Properties

Label 18.2.109...889.1
Degree $18$
Signature $[2, 8]$
Discriminant $1.099\times 10^{19}$
Root discriminant \(11.42\)
Ramified primes $23,379,719$
Class number $1$
Class group trivial
Galois group $A_4^3.(C_2\times S_4)$ (as 18T776)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^16 - 4*x^15 - x^14 + 3*x^13 + 9*x^12 + x^11 - 2*x^10 - 13*x^9 - 2*x^8 + x^7 + 9*x^6 + 3*x^5 - x^4 - 4*x^3 - x^2 + 1)
 
gp: K = bnfinit(y^18 - y^16 - 4*y^15 - y^14 + 3*y^13 + 9*y^12 + y^11 - 2*y^10 - 13*y^9 - 2*y^8 + y^7 + 9*y^6 + 3*y^5 - y^4 - 4*y^3 - y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^16 - 4*x^15 - x^14 + 3*x^13 + 9*x^12 + x^11 - 2*x^10 - 13*x^9 - 2*x^8 + x^7 + 9*x^6 + 3*x^5 - x^4 - 4*x^3 - x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^16 - 4*x^15 - x^14 + 3*x^13 + 9*x^12 + x^11 - 2*x^10 - 13*x^9 - 2*x^8 + x^7 + 9*x^6 + 3*x^5 - x^4 - 4*x^3 - x^2 + 1)
 

\( x^{18} - x^{16} - 4 x^{15} - x^{14} + 3 x^{13} + 9 x^{12} + x^{11} - 2 x^{10} - 13 x^{9} - 2 x^{8} + x^{7} + 9 x^{6} + 3 x^{5} - x^{4} - 4 x^{3} - x^{2} + 1 \) Copy content Toggle raw display

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:   \(10992670662263790889\) \(\medspace = 23^{6}\cdot 379^{2}\cdot 719^{2}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(11.42\)
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:  $23^{1/2}379^{1/2}719^{1/2}\approx 2503.5021469932876$
Ramified primes:   \(23\), \(379\), \(719\) Copy content Toggle raw display
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7}a^{16}+\frac{1}{7}a^{15}-\frac{1}{7}a^{14}+\frac{1}{7}a^{13}+\frac{1}{7}a^{12}+\frac{3}{7}a^{11}-\frac{3}{7}a^{10}+\frac{2}{7}a^{9}+\frac{3}{7}a^{8}+\frac{2}{7}a^{7}-\frac{3}{7}a^{6}+\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{1}{7}a^{3}-\frac{1}{7}a^{2}+\frac{1}{7}a+\frac{1}{7}$, $\frac{1}{35}a^{17}-\frac{1}{35}a^{16}-\frac{2}{7}a^{15}-\frac{4}{35}a^{14}+\frac{13}{35}a^{13}+\frac{3}{7}a^{12}-\frac{16}{35}a^{11}-\frac{13}{35}a^{10}+\frac{6}{35}a^{9}-\frac{4}{35}a^{8}+\frac{1}{5}a^{7}+\frac{9}{35}a^{6}-\frac{1}{7}a^{5}+\frac{13}{35}a^{4}+\frac{11}{35}a^{3}+\frac{2}{7}a^{2}-\frac{1}{35}a-\frac{9}{35}$ Copy content Toggle raw display

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 $2$) Copy content Toggle raw display
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:   $\frac{3}{7}a^{17}+\frac{10}{7}a^{16}-\frac{3}{7}a^{15}-\frac{25}{7}a^{14}-\frac{39}{7}a^{13}+\frac{2}{7}a^{12}+\frac{61}{7}a^{11}+\frac{83}{7}a^{10}-\frac{5}{7}a^{9}-\frac{64}{7}a^{8}-\frac{107}{7}a^{7}-\frac{19}{7}a^{6}+\frac{45}{7}a^{5}+\frac{66}{7}a^{4}+\frac{18}{7}a^{3}-\frac{11}{7}a^{2}-\frac{25}{7}a$, $a$, $\frac{23}{35}a^{17}+\frac{27}{35}a^{16}-\frac{1}{7}a^{15}-\frac{107}{35}a^{14}-\frac{141}{35}a^{13}-\frac{5}{7}a^{12}+\frac{202}{35}a^{11}+\frac{251}{35}a^{10}+\frac{19}{5}a^{9}-\frac{187}{35}a^{8}-\frac{299}{35}a^{7}-\frac{188}{35}a^{6}+\frac{209}{35}a^{4}+\frac{93}{35}a^{3}+\frac{1}{7}a^{2}-\frac{43}{35}a-\frac{52}{35}$, $a^{16}-a^{14}-3a^{13}-a^{12}+2a^{11}+6a^{10}-7a^{7}-2a^{6}+a^{5}+2a^{4}+a^{3}-2a$, $\frac{5}{7}a^{16}+\frac{12}{7}a^{15}-\frac{5}{7}a^{14}-\frac{30}{7}a^{13}-\frac{44}{7}a^{12}+\frac{1}{7}a^{11}+\frac{62}{7}a^{10}+\frac{80}{7}a^{9}+\frac{1}{7}a^{8}-\frac{46}{7}a^{7}-\frac{99}{7}a^{6}-\frac{20}{7}a^{5}+\frac{19}{7}a^{4}+\frac{40}{7}a^{3}+\frac{16}{7}a^{2}-\frac{9}{7}a-\frac{9}{7}$, $\frac{1}{35}a^{17}+\frac{34}{35}a^{16}-\frac{9}{7}a^{15}-\frac{39}{35}a^{14}-\frac{57}{35}a^{13}+\frac{24}{7}a^{12}+\frac{124}{35}a^{11}+\frac{92}{35}a^{10}-\frac{309}{35}a^{9}-\frac{39}{35}a^{8}-\frac{29}{5}a^{7}+\frac{324}{35}a^{6}+\frac{27}{7}a^{5}+\frac{13}{35}a^{4}-\frac{129}{35}a^{3}-\frac{26}{7}a^{2}-\frac{1}{35}a+\frac{96}{35}$, $\frac{2}{35}a^{17}-\frac{2}{35}a^{16}-\frac{4}{7}a^{15}-\frac{8}{35}a^{14}+\frac{26}{35}a^{13}+\frac{13}{7}a^{12}+\frac{38}{35}a^{11}-\frac{61}{35}a^{10}-\frac{128}{35}a^{9}-\frac{43}{35}a^{8}+\frac{7}{5}a^{7}+\frac{158}{35}a^{6}+\frac{19}{7}a^{5}-\frac{79}{35}a^{4}-\frac{83}{35}a^{3}-\frac{10}{7}a^{2}+\frac{33}{35}a+\frac{52}{35}$, $\frac{51}{35}a^{17}-\frac{16}{35}a^{16}-\frac{4}{7}a^{15}-\frac{169}{35}a^{14}-\frac{37}{35}a^{13}+\frac{6}{7}a^{12}+\frac{339}{35}a^{11}+\frac{37}{35}a^{10}+\frac{201}{35}a^{9}-\frac{519}{35}a^{8}-\frac{9}{5}a^{7}-\frac{276}{35}a^{6}+\frac{47}{7}a^{5}+\frac{138}{35}a^{4}+\frac{141}{35}a^{3}-\frac{10}{7}a^{2}-\frac{51}{35}a-\frac{74}{35}$, $\frac{44}{35}a^{17}+\frac{16}{35}a^{16}-\frac{6}{7}a^{15}-\frac{166}{35}a^{14}-\frac{103}{35}a^{13}+\frac{4}{7}a^{12}+\frac{316}{35}a^{11}+\frac{158}{35}a^{10}+\frac{139}{35}a^{9}-\frac{346}{35}a^{8}-\frac{202}{35}a^{7}-\frac{239}{35}a^{6}+\frac{27}{7}a^{5}+\frac{72}{35}a^{4}+\frac{124}{35}a^{3}-\frac{1}{7}a^{2}+\frac{16}{35}a-\frac{3}{5}$ Copy content Toggle raw display
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:  \( 107.172918917 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 107.172918917 \cdot 1}{2\cdot\sqrt{10992670662263790889}}\cr\approx \mathstrut & 0.157037145814 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - x^16 - 4*x^15 - x^14 + 3*x^13 + 9*x^12 + x^11 - 2*x^10 - 13*x^9 - 2*x^8 + x^7 + 9*x^6 + 3*x^5 - x^4 - 4*x^3 - 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 - x^16 - 4*x^15 - x^14 + 3*x^13 + 9*x^12 + x^11 - 2*x^10 - 13*x^9 - 2*x^8 + x^7 + 9*x^6 + 3*x^5 - x^4 - 4*x^3 - 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 - x^16 - 4*x^15 - x^14 + 3*x^13 + 9*x^12 + x^11 - 2*x^10 - 13*x^9 - 2*x^8 + x^7 + 9*x^6 + 3*x^5 - x^4 - 4*x^3 - 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 - x^16 - 4*x^15 - x^14 + 3*x^13 + 9*x^12 + x^11 - 2*x^10 - 13*x^9 - 2*x^8 + x^7 + 9*x^6 + 3*x^5 - x^4 - 4*x^3 - 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

$A_4^3.(C_2\times S_4)$ (as 18T776):

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 82944
The 65 conjugacy class representatives for $A_4^3.(C_2\times S_4)$ are not computed
Character table for $A_4^3.(C_2\times S_4)$ is not computed

Intermediate fields

3.1.23.1, 9.3.3315519667.1

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 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.0.76256952341.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{7}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ R ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(23\) Copy content Toggle raw display 23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
23.6.3.2$x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(379\) Copy content Toggle raw display $\Q_{379}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{379}$$x$$1$$1$$0$Trivial$[\ ]$
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 $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
\(719\) Copy content Toggle raw display $\Q_{719}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{719}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$2$$2$$2$