a^{3}+\frac{28\!\cdots\!45}{34\!\cdots\!86}a^{2}-\frac{12\!\cdots\!47}{34\!\cdots\!86}a-\frac{21\!\cdots\!91}{34\!\cdots\!86}$, $\frac{44\!\cdots\!22}{17\!\cdots\!43}a^{11}-\frac{51\!\cdots\!17}{34\!\cdots\!86}a^{10}-\frac{74\!\cdots\!89}{34\!\cdots\!86}a^{9}+\frac{30\!\cdots\!59}{34\!\cdots\!86}a^{8}+\frac{24\!\cdots\!29}{34\!\cdots\!86}a^{7}-\frac{52\!\cdots\!01}{34\!\cdots\!86}a^{6}-\frac{17\!\cdots\!03}{17\!\cdots\!43}a^{5}+\frac{20\!\cdots\!61}{34\!\cdots\!86}a^{4}+\frac{97\!\cdots\!45}{17\!\cdots\!43}a^{3}+\frac{41\!\cdots\!88}{17\!\cdots\!43}a^{2}-\frac{16\!\cdots\!53}{34\!\cdots\!86}a-\frac{39\!\cdots\!71}{34\!\cdots\!86}$, $\frac{49\!\cdots\!82}{17\!\cdots\!43}a^{11}-\frac{26\!\cdots\!16}{17\!\cdots\!43}a^{10}-\frac{84\!\cdots\!23}{34\!\cdots\!86}a^{9}+\frac{15\!\cdots\!27}{17\!\cdots\!43}a^{8}+\frac{27\!\cdots\!05}{34\!\cdots\!86}a^{7}-\frac{25\!\cdots\!31}{17\!\cdots\!43}a^{6}-\frac{41\!\cdots\!33}{34\!\cdots\!86}a^{5}+\frac{13\!\cdots\!19}{34\!\cdots\!86}a^{4}+\frac{11\!\cdots\!31}{17\!\cdots\!43}a^{3}+\frac{65\!\cdots\!97}{17\!\cdots\!43}a^{2}-\frac{87\!\cdots\!08}{17\!\cdots\!43}a-\frac{45\!\cdots\!73}{34\!\cdots\!86}$, $\frac{46\!\cdots\!47}{34\!\cdots\!86}a^{11}-\frac{50\!\cdots\!13}{17\!\cdots\!43}a^{10}-\frac{24\!\cdots\!85}{17\!\cdots\!43}a^{9}+\frac{10\!\cdots\!81}{17\!\cdots\!43}a^{8}+\frac{89\!\cdots\!35}{17\!\cdots\!43}a^{7}+\frac{15\!\cdots\!71}{34\!\cdots\!86}a^{6}-\frac{26\!\cdots\!11}{34\!\cdots\!86}a^{5}-\frac{50\!\cdots\!97}{34\!\cdots\!86}a^{4}+\frac{55\!\cdots\!66}{17\!\cdots\!43}a^{3}+\frac{33\!\cdots\!91}{34\!\cdots\!86}a^{2}+\frac{10\!\cdots\!86}{17\!\cdots\!43}a+\frac{33\!\cdots\!33}{34\!\cdots\!86}$, $\frac{44\!\cdots\!31}{17\!\cdots\!43}a^{11}-\frac{19\!\cdots\!28}{17\!\cdots\!43}a^{10}-\frac{40\!\cdots\!17}{17\!\cdots\!43}a^{9}+\frac{10\!\cdots\!49}{17\!\cdots\!43}a^{8}+\frac{14\!\cdots\!73}{17\!\cdots\!43}a^{7}-\frac{13\!\cdots\!74}{17\!\cdots\!43}a^{6}-\frac{21\!\cdots\!48}{17\!\cdots\!43}a^{5}-\frac{47\!\cdots\!27}{17\!\cdots\!43}a^{4}+\frac{10\!\cdots\!48}{17\!\cdots\!43}a^{3}+\frac{89\!\cdots\!32}{17\!\cdots\!43}a^{2}-\frac{29\!\cdots\!83}{17\!\cdots\!43}a+\frac{10\!\cdots\!03}{17\!\cdots\!43}$
| 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: | \( 144132247.105 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 144132247.105 \cdot 3}{2\cdot\sqrt{29428267022381449968353937}}\cr\approx \mathstrut & 0.163241290942 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 99*x^10 + 108*x^9 + 3615*x^8 + 1617*x^7 - 55041*x^6 - 87213*x^5 + 268551*x^4 + 688433*x^3 + 130815*x^2 - 469788*x - 108289)
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 - 3*x^11 - 99*x^10 + 108*x^9 + 3615*x^8 + 1617*x^7 - 55041*x^6 - 87213*x^5 + 268551*x^4 + 688433*x^3 + 130815*x^2 - 469788*x - 108289, 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 - 3*x^11 - 99*x^10 + 108*x^9 + 3615*x^8 + 1617*x^7 - 55041*x^6 - 87213*x^5 + 268551*x^4 + 688433*x^3 + 130815*x^2 - 469788*x - 108289);
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 - 3*x^11 - 99*x^10 + 108*x^9 + 3615*x^8 + 1617*x^7 - 55041*x^6 - 87213*x^5 + 268551*x^4 + 688433*x^3 + 130815*x^2 - 469788*x - 108289);
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
| \(\Q(\sqrt{17}) \), 3.3.3969.2, 4.4.4913.1, 6.6.77394297393.2 |
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 | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.12.16.25 | $x^{12} + 24 x^{11} + 216 x^{10} + 804 x^{9} + 216 x^{8} - 6480 x^{7} - 11610 x^{6} + 16200 x^{5} + 48600 x^{4} + 33156 x^{3} + 198936 x^{2} + 190593$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ |
|
\(7\)
| 7.12.8.2 | $x^{12} - 70 x^{9} + 1519 x^{6} - 4802 x^{3} + 21609$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |