Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 6*x^10 + 30*x^9 - 17*x^8 - 50*x^7 + 172*x^6 - 204*x^5 + 266*x^4 - 228*x^3 + 384*x^2 - 206*x + 125)
gp: K = bnfinit(y^12 - 2*y^11 - 6*y^10 + 30*y^9 - 17*y^8 - 50*y^7 + 172*y^6 - 204*y^5 + 266*y^4 - 228*y^3 + 384*y^2 - 206*y + 125, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 6*x^10 + 30*x^9 - 17*x^8 - 50*x^7 + 172*x^6 - 204*x^5 + 266*x^4 - 228*x^3 + 384*x^2 - 206*x + 125);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - 6*x^10 + 30*x^9 - 17*x^8 - 50*x^7 + 172*x^6 - 204*x^5 + 266*x^4 - 228*x^3 + 384*x^2 - 206*x + 125)
\( x^{12} - 2 x^{11} - 6 x^{10} + 30 x^{9} - 17 x^{8} - 50 x^{7} + 172 x^{6} - 204 x^{5} + 266 x^{4} + \cdots + 125 \)
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: |
\(2779905883635712\)
\(\medspace = 2^{18}\cdot 13^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.36\) | 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: | $2^{3/2}13^{5/6}\approx 23.97900291135148$ | ||
| Ramified primes: |
\(2\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{265}a^{9}-\frac{4}{53}a^{8}+\frac{128}{265}a^{7}-\frac{112}{265}a^{6}+\frac{91}{265}a^{5}+\frac{10}{53}a^{4}+\frac{62}{265}a^{3}-\frac{93}{265}a^{2}-\frac{122}{265}a+\frac{3}{53}$, $\frac{1}{265}a^{10}-\frac{7}{265}a^{8}+\frac{63}{265}a^{7}-\frac{29}{265}a^{6}+\frac{3}{53}a^{5}+\frac{2}{265}a^{4}+\frac{87}{265}a^{3}-\frac{127}{265}a^{2}-\frac{8}{53}a+\frac{7}{53}$, $\frac{1}{1783504325}a^{11}-\frac{468629}{356700865}a^{10}-\frac{2448941}{1783504325}a^{9}+\frac{234624333}{1783504325}a^{8}-\frac{639305346}{1783504325}a^{7}+\frac{573936963}{1783504325}a^{6}+\frac{633973138}{1783504325}a^{5}+\frac{137766927}{1783504325}a^{4}+\frac{112233513}{356700865}a^{3}+\frac{168916367}{1783504325}a^{2}+\frac{709522173}{1783504325}a+\frac{17909263}{71340173}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
Class group and class number
$C_{3}$, which has order $3$
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: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{134877}{16362425} a^{11} + \frac{40892}{3272485} a^{10} + \frac{937757}{16362425} a^{9} - \frac{3610431}{16362425} a^{8} + \frac{515152}{16362425} a^{7} + \frac{7421294}{16362425} a^{6} - \frac{20663451}{16362425} a^{5} + \frac{19764911}{16362425} a^{4} - \frac{3928168}{3272485} a^{3} + \frac{10675001}{16362425} a^{2} - \frac{31607796}{16362425} a + \frac{374126}{654497} \)
(order $4$)
| 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: |
$\frac{3235298}{1783504325}a^{11}+\frac{377444}{71340173}a^{10}-\frac{27766448}{1783504325}a^{9}-\frac{22949586}{1783504325}a^{8}+\frac{239953082}{1783504325}a^{7}+\frac{104239644}{1783504325}a^{6}-\frac{41196206}{1783504325}a^{5}+\frac{817777791}{1783504325}a^{4}+\frac{102351406}{356700865}a^{3}+\frac{953806286}{1783504325}a^{2}-\frac{296396436}{1783504325}a+\frac{50539201}{71340173}$, $\frac{3235298}{1783504325}a^{11}+\frac{377444}{71340173}a^{10}-\frac{27766448}{1783504325}a^{9}-\frac{22949586}{1783504325}a^{8}+\frac{239953082}{1783504325}a^{7}+\frac{104239644}{1783504325}a^{6}-\frac{41196206}{1783504325}a^{5}+\frac{817777791}{1783504325}a^{4}+\frac{102351406}{356700865}a^{3}+\frac{953806286}{1783504325}a^{2}-\frac{296396436}{1783504325}a+\frac{121879374}{71340173}$, $\frac{7371528}{1783504325}a^{11}+\frac{1478313}{356700865}a^{10}-\frac{68852193}{1783504325}a^{9}+\frac{48734124}{1783504325}a^{8}+\frac{355820777}{1783504325}a^{7}-\frac{136175871}{1783504325}a^{6}+\frac{374412319}{1783504325}a^{5}+\frac{226232756}{1783504325}a^{4}-\frac{199912374}{356700865}a^{3}+\frac{3223254311}{1783504325}a^{2}-\frac{1110512516}{1783504325}a+\frac{46905812}{71340173}$, $\frac{1188772}{356700865}a^{11}-\frac{2919606}{356700865}a^{10}-\frac{10977529}{356700865}a^{9}+\frac{42501138}{356700865}a^{8}+\frac{9803799}{356700865}a^{7}-\frac{144624356}{356700865}a^{6}+\frac{73051449}{356700865}a^{5}-\frac{45759758}{356700865}a^{4}+\frac{355302844}{356700865}a^{3}-\frac{334250688}{356700865}a^{2}-\frac{15497005}{71340173}a-\frac{61591991}{71340173}$, $\frac{20547937}{1783504325}a^{11}-\frac{1328644}{71340173}a^{10}-\frac{127115797}{1783504325}a^{9}+\frac{580691541}{1783504325}a^{8}-\frac{183361022}{1783504325}a^{7}-\frac{1096654069}{1783504325}a^{6}+\frac{3392747126}{1783504325}a^{5}-\frac{2397767696}{1783504325}a^{4}+\frac{203377326}{71340173}a^{3}-\frac{3332444486}{1783504325}a^{2}+\frac{8368553336}{1783504325}a-\frac{78093671}{71340173}$
| 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: | \( 919.513377747 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 919.513377747 \cdot 3}{4\cdot\sqrt{2779905883635712}}\cr\approx \mathstrut & 0.804791682354 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 6*x^10 + 30*x^9 - 17*x^8 - 50*x^7 + 172*x^6 - 204*x^5 + 266*x^4 - 228*x^3 + 384*x^2 - 206*x + 125)
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 - 2*x^11 - 6*x^10 + 30*x^9 - 17*x^8 - 50*x^7 + 172*x^6 - 204*x^5 + 266*x^4 - 228*x^3 + 384*x^2 - 206*x + 125, 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 - 2*x^11 - 6*x^10 + 30*x^9 - 17*x^8 - 50*x^7 + 172*x^6 - 204*x^5 + 266*x^4 - 228*x^3 + 384*x^2 - 206*x + 125);
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 - 2*x^11 - 6*x^10 + 30*x^9 - 17*x^8 - 50*x^7 + 172*x^6 - 204*x^5 + 266*x^4 - 228*x^3 + 384*x^2 - 206*x + 125);
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
$C_3\times D_4$ (as 12T14):
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 solvable group of order 24 |
| The 15 conjugacy class representatives for $D_4 \times C_3$ |
| Character table for $D_4 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 3.3.169.1, 4.0.832.1, 6.0.1827904.1 |
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]
Sibling fields
| Galois closure: | deg 24 |
| Degree 12 sibling: | 12.6.564668382613504.4 |
| Minimal sibling: | 12.6.564668382613504.4 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{6}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.12.0.1}{12} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.1.0.1}{1} }^{12}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.63 | $x^{12} + 6 x^{11} + 12 x^{10} + 8 x^{9} + 6 x^{8} + 24 x^{7} + 24 x^{6} + 28 x^{4} + 56 x^{3} - 24$ | $4$ | $3$ | $18$ | $D_4 \times C_3$ | $[2, 2]^{6}$ |
|
\(13\)
| 13.6.5.5 | $x^{6} + 65$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |