Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 323*x^10 + 30362*x^8 - 862087*x^6 + 10432900*x^4 - 56754976*x^2 + 113509952)
gp: K = bnfinit(y^12 - 323*y^10 + 30362*y^8 - 862087*y^6 + 10432900*y^4 - 56754976*y^2 + 113509952, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 323*x^10 + 30362*x^8 - 862087*x^6 + 10432900*x^4 - 56754976*x^2 + 113509952);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 323*x^10 + 30362*x^8 - 862087*x^6 + 10432900*x^4 - 56754976*x^2 + 113509952)
\( x^{12} - 323x^{10} + 30362x^{8} - 862087x^{6} + 10432900x^{4} - 56754976x^{2} + 113509952 \)
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: |
\(2978079245004918582489485312\)
\(\medspace = 2^{12}\cdot 17^{9}\cdot 19^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(194.76\) | 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: | $2\cdot 17^{3/4}19^{5/6}\approx 194.75772260536294$ | ||
| Ramified primes: |
\(2\), \(17\), \(19\)
| 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: | \(1292=2^{2}\cdot 17\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1292}(1,·)$, $\chi_{1292}(259,·)$, $\chi_{1292}(1189,·)$, $\chi_{1292}(455,·)$, $\chi_{1292}(939,·)$, $\chi_{1292}(273,·)$, $\chi_{1292}(305,·)$, $\chi_{1292}(531,·)$, $\chi_{1292}(885,·)$, $\chi_{1292}(577,·)$, $\chi_{1292}(183,·)$, $\chi_{1292}(863,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{17}a^{4}$, $\frac{1}{17}a^{5}$, $\frac{1}{323}a^{6}$, $\frac{1}{646}a^{7}-\frac{1}{34}a^{5}-\frac{1}{2}a$, $\frac{1}{153748}a^{8}+\frac{5}{9044}a^{6}+\frac{1}{238}a^{4}-\frac{5}{28}a^{2}+\frac{3}{7}$, $\frac{1}{307496}a^{9}+\frac{5}{18088}a^{7}-\frac{13}{476}a^{5}-\frac{5}{56}a^{3}+\frac{3}{14}a$, $\frac{1}{51044336}a^{10}+\frac{129}{51044336}a^{8}+\frac{1319}{1501304}a^{6}+\frac{25}{1904}a^{4}-\frac{551}{1162}a^{2}+\frac{75}{581}$, $\frac{1}{102088672}a^{11}+\frac{129}{102088672}a^{9}+\frac{1319}{3002608}a^{7}-\frac{87}{3808}a^{5}+\frac{611}{2324}a^{3}+\frac{75}{1162}a$
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
$C_{2}\times C_{2}$, which has order $4$ (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: |
$\frac{183}{750652}a^{10}-\frac{244390}{3190271}a^{8}+\frac{37983}{5644}a^{6}-\frac{71931}{476}a^{4}+\frac{2828031}{2324}a^{2}-\frac{1829853}{581}$, $\frac{9683}{51044336}a^{10}-\frac{3042657}{51044336}a^{8}+\frac{59111}{11288}a^{6}-\frac{223885}{1904}a^{4}+\frac{2200559}{2324}a^{2}-\frac{1422396}{581}$, $\frac{8103}{51044336}a^{10}-\frac{2546289}{51044336}a^{8}+\frac{940003}{214472}a^{6}-\frac{187489}{1904}a^{4}+\frac{461596}{581}a^{2}-\frac{1197608}{581}$, $\frac{5827}{12761084}a^{10}-\frac{457757}{3190271}a^{8}+\frac{1351801}{107236}a^{6}-\frac{134763}{476}a^{4}+\frac{5301887}{2324}a^{2}-\frac{3433756}{581}$, $\frac{194}{455753}a^{10}-\frac{3586}{26809}a^{8}+\frac{315062}{26809}a^{6}-264a^{4}+\frac{176704}{83}a^{2}-\frac{458839}{83}$, $\frac{3909}{25522168}a^{11}-\frac{22801}{51044336}a^{10}-\frac{1228447}{25522168}a^{9}+\frac{1023441}{7292048}a^{8}+\frac{453561}{107236}a^{7}-\frac{18508635}{1501304}a^{6}-\frac{90495}{952}a^{5}+\frac{527031}{1904}a^{4}+\frac{890987}{1162}a^{3}-\frac{1295353}{581}a^{2}-\frac{2312055}{1162}a+\frac{3353838}{581}$, $\frac{1467}{14584096}a^{11}-\frac{271}{614992}a^{10}-\frac{3205107}{102088672}a^{9}+\frac{84649}{614992}a^{8}+\frac{8146491}{3002608}a^{7}-\frac{30795}{2584}a^{6}-\frac{218667}{3808}a^{5}+\frac{483627}{1904}a^{4}+\frac{1029589}{2324}a^{3}-\frac{13775}{7}a^{2}-\frac{649217}{581}a+\frac{34773}{7}$, $\frac{45423}{14584096}a^{11}+\frac{469407}{51044336}a^{10}-\frac{99922675}{102088672}a^{9}-\frac{8677385}{3002608}a^{8}+\frac{258255521}{3002608}a^{7}+\frac{381257549}{1501304}a^{6}-\frac{7362231}{3808}a^{5}-\frac{1552607}{272}a^{4}+\frac{72499697}{4648}a^{3}+\frac{26754269}{581}a^{2}-\frac{47009859}{1162}a-\frac{69382443}{581}$, $\frac{6033}{51044336}a^{11}-\frac{4189}{51044336}a^{10}-\frac{1895505}{51044336}a^{9}+\frac{1316495}{51044336}a^{8}+\frac{144007}{44156}a^{7}-\frac{3402317}{1501304}a^{6}-\frac{139283}{1904}a^{5}+\frac{96931}{1904}a^{4}+\frac{2734801}{4648}a^{3}-\frac{951445}{2324}a^{2}-\frac{883244}{581}a+\frac{612520}{581}$, $\frac{699}{25522168}a^{11}-\frac{613}{6380542}a^{10}-\frac{5689}{671636}a^{9}+\frac{22437}{750652}a^{8}+\frac{1072145}{1501304}a^{7}-\frac{1921377}{750652}a^{6}-\frac{12811}{952}a^{5}+\frac{6158}{119}a^{4}+\frac{333013}{4648}a^{3}-\frac{775059}{2324}a^{2}-\frac{2155}{83}a+\frac{41342}{83}$, $\frac{286605}{25522168}a^{11}-\frac{106910}{3190271}a^{10}-\frac{45025951}{12761084}a^{9}+\frac{134365521}{12761084}a^{8}+\frac{465272743}{1501304}a^{7}-\frac{99175749}{107236}a^{6}-\frac{6620561}{952}a^{5}+\frac{4939287}{238}a^{4}+\frac{260169671}{4648}a^{3}-\frac{388204645}{2324}a^{2}-\frac{24051471}{166}a+\frac{251215374}{581}$
| 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: | \( 1286540325.1688519 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 1286540325.1688519 \cdot 4}{2\cdot\sqrt{2978079245004918582489485312}}\cr\approx \mathstrut & 0.193127990607122 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 323*x^10 + 30362*x^8 - 862087*x^6 + 10432900*x^4 - 56754976*x^2 + 113509952)
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 - 323*x^10 + 30362*x^8 - 862087*x^6 + 10432900*x^4 - 56754976*x^2 + 113509952, 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 - 323*x^10 + 30362*x^8 - 862087*x^6 + 10432900*x^4 - 56754976*x^2 + 113509952);
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 - 323*x^10 + 30362*x^8 - 862087*x^6 + 10432900*x^4 - 56754976*x^2 + 113509952);
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
| \(\Q(\sqrt{17}) \), 3.3.361.1, 4.4.28377488.2, 6.6.640267073.1 |
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 | ${\href{/padicField/3.12.0.1}{12} }$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.4.0.1}{4} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | R | R | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(19\)
| 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |