Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 21*x^10 + 33*x^9 + 898*x^8 + 137*x^7 + 14950*x^6 + 14307*x^5 + 235426*x^4 - 424511*x^3 + 3757769*x^2 - 2920010*x + 16177451)
gp: K = bnfinit(y^12 - y^11 + 21*y^10 + 33*y^9 + 898*y^8 + 137*y^7 + 14950*y^6 + 14307*y^5 + 235426*y^4 - 424511*y^3 + 3757769*y^2 - 2920010*y + 16177451, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 21*x^10 + 33*x^9 + 898*x^8 + 137*x^7 + 14950*x^6 + 14307*x^5 + 235426*x^4 - 424511*x^3 + 3757769*x^2 - 2920010*x + 16177451);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 21*x^10 + 33*x^9 + 898*x^8 + 137*x^7 + 14950*x^6 + 14307*x^5 + 235426*x^4 - 424511*x^3 + 3757769*x^2 - 2920010*x + 16177451)
\( x^{12} - x^{11} + 21 x^{10} + 33 x^{9} + 898 x^{8} + 137 x^{7} + 14950 x^{6} + 14307 x^{5} + \cdots + 16177451 \)
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: |
\(40208430206472111189453125\)
\(\medspace = 5^{9}\cdot 17^{6}\cdot 31^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(136.05\) | 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: | $5^{3/4}17^{1/2}31^{2/3}\approx 136.04829178250532$ | ||
| Ramified primes: |
\(5\), \(17\), \(31\)
| 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: | \(2635=5\cdot 17\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2635}(1,·)$, $\chi_{2635}(67,·)$, $\chi_{2635}(2381,·)$, $\chi_{2635}(749,·)$, $\chi_{2635}(1648,·)$, $\chi_{2635}(1427,·)$, $\chi_{2635}(1172,·)$, $\chi_{2635}(373,·)$, $\chi_{2635}(118,·)$, $\chi_{2635}(1276,·)$, $\chi_{2635}(2109,·)$, $\chi_{2635}(1854,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | 4.0.36125.1$^{2}$, 12.0.40208430206472111189453125.1$^{30}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{58}a^{8}+\frac{10}{29}a^{7}-\frac{14}{29}a^{6}+\frac{3}{29}a^{5}+\frac{5}{58}a^{4}+\frac{8}{29}a^{3}+\frac{5}{58}a^{2}-\frac{23}{58}a+\frac{5}{58}$, $\frac{1}{58}a^{9}-\frac{11}{29}a^{7}-\frac{7}{29}a^{6}+\frac{1}{58}a^{5}-\frac{13}{29}a^{4}-\frac{25}{58}a^{3}-\frac{7}{58}a^{2}+\frac{1}{58}a+\frac{8}{29}$, $\frac{1}{58}a^{10}+\frac{10}{29}a^{7}+\frac{23}{58}a^{6}-\frac{5}{29}a^{5}+\frac{27}{58}a^{4}-\frac{3}{58}a^{3}-\frac{5}{58}a^{2}-\frac{13}{29}a-\frac{3}{29}$, $\frac{1}{10\!\cdots\!82}a^{11}+\frac{18\!\cdots\!29}{54\!\cdots\!41}a^{10}-\frac{32\!\cdots\!39}{54\!\cdots\!41}a^{9}+\frac{14\!\cdots\!37}{54\!\cdots\!41}a^{8}+\frac{53\!\cdots\!49}{10\!\cdots\!82}a^{7}-\frac{15\!\cdots\!83}{54\!\cdots\!41}a^{6}-\frac{48\!\cdots\!55}{10\!\cdots\!82}a^{5}-\frac{15\!\cdots\!67}{10\!\cdots\!82}a^{4}+\frac{42\!\cdots\!97}{10\!\cdots\!82}a^{3}-\frac{65\!\cdots\!17}{54\!\cdots\!41}a^{2}-\frac{99\!\cdots\!93}{54\!\cdots\!41}a+\frac{23\!\cdots\!10}{54\!\cdots\!41}$
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
$C_{11714}$, which has order $11714$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $11714$ (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: |
$\frac{149136897810}{13\!\cdots\!41}a^{11}-\frac{106156954490}{13\!\cdots\!41}a^{10}-\frac{622675360068}{13\!\cdots\!41}a^{9}+\frac{223557438657}{13\!\cdots\!41}a^{8}+\frac{110795580342502}{13\!\cdots\!41}a^{7}-\frac{162128648755810}{13\!\cdots\!41}a^{6}-\frac{389498917891596}{13\!\cdots\!41}a^{5}-\frac{10\!\cdots\!76}{13\!\cdots\!41}a^{4}+\frac{19\!\cdots\!60}{13\!\cdots\!41}a^{3}-\frac{25\!\cdots\!73}{47\!\cdots\!29}a^{2}+\frac{14\!\cdots\!84}{13\!\cdots\!41}a+\frac{18\!\cdots\!83}{13\!\cdots\!41}$, $\frac{47\!\cdots\!57}{10\!\cdots\!82}a^{11}-\frac{41\!\cdots\!55}{10\!\cdots\!82}a^{10}-\frac{39\!\cdots\!87}{54\!\cdots\!41}a^{9}+\frac{88\!\cdots\!64}{54\!\cdots\!41}a^{8}+\frac{31\!\cdots\!41}{10\!\cdots\!82}a^{7}-\frac{35\!\cdots\!63}{10\!\cdots\!82}a^{6}-\frac{17\!\cdots\!77}{10\!\cdots\!82}a^{5}+\frac{72\!\cdots\!41}{54\!\cdots\!41}a^{4}+\frac{27\!\cdots\!02}{54\!\cdots\!41}a^{3}-\frac{17\!\cdots\!97}{10\!\cdots\!82}a^{2}+\frac{21\!\cdots\!27}{54\!\cdots\!41}a-\frac{39\!\cdots\!91}{54\!\cdots\!41}$, $\frac{68\!\cdots\!75}{10\!\cdots\!82}a^{11}+\frac{21\!\cdots\!17}{10\!\cdots\!82}a^{10}+\frac{53\!\cdots\!07}{54\!\cdots\!41}a^{9}+\frac{82\!\cdots\!51}{10\!\cdots\!82}a^{8}+\frac{71\!\cdots\!33}{10\!\cdots\!82}a^{7}+\frac{24\!\cdots\!61}{10\!\cdots\!82}a^{6}+\frac{89\!\cdots\!17}{10\!\cdots\!82}a^{5}+\frac{44\!\cdots\!71}{10\!\cdots\!82}a^{4}+\frac{84\!\cdots\!04}{54\!\cdots\!41}a^{3}+\frac{70\!\cdots\!83}{54\!\cdots\!41}a^{2}+\frac{81\!\cdots\!33}{10\!\cdots\!82}a+\frac{77\!\cdots\!15}{10\!\cdots\!82}$, $\frac{66\!\cdots\!45}{10\!\cdots\!82}a^{11}+\frac{10\!\cdots\!13}{10\!\cdots\!82}a^{10}+\frac{70\!\cdots\!82}{54\!\cdots\!41}a^{9}+\frac{25\!\cdots\!38}{54\!\cdots\!41}a^{8}+\frac{70\!\cdots\!93}{10\!\cdots\!82}a^{7}+\frac{15\!\cdots\!41}{10\!\cdots\!82}a^{6}+\frac{10\!\cdots\!83}{10\!\cdots\!82}a^{5}+\frac{11\!\cdots\!38}{54\!\cdots\!41}a^{4}+\frac{94\!\cdots\!69}{54\!\cdots\!41}a^{3}-\frac{46\!\cdots\!39}{10\!\cdots\!82}a^{2}+\frac{44\!\cdots\!99}{54\!\cdots\!41}a-\frac{15\!\cdots\!91}{54\!\cdots\!41}$, $\frac{71\!\cdots\!06}{54\!\cdots\!41}a^{11}+\frac{33\!\cdots\!72}{54\!\cdots\!41}a^{10}+\frac{72\!\cdots\!37}{54\!\cdots\!41}a^{9}+\frac{24\!\cdots\!59}{10\!\cdots\!82}a^{8}+\frac{32\!\cdots\!15}{54\!\cdots\!41}a^{7}+\frac{44\!\cdots\!13}{54\!\cdots\!41}a^{6}+\frac{64\!\cdots\!00}{54\!\cdots\!41}a^{5}+\frac{16\!\cdots\!45}{10\!\cdots\!82}a^{4}+\frac{10\!\cdots\!23}{54\!\cdots\!41}a^{3}+\frac{85\!\cdots\!75}{10\!\cdots\!82}a^{2}+\frac{29\!\cdots\!65}{37\!\cdots\!58}a+\frac{26\!\cdots\!47}{10\!\cdots\!82}$
| 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: | \( 4882.16021651 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 4882.16021651 \cdot 11714}{2\cdot\sqrt{40208430206472111189453125}}\cr\approx \mathstrut & 0.277464804825 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 21*x^10 + 33*x^9 + 898*x^8 + 137*x^7 + 14950*x^6 + 14307*x^5 + 235426*x^4 - 424511*x^3 + 3757769*x^2 - 2920010*x + 16177451)
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 + 21*x^10 + 33*x^9 + 898*x^8 + 137*x^7 + 14950*x^6 + 14307*x^5 + 235426*x^4 - 424511*x^3 + 3757769*x^2 - 2920010*x + 16177451, 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 + 21*x^10 + 33*x^9 + 898*x^8 + 137*x^7 + 14950*x^6 + 14307*x^5 + 235426*x^4 - 424511*x^3 + 3757769*x^2 - 2920010*x + 16177451);
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 + 21*x^10 + 33*x^9 + 898*x^8 + 137*x^7 + 14950*x^6 + 14307*x^5 + 235426*x^4 - 424511*x^3 + 3757769*x^2 - 2920010*x + 16177451);
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
| \(\Q(\sqrt{5}) \), 3.3.961.1, 4.0.36125.1, 6.6.115440125.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{3}$ | ${\href{/padicField/3.12.0.1}{12} }$ | R | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.1.0.1}{1} }^{12}$ | R | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(17\)
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
|
\(31\)
| 31.6.4.1 | $x^{6} + 87 x^{5} + 2532 x^{4} + 24973 x^{3} + 10293 x^{2} + 78438 x + 748956$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 31.6.4.1 | $x^{6} + 87 x^{5} + 2532 x^{4} + 24973 x^{3} + 10293 x^{2} + 78438 x + 748956$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |