Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 17*x^10 + 140*x^9 + 20*x^8 - 986*x^7 + 323*x^6 + 3136*x^5 - 920*x^4 - 4850*x^3 + 115*x^2 + 3044*x + 1055)
gp: K = bnfinit(y^12 - 6*y^11 - 17*y^10 + 140*y^9 + 20*y^8 - 986*y^7 + 323*y^6 + 3136*y^5 - 920*y^4 - 4850*y^3 + 115*y^2 + 3044*y + 1055, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 - 17*x^10 + 140*x^9 + 20*x^8 - 986*x^7 + 323*x^6 + 3136*x^5 - 920*x^4 - 4850*x^3 + 115*x^2 + 3044*x + 1055);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 - 17*x^10 + 140*x^9 + 20*x^8 - 986*x^7 + 323*x^6 + 3136*x^5 - 920*x^4 - 4850*x^3 + 115*x^2 + 3044*x + 1055)
\( x^{12} - 6 x^{11} - 17 x^{10} + 140 x^{9} + 20 x^{8} - 986 x^{7} + 323 x^{6} + 3136 x^{5} - 920 x^{4} + \cdots + 1055 \)
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: |
\(629599793037598310656\)
\(\medspace = 2^{8}\cdot 199^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(54.11\) | 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: | $2^{2/3}199^{4/5}\approx 109.58954527670048$ | ||
| Ramified primes: |
\(2\), \(199\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{188}a^{10}-\frac{5}{188}a^{9}-\frac{15}{188}a^{8}-\frac{1}{47}a^{7}+\frac{5}{188}a^{6}-\frac{69}{188}a^{5}+\frac{7}{188}a^{4}+\frac{7}{47}a^{3}+\frac{3}{188}a^{2}-\frac{45}{188}a+\frac{91}{188}$, $\frac{1}{543132}a^{11}+\frac{1439}{543132}a^{10}-\frac{32521}{543132}a^{9}-\frac{7261}{90522}a^{8}-\frac{44875}{543132}a^{7}+\frac{87803}{543132}a^{6}-\frac{61817}{181044}a^{5}+\frac{110489}{271566}a^{4}+\frac{34597}{181044}a^{3}-\frac{133517}{543132}a^{2}+\frac{20639}{181044}a+\frac{56725}{271566}$
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 $2$
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: |
$\frac{8545}{271566}a^{11}-\frac{99773}{543132}a^{10}-\frac{260969}{543132}a^{9}+\frac{349333}{90522}a^{8}-\frac{46129}{271566}a^{7}-\frac{11425103}{543132}a^{6}+\frac{1646609}{181044}a^{5}+\frac{5801528}{135783}a^{4}-\frac{21391}{1926}a^{3}-\frac{313019}{11556}a^{2}-\frac{1795421}{181044}a-\frac{486647}{135783}$, $\frac{9575}{181044}a^{11}-\frac{44477}{181044}a^{10}-\frac{218237}{181044}a^{9}+\frac{170803}{30174}a^{8}+\frac{1460827}{181044}a^{7}-\frac{6980507}{181044}a^{6}-\frac{1831723}{60348}a^{5}+\frac{9510223}{90522}a^{4}+\frac{4763693}{60348}a^{3}-\frac{16920313}{181044}a^{2}-\frac{5736359}{60348}a-\frac{943136}{45261}$, $\frac{1052}{15087}a^{11}-\frac{17045}{60348}a^{10}-\frac{54073}{30174}a^{9}+\frac{131269}{20116}a^{8}+\frac{232411}{15087}a^{7}-\frac{2697833}{60348}a^{6}-\frac{367639}{5029}a^{5}+\frac{7102823}{60348}a^{4}+\frac{969628}{5029}a^{3}-\frac{4638427}{60348}a^{2}-\frac{2030107}{10058}a-\frac{91663}{1284}$, $\frac{197}{90522}a^{11}-\frac{2167}{181044}a^{10}-\frac{10189}{181044}a^{9}+\frac{20701}{60348}a^{8}+\frac{18761}{45261}a^{7}-\frac{567637}{181044}a^{6}-\frac{41111}{60348}a^{5}+\frac{1883167}{181044}a^{4}-\frac{34531}{30174}a^{3}-\frac{1900859}{181044}a^{2}+\frac{223931}{60348}a-\frac{31087}{181044}$, $\frac{197}{90522}a^{11}-\frac{2167}{181044}a^{10}-\frac{10189}{181044}a^{9}+\frac{20701}{60348}a^{8}+\frac{18761}{45261}a^{7}-\frac{567637}{181044}a^{6}-\frac{41111}{60348}a^{5}+\frac{1883167}{181044}a^{4}-\frac{34531}{30174}a^{3}-\frac{1900859}{181044}a^{2}+\frac{223931}{60348}a+\frac{149957}{181044}$, $\frac{2251}{543132}a^{11}+\frac{3509}{543132}a^{10}-\frac{66647}{271566}a^{9}+\frac{9643}{181044}a^{8}+\frac{2089007}{543132}a^{7}-\frac{1840897}{543132}a^{6}-\frac{1836169}{90522}a^{5}+\frac{10258075}{543132}a^{4}+\frac{7486399}{181044}a^{3}-\frac{12889037}{543132}a^{2}-\frac{2720285}{90522}a-\frac{2816869}{543132}$, $\frac{2251}{543132}a^{11}-\frac{14135}{271566}a^{10}+\frac{25601}{543132}a^{9}+\frac{213799}{181044}a^{8}-\frac{1314235}{543132}a^{7}-\frac{1043080}{135783}a^{6}+\frac{2851987}{181044}a^{5}+\frac{12615499}{543132}a^{4}-\frac{6388505}{181044}a^{3}-\frac{9207847}{271566}a^{2}+\frac{4631447}{181044}a+\frac{10177853}{543132}$, $\frac{1229}{45261}a^{11}-\frac{13519}{90522}a^{10}-\frac{20180}{45261}a^{9}+\frac{188675}{60348}a^{8}+\frac{16969}{45261}a^{7}-\frac{1534141}{90522}a^{6}+\frac{203641}{30174}a^{5}+\frac{6072005}{181044}a^{4}-\frac{436133}{30174}a^{3}-\frac{1024714}{45261}a^{2}+\frac{42604}{15087}a+\frac{633367}{181044}$, $\frac{25199}{543132}a^{11}-\frac{140039}{543132}a^{10}-\frac{395843}{543132}a^{9}+\frac{958307}{181044}a^{8}+\frac{41923}{543132}a^{7}-\frac{14894315}{543132}a^{6}+\frac{2323787}{181044}a^{5}+\frac{28533113}{543132}a^{4}-\frac{4641535}{181044}a^{3}-\frac{20737393}{543132}a^{2}+\frac{1745485}{181044}a+\frac{5109907}{543132}$, $\frac{5957}{543132}a^{11}-\frac{5663}{135783}a^{10}-\frac{3625}{11556}a^{9}+\frac{188447}{181044}a^{8}+\frac{1691749}{543132}a^{7}-\frac{1101287}{135783}a^{6}-\frac{2781517}{181044}a^{5}+\frac{12973373}{543132}a^{4}+\frac{7111565}{181044}a^{3}-\frac{4852943}{271566}a^{2}-\frac{7160555}{181044}a-\frac{6909719}{543132}$, $\frac{38275}{271566}a^{11}-\frac{421025}{543132}a^{10}-\frac{1420859}{543132}a^{9}+\frac{3183851}{181044}a^{8}+\frac{1099648}{135783}a^{7}-\frac{62916161}{543132}a^{6}+\frac{1166321}{181044}a^{5}+\frac{172935077}{543132}a^{4}-\frac{1110790}{45261}a^{3}-\frac{191398177}{543132}a^{2}-\frac{3573377}{181044}a+\frac{44872549}{543132}$
| 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: | \( 6474868.58794 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 6474868.58794 \cdot 2}{2\cdot\sqrt{629599793037598310656}}\cr\approx \mathstrut & 1.05696016109 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 17*x^10 + 140*x^9 + 20*x^8 - 986*x^7 + 323*x^6 + 3136*x^5 - 920*x^4 - 4850*x^3 + 115*x^2 + 3044*x + 1055)
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 - 17*x^10 + 140*x^9 + 20*x^8 - 986*x^7 + 323*x^6 + 3136*x^5 - 920*x^4 - 4850*x^3 + 115*x^2 + 3044*x + 1055, 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 - 17*x^10 + 140*x^9 + 20*x^8 - 986*x^7 + 323*x^6 + 3136*x^5 - 920*x^4 - 4850*x^3 + 115*x^2 + 3044*x + 1055);
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 - 17*x^10 + 140*x^9 + 20*x^8 - 986*x^7 + 323*x^6 + 3136*x^5 - 920*x^4 - 4850*x^3 + 115*x^2 + 3044*x + 1055);
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 non-solvable group of order 60 |
| The 5 conjugacy class representatives for $A_5$ |
| Character table for $A_5$ |
Intermediate fields
| 6.6.25091827216.1 |
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
| Degree 5 sibling: | 5.5.6272956804.1 |
| Degree 6 sibling: | 6.6.25091827216.1 |
| Degree 10 sibling: | 10.10.157399948259399577664.1 |
| Degree 15 sibling: | deg 15 |
| Degree 20 sibling: | deg 20 |
| Degree 30 sibling: | data not computed |
| Minimal sibling: | 5.5.6272956804.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(199\)
| 199.2.0.1 | $x^{2} + 193 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 199.10.8.1 | $x^{10} + 965 x^{9} + 372505 x^{8} + 71902150 x^{7} + 6940792505 x^{6} + 268216580121 x^{5} + 20822569550 x^{4} + 721232920 x^{3} + 14312824435 x^{2} + 1380105904010 x + 53267794087421$ | $5$ | $2$ | $8$ | $D_5$ | $[\ ]_{5}^{2}$ |