Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 7*x^16 + 23*x^14 + 45*x^12 + 71*x^10 + 80*x^8 + 66*x^6 + 37*x^4 + 8*x^2 - 1)
gp: K = bnfinit(y^18 + 7*y^16 + 23*y^14 + 45*y^12 + 71*y^10 + 80*y^8 + 66*y^6 + 37*y^4 + 8*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 7*x^16 + 23*x^14 + 45*x^12 + 71*x^10 + 80*x^8 + 66*x^6 + 37*x^4 + 8*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 7*x^16 + 23*x^14 + 45*x^12 + 71*x^10 + 80*x^8 + 66*x^6 + 37*x^4 + 8*x^2 - 1)
\( x^{18} + 7x^{16} + 23x^{14} + 45x^{12} + 71x^{10} + 80x^{8} + 66x^{6} + 37x^{4} + 8x^{2} - 1 \)
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: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3511479662680207261696\)
\(\medspace = 2^{22}\cdot 307^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.74\) | 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: |
\(2\), \(307\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{14}a^{14}-\frac{1}{14}a^{12}-\frac{1}{2}a^{11}-\frac{3}{7}a^{10}-\frac{1}{2}a^{9}+\frac{2}{7}a^{8}+\frac{1}{7}a^{6}-\frac{1}{2}a^{4}+\frac{3}{7}a^{2}-\frac{1}{2}a-\frac{2}{7}$, $\frac{1}{14}a^{15}-\frac{1}{14}a^{13}+\frac{1}{14}a^{11}-\frac{1}{2}a^{10}-\frac{3}{14}a^{9}-\frac{1}{2}a^{8}+\frac{1}{7}a^{7}-\frac{1}{2}a^{5}+\frac{3}{7}a^{3}+\frac{3}{14}a-\frac{1}{2}$, $\frac{1}{98}a^{16}+\frac{3}{98}a^{14}+\frac{11}{98}a^{12}-\frac{1}{2}a^{11}+\frac{1}{98}a^{10}-\frac{1}{2}a^{9}+\frac{9}{49}a^{8}-\frac{41}{98}a^{6}+\frac{17}{49}a^{4}-\frac{1}{98}a^{2}-\frac{1}{2}a+\frac{6}{49}$, $\frac{1}{98}a^{17}+\frac{3}{98}a^{15}+\frac{11}{98}a^{13}-\frac{24}{49}a^{11}-\frac{1}{2}a^{10}-\frac{31}{98}a^{9}-\frac{1}{2}a^{8}-\frac{41}{98}a^{7}+\frac{17}{49}a^{5}-\frac{1}{98}a^{3}-\frac{37}{98}a-\frac{1}{2}$
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: | $9$ | 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: |
$\frac{327}{98}a^{16}+\frac{2031}{98}a^{14}+\frac{2964}{49}a^{12}+\frac{10099}{98}a^{10}+\frac{15427}{98}a^{8}+\frac{14271}{98}a^{6}+\frac{5363}{49}a^{4}+\frac{2031}{49}a^{2}-\frac{325}{98}$, $\frac{69}{49}a^{17}+\frac{841}{98}a^{15}+\frac{1207}{49}a^{13}+\frac{2022}{49}a^{11}+\frac{6201}{98}a^{9}+\frac{2743}{49}a^{7}+\frac{4055}{98}a^{5}+\frac{1297}{98}a^{3}-\frac{395}{98}a$, $\frac{197}{98}a^{16}+\frac{607}{49}a^{14}+\frac{1752}{49}a^{12}+\frac{5867}{98}a^{10}+\frac{4440}{49}a^{8}+\frac{7967}{98}a^{6}+\frac{5767}{98}a^{4}+\frac{1973}{98}a^{2}-\frac{211}{49}$, $\frac{16}{49}a^{17}+\frac{187}{98}a^{15}+\frac{253}{49}a^{13}+\frac{380}{49}a^{11}+\frac{1087}{98}a^{9}+\frac{366}{49}a^{7}+\frac{451}{98}a^{5}-\frac{123}{98}a^{3}-\frac{225}{98}a$, $\frac{69}{49}a^{16}+\frac{841}{98}a^{14}+\frac{1207}{49}a^{12}+\frac{2022}{49}a^{10}+\frac{6201}{98}a^{8}+\frac{2743}{49}a^{6}+\frac{4055}{98}a^{4}+\frac{1297}{98}a^{2}-\frac{297}{98}$, $\frac{37}{14}a^{17}-\frac{16}{49}a^{16}+\frac{118}{7}a^{15}-\frac{215}{98}a^{14}+\frac{709}{14}a^{13}-\frac{337}{49}a^{12}+\frac{627}{7}a^{11}-\frac{1229}{98}a^{10}+\frac{968}{7}a^{9}-\frac{918}{49}a^{8}+\frac{273}{2}a^{7}-\frac{933}{49}a^{6}+\frac{1489}{14}a^{5}-\frac{1431}{98}a^{4}+\frac{332}{7}a^{3}-\frac{633}{98}a^{2}+3a-\frac{52}{49}$, $\frac{289}{98}a^{17}+\frac{62}{49}a^{16}+\frac{885}{49}a^{15}+\frac{771}{98}a^{14}+\frac{5069}{98}a^{13}+\frac{2239}{98}a^{12}+\frac{8395}{98}a^{11}+\frac{3757}{98}a^{10}+\frac{12685}{98}a^{9}+\frac{5641}{98}a^{8}+\frac{11223}{98}a^{7}+\frac{2561}{49}a^{6}+\frac{8013}{98}a^{5}+\frac{3677}{98}a^{4}+\frac{1266}{49}a^{3}+\frac{596}{49}a^{2}-\frac{781}{98}a-\frac{255}{98}$, $\frac{18}{49}a^{17}+\frac{253}{98}a^{16}+\frac{96}{49}a^{15}+\frac{789}{49}a^{14}+\frac{459}{98}a^{13}+\frac{2305}{49}a^{12}+\frac{561}{98}a^{11}+\frac{3917}{49}a^{10}+\frac{394}{49}a^{9}+\frac{11897}{98}a^{8}+\frac{81}{49}a^{7}+\frac{11061}{98}a^{6}-\frac{74}{49}a^{5}+\frac{8161}{98}a^{4}-\frac{463}{98}a^{3}+\frac{2995}{98}a^{2}-\frac{197}{49}a-\frac{289}{98}$, $\frac{18}{49}a^{17}-\frac{37}{98}a^{16}+\frac{96}{49}a^{15}-\frac{223}{98}a^{14}+\frac{459}{98}a^{13}-\frac{319}{49}a^{12}+\frac{561}{98}a^{11}-\frac{540}{49}a^{10}+\frac{394}{49}a^{9}-\frac{851}{49}a^{8}+\frac{81}{49}a^{7}-\frac{1549}{98}a^{6}-\frac{74}{49}a^{5}-\frac{629}{49}a^{4}-\frac{561}{98}a^{3}-\frac{244}{49}a^{2}-\frac{246}{49}a+\frac{2}{49}$
| 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: | \( 3198.65544505 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 3198.65544505 \cdot 1}{2\cdot\sqrt{3511479662680207261696}}\cr\approx \mathstrut & 0.262235404014 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 7*x^16 + 23*x^14 + 45*x^12 + 71*x^10 + 80*x^8 + 66*x^6 + 37*x^4 + 8*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 + 7*x^16 + 23*x^14 + 45*x^12 + 71*x^10 + 80*x^8 + 66*x^6 + 37*x^4 + 8*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 + 7*x^16 + 23*x^14 + 45*x^12 + 71*x^10 + 80*x^8 + 66*x^6 + 37*x^4 + 8*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 + 7*x^16 + 23*x^14 + 45*x^12 + 71*x^10 + 80*x^8 + 66*x^6 + 37*x^4 + 8*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
$C_2^2:A_4^2.S_4$ (as 18T595):
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 13824 |
| The 48 conjugacy class representatives for $C_2^2:A_4^2.S_4$ |
| Character table for $C_2^2:A_4^2.S_4$ |
Intermediate fields
| 3.1.307.1, 9.3.462951088.1 |
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 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.4.101602708145479684241728995328.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.6 | $x^{6} - 4 x^{5} + 30 x^{4} - 16 x^{3} + 164 x^{2} + 160 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
| 2.12.16.1 | $x^{12} - 2 x^{11} + 4 x^{10} + 4 x^{9} - 4 x^{7} + 10 x^{6} + 8 x^{5} + 4 x^{4} + 12 x^{3} + 12 x^{2} + 12$ | $6$ | $2$ | $16$ | 12T208 | $[4/3, 4/3, 4/3, 4/3, 2, 2]_{3}^{6}$ | |
|
\(307\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ | ||||
| Deg $6$ | $2$ | $3$ | $3$ |