Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 24*x^16 + 220*x^14 + 1010*x^12 + 2543*x^10 + 3581*x^8 + 2690*x^6 + 905*x^4 + 66*x^2 + 1)
gp: K = bnfinit(y^18 + 24*y^16 + 220*y^14 + 1010*y^12 + 2543*y^10 + 3581*y^8 + 2690*y^6 + 905*y^4 + 66*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 24*x^16 + 220*x^14 + 1010*x^12 + 2543*x^10 + 3581*x^8 + 2690*x^6 + 905*x^4 + 66*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 24*x^16 + 220*x^14 + 1010*x^12 + 2543*x^10 + 3581*x^8 + 2690*x^6 + 905*x^4 + 66*x^2 + 1)
\( x^{18} + 24x^{16} + 220x^{14} + 1010x^{12} + 2543x^{10} + 3581x^{8} + 2690x^{6} + 905x^{4} + 66x^{2} + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-141204899984457152700528984064\)
\(\medspace = -\,2^{18}\cdot 257^{6}\cdot 43237^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(41.63\) | 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: | $2^{7/4}257^{1/2}43237^{1/2}\approx 11212.354761299932$ | ||
| Ramified primes: |
\(2\), \(257\), \(43237\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5111}a^{16}-\frac{1575}{5111}a^{14}-\frac{1078}{5111}a^{12}+\frac{2325}{5111}a^{10}+\frac{565}{5111}a^{8}-\frac{318}{5111}a^{6}+\frac{72}{5111}a^{4}-\frac{1781}{5111}a^{2}+\frac{1058}{5111}$, $\frac{1}{5111}a^{17}-\frac{1575}{5111}a^{15}-\frac{1078}{5111}a^{13}+\frac{2325}{5111}a^{11}+\frac{565}{5111}a^{9}-\frac{318}{5111}a^{7}+\frac{72}{5111}a^{5}-\frac{1781}{5111}a^{3}+\frac{1058}{5111}a$
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
$C_{2}\times C_{68}$, which has order $136$ (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: | $8$ | 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: |
$\frac{32858}{5111}a^{17}+\frac{748842}{5111}a^{15}+\frac{6320613}{5111}a^{13}+\frac{25489290}{5111}a^{11}+\frac{52292259}{5111}a^{9}+\frac{52666895}{5111}a^{7}+\frac{21087369}{5111}a^{5}+\frac{537507}{5111}a^{3}-\frac{72812}{5111}a$, $\frac{7674}{5111}a^{16}+\frac{174739}{5111}a^{14}+\frac{1474105}{5111}a^{12}+\frac{5953864}{5111}a^{10}+\frac{12324303}{5111}a^{8}+\frac{12836447}{5111}a^{6}+\frac{5862857}{5111}a^{4}+\frac{694516}{5111}a^{2}+\frac{18157}{5111}$, $\frac{3191}{5111}a^{17}+\frac{74953}{5111}a^{15}+\frac{664235}{5111}a^{13}+\frac{2895840}{5111}a^{11}+\frac{6745252}{5111}a^{9}+\frac{8455945}{5111}a^{7}+\frac{5310086}{5111}a^{5}+\frac{1288233}{5111}a^{3}+\frac{2818}{5111}a$, $\frac{19845}{5111}a^{16}+\frac{452769}{5111}a^{14}+\frac{3829875}{5111}a^{12}+\frac{15514513}{5111}a^{10}+\frac{32152192}{5111}a^{8}+\frac{33217764}{5111}a^{6}+\frac{14461890}{5111}a^{4}+\frac{1179261}{5111}a^{2}+\frac{15355}{5111}$, $\frac{9594}{5111}a^{17}+\frac{217339}{5111}a^{15}+\frac{1816737}{5111}a^{13}+\frac{7213267}{5111}a^{11}+\frac{14421081}{5111}a^{9}+\frac{13840963}{5111}a^{7}+\frac{4840900}{5111}a^{5}-\frac{261502}{5111}a^{3}-\frac{20438}{5111}a$, $\frac{12171}{5111}a^{16}+\frac{278030}{5111}a^{14}+\frac{2355770}{5111}a^{12}+\frac{9560649}{5111}a^{10}+\frac{19827889}{5111}a^{8}+\frac{20381317}{5111}a^{6}+\frac{8599033}{5111}a^{4}+\frac{484745}{5111}a^{2}-\frac{2802}{5111}$, $\frac{27358}{5111}a^{17}+\frac{630644}{5111}a^{15}+\frac{5426428}{5111}a^{13}+\frac{22617130}{5111}a^{11}+\frac{49235869}{5111}a^{9}+\frac{55770310}{5111}a^{7}+\frac{29947390}{5111}a^{5}+\frac{5758662}{5111}a^{3}+\frac{261832}{5111}a$, $\frac{6893}{5111}a^{16}+\frac{157730}{5111}a^{14}+\frac{1339822}{5111}a^{12}+\frac{5456677}{5111}a^{10}+\frac{11371938}{5111}a^{8}+\frac{11776389}{5111}a^{6}+\frac{5045086}{5111}a^{4}+\frac{317071}{5111}a^{2}-\frac{5714}{5111}$
| 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: | \( 95074.4744327 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 95074.4744327 \cdot 136}{2\cdot\sqrt{141204899984457152700528984064}}\cr\approx \mathstrut & 0.262583103800 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 24*x^16 + 220*x^14 + 1010*x^12 + 2543*x^10 + 3581*x^8 + 2690*x^6 + 905*x^4 + 66*x^2 + 1)
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^18 + 24*x^16 + 220*x^14 + 1010*x^12 + 2543*x^10 + 3581*x^8 + 2690*x^6 + 905*x^4 + 66*x^2 + 1, 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^18 + 24*x^16 + 220*x^14 + 1010*x^12 + 2543*x^10 + 3581*x^8 + 2690*x^6 + 905*x^4 + 66*x^2 + 1);
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^18 + 24*x^16 + 220*x^14 + 1010*x^12 + 2543*x^10 + 3581*x^8 + 2690*x^6 + 905*x^4 + 66*x^2 + 1);
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
$D_6\wr S_3$ (as 18T556):
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 10368 |
| The 98 conjugacy class representatives for $D_6\wr S_3$ |
| Character table for $D_6\wr S_3$ |
Intermediate fields
| 3.3.257.1, 6.0.4227136.1, 9.9.733930477541.1 |
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 18 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.4 | $x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
| 2.12.12.23 | $x^{12} + 56 x^{10} - 152 x^{9} - 764 x^{8} + 2976 x^{7} - 960 x^{6} - 11008 x^{5} + 20336 x^{4} - 14976 x^{3} + 80768 x^{2} + 6016 x + 38848$ | $2$ | $6$ | $12$ | $C_2^2 \times A_4$ | $[2, 2, 2]^{6}$ | |
|
\(257\)
| $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
|
\(43237\)
| $\Q_{43237}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{43237}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |