Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 75*x^10 + 58*x^9 + 1982*x^8 - 987*x^7 - 22024*x^6 + 6367*x^5 + 89299*x^4 - 25629*x^3 - 66876*x^2 - 18568*x - 883)
gp: K = bnfinit(y^12 - y^11 - 75*y^10 + 58*y^9 + 1982*y^8 - 987*y^7 - 22024*y^6 + 6367*y^5 + 89299*y^4 - 25629*y^3 - 66876*y^2 - 18568*y - 883, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 75*x^10 + 58*x^9 + 1982*x^8 - 987*x^7 - 22024*x^6 + 6367*x^5 + 89299*x^4 - 25629*x^3 - 66876*x^2 - 18568*x - 883);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 75*x^10 + 58*x^9 + 1982*x^8 - 987*x^7 - 22024*x^6 + 6367*x^5 + 89299*x^4 - 25629*x^3 - 66876*x^2 - 18568*x - 883)
\( x^{12} - x^{11} - 75 x^{10} + 58 x^{9} + 1982 x^{8} - 987 x^{7} - 22024 x^{6} + 6367 x^{5} + 89299 x^{4} + \cdots - 883 \)
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: |
\(16348345805451893172953\)
\(\medspace = 13^{10}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(70.98\) | 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: | $13^{5/6}17^{3/4}\approx 70.9778464077956$ | ||
| Ramified primes: |
\(13\), \(17\)
| 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: | \(221=13\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{221}(64,·)$, $\chi_{221}(1,·)$, $\chi_{221}(35,·)$, $\chi_{221}(4,·)$, $\chi_{221}(38,·)$, $\chi_{221}(166,·)$, $\chi_{221}(140,·)$, $\chi_{221}(16,·)$, $\chi_{221}(152,·)$, $\chi_{221}(118,·)$, $\chi_{221}(120,·)$, $\chi_{221}(30,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{69\!\cdots\!88}a^{11}+\frac{55\!\cdots\!13}{34\!\cdots\!94}a^{10}+\frac{12\!\cdots\!47}{69\!\cdots\!88}a^{9}-\frac{11\!\cdots\!39}{69\!\cdots\!88}a^{8}-\frac{20\!\cdots\!61}{69\!\cdots\!88}a^{7}-\frac{15\!\cdots\!99}{34\!\cdots\!94}a^{6}+\frac{25\!\cdots\!05}{34\!\cdots\!94}a^{5}+\frac{22\!\cdots\!01}{69\!\cdots\!88}a^{4}-\frac{91\!\cdots\!95}{34\!\cdots\!94}a^{3}-\frac{12\!\cdots\!11}{69\!\cdots\!88}a^{2}+\frac{68\!\cdots\!29}{69\!\cdots\!88}a+\frac{33\!\cdots\!37}{69\!\cdots\!88}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$ (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: |
$\frac{185040226691917}{16\!\cdots\!99}a^{11}-\frac{759848391571835}{33\!\cdots\!98}a^{10}-\frac{26\!\cdots\!61}{33\!\cdots\!98}a^{9}+\frac{23\!\cdots\!19}{16\!\cdots\!99}a^{8}+\frac{67\!\cdots\!11}{33\!\cdots\!98}a^{7}-\frac{46\!\cdots\!57}{16\!\cdots\!99}a^{6}-\frac{35\!\cdots\!22}{16\!\cdots\!99}a^{5}+\frac{38\!\cdots\!24}{16\!\cdots\!99}a^{4}+\frac{26\!\cdots\!07}{33\!\cdots\!98}a^{3}-\frac{25\!\cdots\!17}{33\!\cdots\!98}a^{2}-\frac{60\!\cdots\!34}{16\!\cdots\!99}a-\frac{15\!\cdots\!95}{33\!\cdots\!98}$, $\frac{17\!\cdots\!07}{67\!\cdots\!96}a^{11}-\frac{764629925327854}{16\!\cdots\!99}a^{10}-\frac{12\!\cdots\!73}{67\!\cdots\!96}a^{9}+\frac{19\!\cdots\!81}{67\!\cdots\!96}a^{8}+\frac{33\!\cdots\!79}{67\!\cdots\!96}a^{7}-\frac{20\!\cdots\!53}{33\!\cdots\!98}a^{6}-\frac{18\!\cdots\!61}{33\!\cdots\!98}a^{5}+\frac{35\!\cdots\!43}{67\!\cdots\!96}a^{4}+\frac{35\!\cdots\!93}{16\!\cdots\!99}a^{3}-\frac{13\!\cdots\!75}{67\!\cdots\!96}a^{2}-\frac{63\!\cdots\!99}{67\!\cdots\!96}a+\frac{14\!\cdots\!89}{67\!\cdots\!96}$, $\frac{67\!\cdots\!55}{69\!\cdots\!88}a^{11}-\frac{32\!\cdots\!45}{34\!\cdots\!94}a^{10}-\frac{50\!\cdots\!15}{69\!\cdots\!88}a^{9}+\frac{37\!\cdots\!77}{69\!\cdots\!88}a^{8}+\frac{13\!\cdots\!05}{69\!\cdots\!88}a^{7}-\frac{31\!\cdots\!67}{34\!\cdots\!94}a^{6}-\frac{73\!\cdots\!31}{34\!\cdots\!94}a^{5}+\frac{41\!\cdots\!19}{69\!\cdots\!88}a^{4}+\frac{29\!\cdots\!31}{34\!\cdots\!94}a^{3}-\frac{18\!\cdots\!29}{69\!\cdots\!88}a^{2}-\frac{43\!\cdots\!91}{69\!\cdots\!88}a-\frac{10\!\cdots\!69}{69\!\cdots\!88}$, $\frac{34\!\cdots\!09}{69\!\cdots\!88}a^{11}-\frac{11\!\cdots\!35}{17\!\cdots\!97}a^{10}-\frac{26\!\cdots\!15}{69\!\cdots\!88}a^{9}+\frac{27\!\cdots\!19}{69\!\cdots\!88}a^{8}+\frac{68\!\cdots\!69}{69\!\cdots\!88}a^{7}-\frac{26\!\cdots\!29}{34\!\cdots\!94}a^{6}-\frac{38\!\cdots\!31}{34\!\cdots\!94}a^{5}+\frac{42\!\cdots\!09}{69\!\cdots\!88}a^{4}+\frac{79\!\cdots\!18}{17\!\cdots\!97}a^{3}-\frac{16\!\cdots\!53}{69\!\cdots\!88}a^{2}-\frac{25\!\cdots\!53}{69\!\cdots\!88}a-\frac{52\!\cdots\!97}{69\!\cdots\!88}$, $\frac{41\!\cdots\!89}{69\!\cdots\!88}a^{11}-\frac{16\!\cdots\!72}{17\!\cdots\!97}a^{10}-\frac{25\!\cdots\!07}{69\!\cdots\!88}a^{9}+\frac{34\!\cdots\!89}{69\!\cdots\!88}a^{8}+\frac{50\!\cdots\!57}{69\!\cdots\!88}a^{7}-\frac{22\!\cdots\!37}{34\!\cdots\!94}a^{6}-\frac{15\!\cdots\!69}{34\!\cdots\!94}a^{5}+\frac{55\!\cdots\!41}{69\!\cdots\!88}a^{4}-\frac{83\!\cdots\!05}{17\!\cdots\!97}a^{3}+\frac{90\!\cdots\!07}{69\!\cdots\!88}a^{2}+\frac{35\!\cdots\!57}{69\!\cdots\!88}a+\frac{36\!\cdots\!75}{69\!\cdots\!88}$, $\frac{60\!\cdots\!33}{17\!\cdots\!97}a^{11}-\frac{11\!\cdots\!19}{17\!\cdots\!97}a^{10}-\frac{85\!\cdots\!39}{34\!\cdots\!94}a^{9}+\frac{13\!\cdots\!93}{34\!\cdots\!94}a^{8}+\frac{21\!\cdots\!43}{34\!\cdots\!94}a^{7}-\frac{11\!\cdots\!46}{17\!\cdots\!97}a^{6}-\frac{10\!\cdots\!94}{17\!\cdots\!97}a^{5}+\frac{73\!\cdots\!14}{17\!\cdots\!97}a^{4}+\frac{38\!\cdots\!73}{17\!\cdots\!97}a^{3}-\frac{31\!\cdots\!97}{34\!\cdots\!94}a^{2}-\frac{56\!\cdots\!39}{34\!\cdots\!94}a-\frac{12\!\cdots\!21}{34\!\cdots\!94}$, $\frac{20\!\cdots\!99}{17\!\cdots\!97}a^{11}+\frac{12937572763721}{34\!\cdots\!94}a^{10}-\frac{27\!\cdots\!73}{34\!\cdots\!94}a^{9}-\frac{11\!\cdots\!51}{34\!\cdots\!94}a^{8}+\frac{32\!\cdots\!00}{17\!\cdots\!97}a^{7}+\frac{25\!\cdots\!30}{17\!\cdots\!97}a^{6}-\frac{29\!\cdots\!75}{17\!\cdots\!97}a^{5}+\frac{30\!\cdots\!07}{17\!\cdots\!97}a^{4}+\frac{20\!\cdots\!07}{34\!\cdots\!94}a^{3}-\frac{14\!\cdots\!63}{34\!\cdots\!94}a^{2}-\frac{86\!\cdots\!11}{34\!\cdots\!94}a-\frac{34\!\cdots\!05}{17\!\cdots\!97}$, $\frac{50\!\cdots\!95}{34\!\cdots\!94}a^{11}-\frac{38\!\cdots\!26}{17\!\cdots\!97}a^{10}-\frac{18\!\cdots\!46}{17\!\cdots\!97}a^{9}+\frac{49\!\cdots\!17}{34\!\cdots\!94}a^{8}+\frac{97\!\cdots\!73}{34\!\cdots\!94}a^{7}-\frac{50\!\cdots\!97}{17\!\cdots\!97}a^{6}-\frac{53\!\cdots\!85}{17\!\cdots\!97}a^{5}+\frac{89\!\cdots\!01}{34\!\cdots\!94}a^{4}+\frac{20\!\cdots\!70}{17\!\cdots\!97}a^{3}-\frac{17\!\cdots\!26}{17\!\cdots\!97}a^{2}-\frac{14\!\cdots\!63}{34\!\cdots\!94}a-\frac{76\!\cdots\!95}{34\!\cdots\!94}$, $\frac{25\!\cdots\!52}{17\!\cdots\!97}a^{11}-\frac{45\!\cdots\!15}{34\!\cdots\!94}a^{10}-\frac{38\!\cdots\!53}{34\!\cdots\!94}a^{9}+\frac{25\!\cdots\!55}{34\!\cdots\!94}a^{8}+\frac{51\!\cdots\!41}{17\!\cdots\!97}a^{7}-\frac{20\!\cdots\!01}{17\!\cdots\!97}a^{6}-\frac{57\!\cdots\!08}{17\!\cdots\!97}a^{5}+\frac{10\!\cdots\!09}{17\!\cdots\!97}a^{4}+\frac{47\!\cdots\!79}{34\!\cdots\!94}a^{3}-\frac{87\!\cdots\!17}{34\!\cdots\!94}a^{2}-\frac{39\!\cdots\!13}{34\!\cdots\!94}a-\frac{50\!\cdots\!87}{17\!\cdots\!97}$, $\frac{18\!\cdots\!49}{69\!\cdots\!88}a^{11}-\frac{35\!\cdots\!57}{34\!\cdots\!94}a^{10}-\frac{11\!\cdots\!53}{69\!\cdots\!88}a^{9}+\frac{42\!\cdots\!55}{69\!\cdots\!88}a^{8}+\frac{21\!\cdots\!81}{69\!\cdots\!88}a^{7}-\frac{37\!\cdots\!67}{34\!\cdots\!94}a^{6}-\frac{68\!\cdots\!23}{34\!\cdots\!94}a^{5}+\frac{42\!\cdots\!17}{69\!\cdots\!88}a^{4}+\frac{67\!\cdots\!71}{34\!\cdots\!94}a^{3}-\frac{35\!\cdots\!27}{69\!\cdots\!88}a^{2}-\frac{19\!\cdots\!85}{69\!\cdots\!88}a-\frac{13\!\cdots\!69}{69\!\cdots\!88}$, $\frac{10374701846}{6562535158843}a^{11}-\frac{13664122216}{6562535158843}a^{10}-\frac{771187729924}{6562535158843}a^{9}+\frac{836515769096}{6562535158843}a^{8}+\frac{20125313799018}{6562535158843}a^{7}-\frac{16052450606834}{6562535158843}a^{6}-\frac{219650139498882}{6562535158843}a^{5}+\frac{126092356700410}{6562535158843}a^{4}+\frac{856640592578358}{6562535158843}a^{3}-\frac{491364536485740}{6562535158843}a^{2}-\frac{492920306351840}{6562535158843}a-\frac{28547523974173}{6562535158843}$
| 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: | \( 9960963.64834 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 9960963.64834 \cdot 2}{2\cdot\sqrt{16348345805451893172953}}\cr\approx \mathstrut & 0.319098230142 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 75*x^10 + 58*x^9 + 1982*x^8 - 987*x^7 - 22024*x^6 + 6367*x^5 + 89299*x^4 - 25629*x^3 - 66876*x^2 - 18568*x - 883)
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 - 75*x^10 + 58*x^9 + 1982*x^8 - 987*x^7 - 22024*x^6 + 6367*x^5 + 89299*x^4 - 25629*x^3 - 66876*x^2 - 18568*x - 883, 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 - 75*x^10 + 58*x^9 + 1982*x^8 - 987*x^7 - 22024*x^6 + 6367*x^5 + 89299*x^4 - 25629*x^3 - 66876*x^2 - 18568*x - 883);
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 - 75*x^10 + 58*x^9 + 1982*x^8 - 987*x^7 - 22024*x^6 + 6367*x^5 + 89299*x^4 - 25629*x^3 - 66876*x^2 - 18568*x - 883);
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
| \(\Q(\sqrt{17}) \), 3.3.169.1, 4.4.830297.1, 6.6.140320193.1 |
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 | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.12.0.1}{12} }$ | R | R | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |