Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 24*x^16 + 228*x^14 + 1129*x^12 + 3177*x^10 + 5133*x^8 + 4523*x^6 + 1874*x^4 + 231*x^2 + 1)
gp: K = bnfinit(y^18 + 24*y^16 + 228*y^14 + 1129*y^12 + 3177*y^10 + 5133*y^8 + 4523*y^6 + 1874*y^4 + 231*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 24*x^16 + 228*x^14 + 1129*x^12 + 3177*x^10 + 5133*x^8 + 4523*x^6 + 1874*x^4 + 231*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 24*x^16 + 228*x^14 + 1129*x^12 + 3177*x^10 + 5133*x^8 + 4523*x^6 + 1874*x^4 + 231*x^2 + 1)
\( x^{18} + 24x^{16} + 228x^{14} + 1129x^{12} + 3177x^{10} + 5133x^{8} + 4523x^{6} + 1874x^{4} + 231x^{2} + 1 \)
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: |
\(-333998778547472552821465808896\)
\(\medspace = -\,2^{30}\cdot 19^{6}\cdot 137^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(43.67\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(19\), \(137\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{64549}a^{16}-\frac{4681}{64549}a^{14}+\frac{772}{3797}a^{12}+\frac{26102}{64549}a^{10}+\frac{30014}{64549}a^{8}+\frac{22475}{64549}a^{6}-\frac{9090}{64549}a^{4}-\frac{25663}{64549}a^{2}-\frac{26533}{64549}$, $\frac{1}{64549}a^{17}-\frac{4681}{64549}a^{15}+\frac{772}{3797}a^{13}+\frac{26102}{64549}a^{11}+\frac{30014}{64549}a^{9}+\frac{22475}{64549}a^{7}-\frac{9090}{64549}a^{5}-\frac{25663}{64549}a^{3}-\frac{26533}{64549}a$
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
$C_{2}\times C_{64}$, which has order $128$ (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: |
$\frac{812}{64549}a^{16}+\frac{7419}{64549}a^{14}-\frac{3438}{3797}a^{12}-\frac{880934}{64549}a^{10}-\frac{3707447}{64549}a^{8}-\frac{6795312}{64549}a^{6}-\frac{5315512}{64549}a^{4}-\frac{1344558}{64549}a^{2}-\frac{49979}{64549}$, $\frac{1867}{64549}a^{16}+\frac{39237}{64549}a^{14}+\frac{17449}{3797}a^{12}+\frac{966174}{64549}a^{10}+\frac{975841}{64549}a^{8}-\frac{1545201}{64549}a^{6}-\frac{3932132}{64549}a^{4}-\frac{2276678}{64549}a^{2}-\frac{92577}{64549}$, $\frac{8535}{64549}a^{16}+\frac{197143}{64549}a^{14}+\frac{103744}{3797}a^{12}+\frac{7961498}{64549}a^{10}+\frac{19403758}{64549}a^{8}+\frac{24900411}{64549}a^{6}+\frac{14851018}{64549}a^{4}+\frac{3014855}{64549}a^{2}+\frac{172384}{64549}$, $\frac{11736}{64549}a^{17}+\frac{253179}{64549}a^{15}+\frac{122054}{3797}a^{13}+\frac{8503986}{64549}a^{11}+\frac{18977817}{64549}a^{9}+\frac{23450673}{64549}a^{7}+\frac{15769213}{64549}a^{5}+\frac{5491331}{64549}a^{3}+\frac{896774}{64549}a$, $\frac{9347}{64549}a^{16}+\frac{204562}{64549}a^{14}+\frac{100306}{3797}a^{12}+\frac{7080564}{64549}a^{10}+\frac{15696311}{64549}a^{8}+\frac{18105099}{64549}a^{6}+\frac{9535506}{64549}a^{4}+\frac{1670297}{64549}a^{2}+\frac{122405}{64549}$, $a$, $\frac{19062}{64549}a^{17}+\frac{429339}{64549}a^{15}+\frac{218918}{3797}a^{13}+\frac{16214431}{64549}a^{11}+\frac{37983893}{64549}a^{9}+\frac{46352919}{64549}a^{7}+\frac{25085497}{64549}a^{5}+\frac{3191666}{64549}a^{3}-\frac{482474}{64549}a$, $\frac{19298}{64549}a^{17}+\frac{421956}{64549}a^{15}+\frac{207463}{3797}a^{13}+\frac{14822270}{64549}a^{11}+\frac{33900220}{64549}a^{9}+\frac{41974669}{64549}a^{7}+\frac{25974060}{64549}a^{5}+\frac{6688450}{64549}a^{3}+\frac{549775}{64549}a$
| 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: | \( 138588.252508 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 138588.252508 \cdot 128}{2\cdot\sqrt{333998778547472552821465808896}}\cr\approx \mathstrut & 0.234235528349 \end{aligned}\] (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 + 228*x^14 + 1129*x^12 + 3177*x^10 + 5133*x^8 + 4523*x^6 + 1874*x^4 + 231*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 + 228*x^14 + 1129*x^12 + 3177*x^10 + 5133*x^8 + 4523*x^6 + 1874*x^4 + 231*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 + 228*x^14 + 1129*x^12 + 3177*x^10 + 5133*x^8 + 4523*x^6 + 1874*x^4 + 231*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 + 228*x^14 + 1129*x^12 + 3177*x^10 + 5133*x^8 + 4523*x^6 + 1874*x^4 + 231*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
$S_4^2:C_2^2$ (as 18T370):
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 2304 |
| The 40 conjugacy class representatives for $S_4^2:C_2^2$ |
| Character table for $S_4^2:C_2^2$ |
Intermediate fields
| 9.9.1128762254528.1 |
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 16 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.743026229623638114304.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 | $x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
| 2.12.24.259 | $x^{12} - 8 x^{11} + 26 x^{10} + 12 x^{9} + 254 x^{8} + 208 x^{7} + 352 x^{6} + 768 x^{5} + 372 x^{4} + 896 x^{3} + 776 x^{2} - 80 x + 536$ | $4$ | $3$ | $24$ | $C_2^2 \times A_4$ | $[2, 2, 2, 3]^{3}$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.8.4.1 | $x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(137\)
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.4.2.1 | $x^{4} + 20538 x^{3} + 106780719 x^{2} + 13640908302 x + 496637065$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 137.8.4.1 | $x^{8} + 40552 x^{7} + 616674814 x^{6} + 4167912520178 x^{5} + 10563701578684025 x^{4} + 575230492397974 x^{3} + 10971113842265417 x^{2} + 1003833647367390048 x + 45253593243784800$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |