Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 17*x^10 + 85*x^8 - 306*x^6 + 748*x^4 - 1003*x^2 + 544)
gp: K = bnfinit(y^12 - 17*y^10 + 85*y^8 - 306*y^6 + 748*y^4 - 1003*y^2 + 544, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 17*x^10 + 85*x^8 - 306*x^6 + 748*x^4 - 1003*x^2 + 544);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 17*x^10 + 85*x^8 - 306*x^6 + 748*x^4 - 1003*x^2 + 544)
\( x^{12} - 17x^{10} + 85x^{8} - 306x^{6} + 748x^{4} - 1003x^{2} + 544 \)
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: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4492085992834072576\)
\(\medspace = 2^{17}\cdot 17^{11}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.84\) | 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: | $2^{2}17^{11/12}\approx 53.69965587306223$ | ||
| Ramified primes: |
\(2\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{34}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{252304}a^{10}+\frac{21441}{126152}a^{8}+\frac{46539}{252304}a^{6}-\frac{5297}{252304}a^{4}+\frac{6973}{19408}a^{2}-\frac{694}{15769}$, $\frac{1}{504608}a^{11}-\frac{1}{504608}a^{10}+\frac{21441}{252304}a^{9}+\frac{104711}{252304}a^{8}+\frac{46539}{504608}a^{7}-\frac{46539}{504608}a^{6}+\frac{247007}{504608}a^{5}+\frac{5297}{504608}a^{4}-\frac{12435}{38816}a^{3}-\frac{6973}{38816}a^{2}+\frac{15075}{31538}a+\frac{347}{15769}$
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
$C_{2}$, 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: | $7$ | 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: |
$\frac{705}{126152}a^{11}-\frac{1929}{252304}a^{10}-\frac{3283}{31538}a^{9}+\frac{18167}{126152}a^{8}+\frac{73551}{126152}a^{7}-\frac{205811}{252304}a^{6}-\frac{233667}{126152}a^{5}+\frac{630361}{252304}a^{4}+\frac{51835}{9704}a^{3}-\frac{137029}{19408}a^{2}-\frac{366135}{63076}a+\frac{124513}{15769}$, $\frac{705}{126152}a^{11}+\frac{1929}{252304}a^{10}-\frac{3283}{31538}a^{9}-\frac{18167}{126152}a^{8}+\frac{73551}{126152}a^{7}+\frac{205811}{252304}a^{6}-\frac{233667}{126152}a^{5}-\frac{630361}{252304}a^{4}+\frac{51835}{9704}a^{3}+\frac{137029}{19408}a^{2}-\frac{366135}{63076}a-\frac{124513}{15769}$, $\frac{2021}{38816}a^{11}-\frac{41365}{504608}a^{10}-\frac{15507}{19408}a^{9}+\frac{320051}{252304}a^{8}+\frac{120599}{38816}a^{7}-\frac{2529255}{504608}a^{6}-\frac{418997}{38816}a^{5}+\frac{8688869}{504608}a^{4}+\frac{805245}{38816}a^{3}-\frac{1355009}{38816}a^{2}-\frac{20794}{1213}a+\frac{461166}{15769}$, $\frac{18045}{252304}a^{10}-\frac{131491}{126152}a^{8}+\frac{885455}{252304}a^{6}-\frac{3241101}{252304}a^{4}+\frac{413289}{19408}a^{2}-\frac{239179}{15769}$, $\frac{17161}{252304}a^{11}+\frac{85}{1213}a^{10}-\frac{130997}{126152}a^{9}-\frac{1308}{1213}a^{8}+\frac{996683}{252304}a^{7}+\frac{5074}{1213}a^{6}-\frac{3415405}{252304}a^{5}-\frac{17204}{1213}a^{4}+\frac{513089}{19408}a^{3}+\frac{34153}{1213}a^{2}-\frac{1199231}{63076}a-\frac{25599}{1213}$, $\frac{17161}{252304}a^{11}-\frac{85}{1213}a^{10}-\frac{130997}{126152}a^{9}+\frac{1308}{1213}a^{8}+\frac{996683}{252304}a^{7}-\frac{5074}{1213}a^{6}-\frac{3415405}{252304}a^{5}+\frac{17204}{1213}a^{4}+\frac{513089}{19408}a^{3}-\frac{34153}{1213}a^{2}-\frac{1199231}{63076}a+\frac{25599}{1213}$, $\frac{1425}{38816}a^{11}-\frac{40249}{504608}a^{10}-\frac{9323}{19408}a^{9}+\frac{279327}{252304}a^{8}+\frac{39755}{38816}a^{7}-\frac{1557139}{504608}a^{6}-\frac{163481}{38816}a^{5}+\frac{5805065}{504608}a^{4}+\frac{82513}{38816}a^{3}-\frac{598837}{38816}a^{2}+\frac{24763}{4852}a+\frac{42376}{15769}$
| 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: | \( 131794.878234 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 131794.878234 \cdot 2}{2\cdot\sqrt{4492085992834072576}}\cr\approx \mathstrut & 1.55065062507 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 17*x^10 + 85*x^8 - 306*x^6 + 748*x^4 - 1003*x^2 + 544)
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 - 17*x^10 + 85*x^8 - 306*x^6 + 748*x^4 - 1003*x^2 + 544, 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 - 17*x^10 + 85*x^8 - 306*x^6 + 748*x^4 - 1003*x^2 + 544);
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 - 17*x^10 + 85*x^8 - 306*x^6 + 748*x^4 - 1003*x^2 + 544);
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
$S_3^2:C_4$ (as 12T80):
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 144 |
| The 18 conjugacy class representatives for $S_3^2:C_4$ |
| Character table for $S_3^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.157216.1, 6.2.11358856.1 |
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 12 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.2246042996417036288.3 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.4.8.3 | $x^{4} + 6 x^{2} + 4 x + 14$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
| 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
|
\(17\)
| 17.12.11.2 | $x^{12} + 34$ | $12$ | $1$ | $11$ | $S_3 \times C_4$ | $[\ ]_{12}^{2}$ |