Properties

Label 18.2.111...312.1
Degree $18$
Signature $[2, 8]$
Discriminant $1.118\times 10^{20}$
Root discriminant \(13.00\)
Ramified primes $2,97$
Class number $1$
Class group trivial
Galois group $\SOPlus(4,2)$ (as 18T34)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 8*x^16 - 8*x^15 + x^14 + 8*x^13 - 10*x^12 + 2*x^11 + 6*x^10 - 4*x^9 - 4*x^8 + 10*x^7 - 6*x^6 + 4*x^5 + x^4 + 2*x^2 + 1)
 
gp: K = bnfinit(y^18 - 4*y^17 + 8*y^16 - 8*y^15 + y^14 + 8*y^13 - 10*y^12 + 2*y^11 + 6*y^10 - 4*y^9 - 4*y^8 + 10*y^7 - 6*y^6 + 4*y^5 + y^4 + 2*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^17 + 8*x^16 - 8*x^15 + x^14 + 8*x^13 - 10*x^12 + 2*x^11 + 6*x^10 - 4*x^9 - 4*x^8 + 10*x^7 - 6*x^6 + 4*x^5 + x^4 + 2*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 8*x^16 - 8*x^15 + x^14 + 8*x^13 - 10*x^12 + 2*x^11 + 6*x^10 - 4*x^9 - 4*x^8 + 10*x^7 - 6*x^6 + 4*x^5 + x^4 + 2*x^2 + 1)
 

\( x^{18} - 4 x^{17} + 8 x^{16} - 8 x^{15} + x^{14} + 8 x^{13} - 10 x^{12} + 2 x^{11} + 6 x^{10} - 4 x^{9} - 4 x^{8} + 10 x^{7} - 6 x^{6} + 4 x^{5} + x^{4} + 2 x^{2} + 1 \) 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:   \(111799609989175181312\) \(\medspace = 2^{27}\cdot 97^{6}\) 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:  \(13.00\)
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))
 
Ramified primes:   \(2\), \(97\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{292}a^{17}-\frac{8}{73}a^{16}-\frac{45}{292}a^{15}-\frac{31}{146}a^{14}+\frac{29}{146}a^{13}+\frac{34}{73}a^{12}+\frac{31}{73}a^{11}+\frac{17}{146}a^{10}-\frac{35}{146}a^{9}+\frac{29}{146}a^{8}-\frac{11}{146}a^{7}-\frac{26}{73}a^{6}+\frac{33}{73}a^{5}+\frac{26}{73}a^{4}+\frac{9}{292}a^{3}+\frac{10}{73}a^{2}-\frac{23}{292}a-\frac{43}{146}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

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:   $\frac{46}{73}a^{17}-\frac{158}{73}a^{16}+\frac{266}{73}a^{15}-\frac{375}{146}a^{14}-\frac{106}{73}a^{13}+\frac{343}{73}a^{12}-\frac{282}{73}a^{11}-\frac{42}{73}a^{10}+\frac{211}{73}a^{9}-\frac{33}{73}a^{8}-\frac{136}{73}a^{7}+\frac{253}{73}a^{6}-\frac{60}{73}a^{5}+\frac{112}{73}a^{4}+\frac{49}{73}a^{3}-\frac{58}{73}a^{2}+\frac{110}{73}a-\frac{101}{146}$, $a$, $\frac{97}{292}a^{17}-\frac{165}{146}a^{16}+\frac{307}{292}a^{15}+\frac{205}{146}a^{14}-\frac{691}{146}a^{13}+\frac{305}{73}a^{12}+\frac{87}{73}a^{11}-\frac{833}{146}a^{10}+\frac{547}{146}a^{9}+\frac{477}{146}a^{8}-\frac{775}{146}a^{7}+\frac{33}{73}a^{6}+\frac{281}{73}a^{5}-\frac{252}{73}a^{4}-\frac{3}{292}a^{3}-\frac{31}{146}a^{2}-\frac{187}{292}a-\frac{83}{146}$, $\frac{45}{146}a^{17}-\frac{136}{73}a^{16}+\frac{338}{73}a^{15}-\frac{446}{73}a^{14}+\frac{210}{73}a^{13}+\frac{286}{73}a^{12}-\frac{568}{73}a^{11}+\frac{327}{73}a^{10}+\frac{177}{73}a^{9}-\frac{301}{73}a^{8}-\frac{57}{73}a^{7}+\frac{361}{73}a^{6}-\frac{388}{73}a^{5}+\frac{150}{73}a^{4}-\frac{179}{146}a^{3}-\frac{49}{73}a^{2}+\frac{30}{73}a-\frac{37}{73}$, $\frac{2}{73}a^{17}-\frac{55}{146}a^{16}+\frac{39}{146}a^{15}+\frac{95}{73}a^{14}-\frac{249}{73}a^{13}+\frac{199}{73}a^{12}+\frac{102}{73}a^{11}-\frac{297}{73}a^{10}+\frac{152}{73}a^{9}+\frac{189}{73}a^{8}-\frac{263}{73}a^{7}-\frac{135}{73}a^{6}+\frac{264}{73}a^{5}-\frac{157}{73}a^{4}-\frac{55}{73}a^{3}-\frac{205}{146}a^{2}-\frac{19}{146}a-\frac{26}{73}$, $\frac{237}{292}a^{17}-\frac{361}{146}a^{16}+\frac{1015}{292}a^{15}-\frac{193}{146}a^{14}-\frac{427}{146}a^{13}+\frac{320}{73}a^{12}-\frac{99}{73}a^{11}-\frac{497}{146}a^{10}+\frac{465}{146}a^{9}+\frac{303}{146}a^{8}-\frac{563}{146}a^{7}+\frac{262}{73}a^{6}+\frac{83}{73}a^{5}+\frac{30}{73}a^{4}+\frac{381}{292}a^{3}+\frac{141}{146}a^{2}-\frac{195}{292}a+\frac{29}{146}$, $\frac{143}{292}a^{17}-\frac{561}{292}a^{16}+\frac{1011}{292}a^{15}-\frac{909}{292}a^{14}+\frac{59}{146}a^{13}+\frac{307}{146}a^{12}-\frac{166}{73}a^{11}+\frac{95}{146}a^{10}+\frac{16}{73}a^{9}+\frac{66}{73}a^{8}-\frac{93}{73}a^{7}+\frac{229}{146}a^{6}-\frac{172}{73}a^{5}+\frac{287}{73}a^{4}-\frac{465}{292}a^{3}-\frac{193}{292}a^{2}+\frac{361}{292}a-\frac{107}{292}$, $\frac{11}{146}a^{17}-\frac{30}{73}a^{16}+\frac{81}{73}a^{15}-\frac{171}{146}a^{14}-\frac{46}{73}a^{13}+\frac{237}{73}a^{12}-\frac{267}{73}a^{11}+\frac{41}{73}a^{10}+\frac{199}{73}a^{9}-\frac{192}{73}a^{8}-\frac{48}{73}a^{7}+\frac{231}{73}a^{6}-\frac{77}{73}a^{5}-\frac{85}{73}a^{4}+\frac{245}{146}a^{3}-\frac{72}{73}a^{2}+\frac{56}{73}a-\frac{143}{146}$, $\frac{16}{73}a^{17}-\frac{369}{292}a^{16}+\frac{385}{146}a^{15}-\frac{683}{292}a^{14}-\frac{94}{73}a^{13}+\frac{775}{146}a^{12}-\frac{352}{73}a^{11}-\frac{40}{73}a^{10}+\frac{753}{146}a^{9}-\frac{407}{146}a^{8}-\frac{485}{146}a^{7}+\frac{687}{146}a^{6}-\frac{151}{73}a^{5}-\frac{161}{73}a^{4}+\frac{71}{73}a^{3}-\frac{725}{292}a^{2}+\frac{67}{146}a-\frac{321}{292}$ 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:  \( 473.705580838 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 473.705580838 \cdot 1}{2\cdot\sqrt{111799609989175181312}}\cr\approx \mathstrut & 0.217649312442 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 8*x^16 - 8*x^15 + x^14 + 8*x^13 - 10*x^12 + 2*x^11 + 6*x^10 - 4*x^9 - 4*x^8 + 10*x^7 - 6*x^6 + 4*x^5 + x^4 + 2*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^17 + 8*x^16 - 8*x^15 + x^14 + 8*x^13 - 10*x^12 + 2*x^11 + 6*x^10 - 4*x^9 - 4*x^8 + 10*x^7 - 6*x^6 + 4*x^5 + x^4 + 2*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^17 + 8*x^16 - 8*x^15 + x^14 + 8*x^13 - 10*x^12 + 2*x^11 + 6*x^10 - 4*x^9 - 4*x^8 + 10*x^7 - 6*x^6 + 4*x^5 + x^4 + 2*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^17 + 8*x^16 - 8*x^15 + x^14 + 8*x^13 - 10*x^12 + 2*x^11 + 6*x^10 - 4*x^9 - 4*x^8 + 10*x^7 - 6*x^6 + 4*x^5 + x^4 + 2*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

$\SOPlus(4,2)$ (as 18T34):

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 72
The 9 conjugacy class representatives for $\SOPlus(4,2)$
Character table for $\SOPlus(4,2)$

Intermediate fields

\(\Q(\sqrt{2}) \), 6.2.49664.1 x2, 9.1.467288576.1

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 6 siblings: 6.2.49664.1, 6.2.7301384.1
Degree 9 sibling: 9.1.467288576.1
Degree 12 siblings: 12.0.239251750912.1, deg 12, deg 12, deg 12, deg 12, deg 12
Degree 18 siblings: deg 18, deg 18
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 6.2.49664.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}$ ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(2\) Copy content Toggle raw display 2.6.9.1$x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$$2$$3$$9$$C_6$$[3]^{3}$
\(97\) Copy content Toggle raw display $\Q_{97}$$x + 92$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 92$$1$$1$$0$Trivial$[\ ]$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{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$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$