Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 4*x^16 + 10*x^14 + 4*x^12 - 24*x^10 - 23*x^8 + 6*x^6 + 15*x^4 + 7*x^2 + 1)
gp: K = bnfinit(y^18 + 4*y^16 + 10*y^14 + 4*y^12 - 24*y^10 - 23*y^8 + 6*y^6 + 15*y^4 + 7*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 4*x^16 + 10*x^14 + 4*x^12 - 24*x^10 - 23*x^8 + 6*x^6 + 15*x^4 + 7*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 4*x^16 + 10*x^14 + 4*x^12 - 24*x^10 - 23*x^8 + 6*x^6 + 15*x^4 + 7*x^2 + 1)
\( x^{18} + 4x^{16} + 10x^{14} + 4x^{12} - 24x^{10} - 23x^{8} + 6x^{6} + 15x^{4} + 7x^{2} + 1 \)
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: | $[4, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-57388858189268996325376\)
\(\medspace = -\,2^{18}\cdot 7^{12}\cdot 41^{2}\cdot 97^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.38\) | 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: |
\(2\), \(7\), \(41\), \(97\)
| 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 not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{167}a^{16}-\frac{5}{167}a^{14}+\frac{55}{167}a^{12}+\frac{10}{167}a^{10}+\frac{53}{167}a^{8}+\frac{1}{167}a^{6}-\frac{3}{167}a^{4}+\frac{42}{167}a^{2}-\frac{37}{167}$, $\frac{1}{167}a^{17}-\frac{5}{167}a^{15}+\frac{55}{167}a^{13}+\frac{10}{167}a^{11}+\frac{53}{167}a^{9}+\frac{1}{167}a^{7}-\frac{3}{167}a^{5}+\frac{42}{167}a^{3}-\frac{37}{167}a$
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
Trivial group, which has order $1$
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: | $10$ | 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: |
$\frac{298}{167}a^{17}+\frac{1182}{167}a^{15}+\frac{3030}{167}a^{13}+\frac{1477}{167}a^{11}-\frac{6083}{167}a^{9}-\frac{5547}{167}a^{7}+\frac{1277}{167}a^{5}+\frac{2997}{167}a^{3}+\frac{998}{167}a$, $a$, $\frac{14}{167}a^{16}+\frac{97}{167}a^{14}+\frac{269}{167}a^{12}+\frac{307}{167}a^{10}-\frac{594}{167}a^{8}-\frac{1656}{167}a^{6}-\frac{209}{167}a^{4}+\frac{1256}{167}a^{2}+\frac{484}{167}$, $\frac{26}{167}a^{16}+\frac{37}{167}a^{14}+\frac{94}{167}a^{12}-\frac{241}{167}a^{10}-\frac{125}{167}a^{8}+\frac{861}{167}a^{6}-\frac{412}{167}a^{4}-\frac{244}{167}a^{2}+\frac{40}{167}$, $\frac{378}{167}a^{17}+\frac{1283}{167}a^{15}+\frac{2921}{167}a^{13}-\frac{562}{167}a^{11}-\frac{9525}{167}a^{9}-\frac{3296}{167}a^{7}+\frac{5546}{167}a^{5}+\frac{3017}{167}a^{3}+\frac{543}{167}a$, $\frac{131}{167}a^{16}+\frac{514}{167}a^{14}+\frac{1360}{167}a^{12}+\frac{809}{167}a^{10}-\frac{2075}{167}a^{8}-\frac{1706}{167}a^{6}+\frac{275}{167}a^{4}+\frac{492}{167}a^{2}-\frac{4}{167}$, $\frac{14}{167}a^{17}+\frac{176}{167}a^{16}+\frac{97}{167}a^{15}+\frac{790}{167}a^{14}+\frac{269}{167}a^{13}+\frac{2165}{167}a^{12}+\frac{307}{167}a^{11}+\frac{1927}{167}a^{10}-\frac{594}{167}a^{9}-\frac{2696}{167}a^{8}-\frac{1656}{167}a^{7}-\frac{4166}{167}a^{6}-\frac{209}{167}a^{5}-\frac{361}{167}a^{4}+\frac{1256}{167}a^{3}+\frac{1046}{167}a^{2}+\frac{484}{167}a+\frac{168}{167}$, $\frac{347}{167}a^{17}-\frac{176}{167}a^{16}+\frac{1104}{167}a^{15}-\frac{790}{167}a^{14}+\frac{2385}{167}a^{13}-\frac{2165}{167}a^{12}-\frac{1373}{167}a^{11}-\frac{1927}{167}a^{10}-\frac{9498}{167}a^{9}+\frac{2696}{167}a^{8}-\frac{2492}{167}a^{7}+\frac{4166}{167}a^{6}+\frac{5973}{167}a^{5}+\frac{361}{167}a^{4}+\frac{3719}{167}a^{3}-\frac{1046}{167}a^{2}+\frac{855}{167}a-\frac{168}{167}$, $\frac{26}{167}a^{17}-\frac{14}{167}a^{16}+\frac{37}{167}a^{15}-\frac{97}{167}a^{14}+\frac{94}{167}a^{13}-\frac{269}{167}a^{12}-\frac{241}{167}a^{11}-\frac{307}{167}a^{10}-\frac{125}{167}a^{9}+\frac{594}{167}a^{8}+\frac{861}{167}a^{7}+\frac{1656}{167}a^{6}-\frac{412}{167}a^{5}+\frac{209}{167}a^{4}-\frac{244}{167}a^{3}-\frac{1256}{167}a^{2}+\frac{207}{167}a-\frac{484}{167}$, $\frac{329}{167}a^{17}+\frac{1361}{167}a^{15}+\frac{3566}{167}a^{13}+\frac{2288}{167}a^{11}-\frac{6110}{167}a^{9}-\frac{6351}{167}a^{7}+\frac{850}{167}a^{5}+\frac{2295}{167}a^{3}+a^{2}+\frac{853}{167}a+1$
| 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: | \( 12064.8420969 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{7}\cdot 12064.8420969 \cdot 1}{2\cdot\sqrt{57388858189268996325376}}\cr\approx \mathstrut & 0.155760350485 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 4*x^16 + 10*x^14 + 4*x^12 - 24*x^10 - 23*x^8 + 6*x^6 + 15*x^4 + 7*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 + 4*x^16 + 10*x^14 + 4*x^12 - 24*x^10 - 23*x^8 + 6*x^6 + 15*x^4 + 7*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 + 4*x^16 + 10*x^14 + 4*x^12 - 24*x^10 - 23*x^8 + 6*x^6 + 15*x^4 + 7*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 + 4*x^16 + 10*x^14 + 4*x^12 - 24*x^10 - 23*x^8 + 6*x^6 + 15*x^4 + 7*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
$D_6\wr C_3$ (as 18T472):
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 5184 |
| The 88 conjugacy class representatives for $D_6\wr C_3$ |
| Character table for $D_6\wr C_3$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 6.4.153664.1, 9.5.467890073.1 |
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: | 18.8.57388858189268996325376.6 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $18$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.18.116 | $x^{18} + 54 x^{16} + 128 x^{15} + 1192 x^{14} + 5296 x^{13} + 25360 x^{12} + 109760 x^{11} + 401024 x^{10} + 1311040 x^{9} + 3636352 x^{8} + 8885760 x^{7} + 18496384 x^{6} + 32649472 x^{5} + 49679104 x^{4} + 60578816 x^{3} + 57839360 x^{2} + 44080128 x + 26046976$ | $2$ | $9$ | $18$ | 18T26 | $[2, 2, 2]^{9}$ |
|
\(7\)
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(41\)
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.3.0.1 | $x^{3} + x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 41.3.0.1 | $x^{3} + x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(97\)
| $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.4.2.1 | $x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |