Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 12*x^10 + 86*x^9 + 402*x^8 - 2586*x^7 + 3933*x^6 - 1749*x^5 - 1353*x^4 + 5309*x^3 - 2481*x^2 - 6120*x - 1985)
gp: K = bnfinit(y^12 - 6*y^11 - 12*y^10 + 86*y^9 + 402*y^8 - 2586*y^7 + 3933*y^6 - 1749*y^5 - 1353*y^4 + 5309*y^3 - 2481*y^2 - 6120*y - 1985, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 - 12*x^10 + 86*x^9 + 402*x^8 - 2586*x^7 + 3933*x^6 - 1749*x^5 - 1353*x^4 + 5309*x^3 - 2481*x^2 - 6120*x - 1985);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^11 - 12*x^10 + 86*x^9 + 402*x^8 - 2586*x^7 + 3933*x^6 - 1749*x^5 - 1353*x^4 + 5309*x^3 - 2481*x^2 - 6120*x - 1985)
\( x^{12} - 6 x^{11} - 12 x^{10} + 86 x^{9} + 402 x^{8} - 2586 x^{7} + 3933 x^{6} - 1749 x^{5} + \cdots - 1985 \)
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: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-2954177797192602046875\)
\(\medspace = -\,3^{13}\cdot 5^{6}\cdot 17^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(61.55\) | 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: | $3^{7/6}5^{1/2}17^{3/4}\approx 67.44708071823815$ | ||
| Ramified primes: |
\(3\), \(5\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-51}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{15}a^{6}-\frac{1}{5}a^{4}-\frac{1}{3}a^{3}+\frac{2}{5}a^{2}-\frac{1}{3}$, $\frac{1}{15}a^{7}-\frac{1}{5}a^{5}-\frac{1}{3}a^{4}+\frac{2}{5}a^{3}-\frac{1}{3}a$, $\frac{1}{45}a^{8}+\frac{1}{45}a^{7}+\frac{1}{45}a^{6}+\frac{7}{45}a^{5}+\frac{4}{45}a^{4}+\frac{1}{45}a^{3}-\frac{11}{45}a^{2}+\frac{2}{9}a-\frac{1}{9}$, $\frac{1}{675}a^{9}-\frac{1}{225}a^{8}+\frac{1}{225}a^{7}+\frac{2}{225}a^{6}-\frac{44}{225}a^{5}-\frac{7}{25}a^{4}+\frac{32}{225}a^{3}+\frac{11}{25}a^{2}+\frac{22}{45}a+\frac{64}{135}$, $\frac{1}{2025}a^{10}-\frac{1}{2025}a^{9}-\frac{1}{675}a^{8}+\frac{19}{675}a^{7}-\frac{2}{135}a^{6}-\frac{196}{675}a^{5}-\frac{34}{675}a^{4}+\frac{328}{675}a^{3}-\frac{187}{675}a^{2}-\frac{119}{405}a+\frac{38}{405}$, $\frac{1}{235514111625}a^{11}+\frac{48426637}{235514111625}a^{10}-\frac{155286746}{235514111625}a^{9}+\frac{354250841}{78504703875}a^{8}+\frac{2445765007}{78504703875}a^{7}-\frac{78509407}{26168234625}a^{6}+\frac{1939485871}{26168234625}a^{5}+\frac{18151072691}{78504703875}a^{4}-\frac{2510399183}{78504703875}a^{3}-\frac{31079037958}{235514111625}a^{2}+\frac{10827996526}{47102822325}a-\frac{2425230791}{47102822325}$
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
$C_{4}$, 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: | $6$ | 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: |
$\frac{309191741}{235514111625}a^{11}-\frac{1994316403}{235514111625}a^{10}-\frac{2829732481}{235514111625}a^{9}+\frac{9343144066}{78504703875}a^{8}+\frac{37191051167}{78504703875}a^{7}-\frac{94680127622}{26168234625}a^{6}+\frac{177447155546}{26168234625}a^{5}-\frac{396943958894}{78504703875}a^{4}-\frac{30694187218}{78504703875}a^{3}+\frac{1885615655137}{235514111625}a^{2}-\frac{310116284794}{47102822325}a-\frac{256318157026}{47102822325}$, $\frac{229459}{64612925}a^{11}-\frac{1549791}{64612925}a^{10}-\frac{64106}{2584517}a^{9}+\frac{62987432}{193838775}a^{8}+\frac{230260268}{193838775}a^{7}-\frac{651575697}{64612925}a^{6}+\frac{4158347864}{193838775}a^{5}-\frac{855171989}{38767755}a^{4}+\frac{740482048}{64612925}a^{3}+\frac{2027912711}{193838775}a^{2}-\frac{128544292}{7753551}a-\frac{122554526}{12922585}$, $\frac{13148891}{26168234625}a^{11}-\frac{82760953}{26168234625}a^{10}-\frac{153633046}{26168234625}a^{9}+\frac{423213406}{8722744875}a^{8}+\frac{1744396427}{8722744875}a^{7}-\frac{4074661582}{2907581625}a^{6}+\frac{5941527991}{2907581625}a^{5}-\frac{35900024}{8722744875}a^{4}-\frac{20820159073}{8722744875}a^{3}+\frac{135133703782}{26168234625}a^{2}-\frac{20351443159}{5233646925}a-\frac{31062712036}{5233646925}$, $\frac{25158884929}{235514111625}a^{11}-\frac{220391374472}{235514111625}a^{10}+\frac{23454278071}{235514111625}a^{9}+\frac{1207914212924}{78504703875}a^{8}+\frac{1662461126533}{78504703875}a^{7}-\frac{11460638340013}{26168234625}a^{6}+\frac{27287969859769}{26168234625}a^{5}-\frac{9050258187586}{78504703875}a^{4}-\frac{126565064785772}{78504703875}a^{3}+\frac{68986186594808}{235514111625}a^{2}+\frac{53659147228144}{47102822325}a+\frac{17300225247991}{47102822325}$, $\frac{182367643337}{235514111625}a^{11}+\frac{1222699094041}{235514111625}a^{10}+\frac{1326527144992}{235514111625}a^{9}-\frac{5532879192727}{78504703875}a^{8}-\frac{20572438691219}{78504703875}a^{7}+\frac{57203478173834}{26168234625}a^{6}-\frac{119834530853747}{26168234625}a^{5}+\frac{362373105578918}{78504703875}a^{4}-\frac{189386478554999}{78504703875}a^{3}-\frac{507787609592239}{235514111625}a^{2}+\frac{166891641773698}{47102822325}a+\frac{95989186071847}{47102822325}$, $\frac{15375533951}{47102822325}a^{11}-\frac{95931637111}{47102822325}a^{10}-\frac{162072328579}{47102822325}a^{9}+\frac{91318441889}{3140188155}a^{8}+\frac{1947113349434}{15700940775}a^{7}-\frac{4588177252811}{5233646925}a^{6}+\frac{312457575179}{209345877}a^{5}-\frac{13388465320109}{15700940775}a^{4}-\frac{7033658462299}{15700940775}a^{3}+\frac{98972081939173}{47102822325}a^{2}-\frac{14099900433208}{9420564465}a-\frac{15973651712689}{9420564465}$
| 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: | \( 2034918.8483919401 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 2034918.8483919401 \cdot 4}{2\cdot\sqrt{2954177797192602046875}}\cr\approx \mathstrut & 2.93304082751766 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 12*x^10 + 86*x^9 + 402*x^8 - 2586*x^7 + 3933*x^6 - 1749*x^5 - 1353*x^4 + 5309*x^3 - 2481*x^2 - 6120*x - 1985)
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 - 6*x^11 - 12*x^10 + 86*x^9 + 402*x^8 - 2586*x^7 + 3933*x^6 - 1749*x^5 - 1353*x^4 + 5309*x^3 - 2481*x^2 - 6120*x - 1985, 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 - 6*x^11 - 12*x^10 + 86*x^9 + 402*x^8 - 2586*x^7 + 3933*x^6 - 1749*x^5 - 1353*x^4 + 5309*x^3 - 2481*x^2 - 6120*x - 1985);
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 - 6*x^11 - 12*x^10 + 86*x^9 + 402*x^8 - 2586*x^7 + 3933*x^6 - 1749*x^5 - 1353*x^4 + 5309*x^3 - 2481*x^2 - 6120*x - 1985);
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 solvable group of order 24 |
| The 9 conjugacy class representatives for $D_{12}$ |
| Character table for $D_{12}$ |
Intermediate fields
| \(\Q(\sqrt{85}) \), 3.1.135.1, 4.2.368475.2, 6.2.447697125.2 |
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]
Sibling fields
| Galois closure: | data not computed |
| Degree 12 sibling: | data not computed |
| Minimal sibling: | 12.0.1772506678315561228125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }$ | R | R | ${\href{/padicField/7.2.0.1}{2} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.2.0.1}{2} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }$ | ${\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
| 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
| 3.6.7.1 | $x^{6} + 6 x^{2} + 6$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ | |
|
\(5\)
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(17\)
| 17.12.9.3 | $x^{12} + 289 x^{4} - 68782$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |