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Properties

Label 18.2.194...928.1
Degree $18$
Signature $[2, 8]$
Discriminant $1.948\times 10^{24}$
Root discriminant \(22.36\)
Ramified primes $2,7$
Class number $3$
Class group [3]
Galois group $C_3^2 : C_4$ (as 18T10)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - 6*x^14 - 16*x^12 + 37*x^10 + 166*x^8 + 372*x^6 - 88*x^4 + 36*x^2 - 8)
 
gp: K = bnfinit(y^18 + 2*y^16 - 6*y^14 - 16*y^12 + 37*y^10 + 166*y^8 + 372*y^6 - 88*y^4 + 36*y^2 - 8, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 2*x^16 - 6*x^14 - 16*x^12 + 37*x^10 + 166*x^8 + 372*x^6 - 88*x^4 + 36*x^2 - 8);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 2*x^16 - 6*x^14 - 16*x^12 + 37*x^10 + 166*x^8 + 372*x^6 - 88*x^4 + 36*x^2 - 8)
 

\( x^{18} + 2x^{16} - 6x^{14} - 16x^{12} + 37x^{10} + 166x^{8} + 372x^{6} - 88x^{4} + 36x^{2} - 8 \) 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:   \(1947987996273487986556928\) \(\medspace = 2^{47}\cdot 7^{12}\) 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:  \(22.36\)
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:  $2^{11/4}7^{2/3}\approx 24.616776431006123$
Ramified primes:   \(2\), \(7\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{10}-\frac{1}{4}a^{8}-\frac{1}{16}a^{6}+\frac{3}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}$, $\frac{1}{16}a^{11}-\frac{1}{4}a^{8}+\frac{3}{16}a^{7}-\frac{1}{4}a^{6}-\frac{1}{8}a^{5}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{8}+\frac{1}{8}a^{6}-\frac{1}{2}a^{5}+\frac{3}{8}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{11}+\frac{3}{32}a^{9}-\frac{1}{4}a^{8}-\frac{5}{32}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{16}a^{3}-\frac{1}{8}a$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{12}-\frac{1}{32}a^{10}+\frac{3}{32}a^{8}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{5}{16}a^{4}-\frac{1}{2}a^{3}-\frac{3}{8}a^{2}-\frac{1}{2}$, $\frac{1}{32}a^{15}-\frac{1}{16}a^{9}-\frac{3}{32}a^{7}-\frac{1}{4}a^{6}+\frac{5}{16}a^{5}+\frac{1}{4}a^{4}+\frac{7}{16}a^{3}-\frac{3}{8}a$, $\frac{1}{1090784}a^{16}-\frac{5653}{545392}a^{14}+\frac{2749}{136348}a^{12}+\frac{825}{68174}a^{10}-\frac{237199}{1090784}a^{8}-\frac{1}{4}a^{7}+\frac{72475}{545392}a^{6}-\frac{1}{4}a^{5}+\frac{6627}{545392}a^{4}+\frac{128881}{272696}a^{2}+\frac{8940}{34087}$, $\frac{1}{2181568}a^{17}-\frac{1}{2181568}a^{16}+\frac{22781}{2181568}a^{15}-\frac{22781}{2181568}a^{14}-\frac{12095}{2181568}a^{13}+\frac{12095}{2181568}a^{12}+\frac{47287}{2181568}a^{11}-\frac{47287}{2181568}a^{10}+\frac{68879}{1090784}a^{9}+\frac{203817}{1090784}a^{8}+\frac{53281}{545392}a^{7}+\frac{83067}{545392}a^{6}+\frac{156705}{545392}a^{5}-\frac{156705}{545392}a^{4}-\frac{7467}{545392}a^{3}+\frac{7467}{545392}a^{2}-\frac{134675}{272696}a+\frac{134675}{272696}$ 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

$C_{3}$, which has order $3$

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:   $\frac{8547}{545392}a^{17}+\frac{9949}{545392}a^{16}+\frac{42613}{1090784}a^{15}+\frac{41499}{1090784}a^{14}-\frac{82449}{1090784}a^{13}-\frac{114351}{1090784}a^{12}-\frac{321923}{1090784}a^{11}-\frac{327605}{1090784}a^{10}+\frac{480249}{1090784}a^{9}+\frac{684319}{1090784}a^{8}+\frac{1563637}{545392}a^{7}+\frac{1693191}{545392}a^{6}+\frac{3896755}{545392}a^{5}+\frac{3867361}{545392}a^{4}+\frac{57256}{34087}a^{3}-\frac{74979}{68174}a^{2}+\frac{135617}{136348}a+\frac{122703}{136348}$, $\frac{8547}{545392}a^{17}-\frac{9949}{545392}a^{16}+\frac{42613}{1090784}a^{15}-\frac{41499}{1090784}a^{14}-\frac{82449}{1090784}a^{13}+\frac{114351}{1090784}a^{12}-\frac{321923}{1090784}a^{11}+\frac{327605}{1090784}a^{10}+\frac{480249}{1090784}a^{9}-\frac{684319}{1090784}a^{8}+\frac{1563637}{545392}a^{7}-\frac{1693191}{545392}a^{6}+\frac{3896755}{545392}a^{5}-\frac{3867361}{545392}a^{4}+\frac{57256}{34087}a^{3}+\frac{74979}{68174}a^{2}+\frac{135617}{136348}a-\frac{122703}{136348}$, $\frac{43}{766}a^{17}+\frac{867}{6128}a^{15}-\frac{1683}{6128}a^{13}-\frac{6559}{6128}a^{11}+\frac{9737}{6128}a^{9}+\frac{7951}{766}a^{7}+\frac{79353}{3064}a^{5}+\frac{9325}{1532}a^{3}+\frac{123}{383}a+1$, $\frac{168851}{2181568}a^{17}+\frac{23599}{2181568}a^{16}+\frac{353899}{2181568}a^{15}+\frac{56829}{2181568}a^{14}-\frac{986937}{2181568}a^{13}-\frac{121715}{2181568}a^{12}-\frac{2808603}{2181568}a^{11}-\frac{423337}{2181568}a^{10}+\frac{3007807}{1090784}a^{9}+\frac{94449}{272696}a^{8}+\frac{3580289}{272696}a^{7}+\frac{1067847}{545392}a^{6}+\frac{16328487}{545392}a^{5}+\frac{1269595}{272696}a^{4}-\frac{2386551}{545392}a^{3}+\frac{288753}{545392}a^{2}+\frac{315537}{272696}a+\frac{76555}{272696}$, $\frac{168851}{2181568}a^{17}-\frac{23599}{2181568}a^{16}+\frac{353899}{2181568}a^{15}-\frac{56829}{2181568}a^{14}-\frac{986937}{2181568}a^{13}+\frac{121715}{2181568}a^{12}-\frac{2808603}{2181568}a^{11}+\frac{423337}{2181568}a^{10}+\frac{3007807}{1090784}a^{9}-\frac{94449}{272696}a^{8}+\frac{3580289}{272696}a^{7}-\frac{1067847}{545392}a^{6}+\frac{16328487}{545392}a^{5}-\frac{1269595}{272696}a^{4}-\frac{2386551}{545392}a^{3}-\frac{288753}{545392}a^{2}+\frac{315537}{272696}a-\frac{76555}{272696}$, $\frac{23855}{545392}a^{17}-\frac{16883}{1090784}a^{16}+\frac{3743}{34087}a^{15}-\frac{42089}{1090784}a^{14}-\frac{58059}{272696}a^{13}+\frac{86929}{1090784}a^{12}-\frac{452837}{545392}a^{11}+\frac{311989}{1090784}a^{10}+\frac{673421}{545392}a^{9}-\frac{244761}{545392}a^{8}+\frac{4394193}{545392}a^{7}-\frac{385597}{136348}a^{6}+\frac{2739793}{136348}a^{5}-\frac{1862795}{272696}a^{4}+\frac{1424321}{272696}a^{3}-\frac{465583}{272696}a^{2}+\frac{89337}{136348}a-\frac{21223}{136348}$, $\frac{65953}{2181568}a^{17}-\frac{4489}{2181568}a^{16}+\frac{124855}{2181568}a^{15}-\frac{2909}{2181568}a^{14}-\frac{406605}{2181568}a^{13}+\frac{27951}{2181568}a^{12}-\frac{990903}{2181568}a^{11}+\frac{22493}{2181568}a^{10}+\frac{157927}{136348}a^{9}-\frac{96915}{1090784}a^{8}+\frac{1318831}{272696}a^{7}-\frac{63139}{272696}a^{6}+\frac{1470879}{136348}a^{5}-\frac{233935}{545392}a^{4}-\frac{1890947}{545392}a^{3}+\frac{318625}{545392}a^{2}+\frac{783751}{272696}a-\frac{351827}{272696}$, $\frac{65953}{2181568}a^{17}+\frac{4489}{2181568}a^{16}+\frac{124855}{2181568}a^{15}+\frac{2909}{2181568}a^{14}-\frac{406605}{2181568}a^{13}-\frac{27951}{2181568}a^{12}-\frac{990903}{2181568}a^{11}-\frac{22493}{2181568}a^{10}+\frac{157927}{136348}a^{9}+\frac{96915}{1090784}a^{8}+\frac{1318831}{272696}a^{7}+\frac{63139}{272696}a^{6}+\frac{1470879}{136348}a^{5}+\frac{233935}{545392}a^{4}-\frac{1890947}{545392}a^{3}-\frac{318625}{545392}a^{2}+\frac{783751}{272696}a+\frac{351827}{272696}$, $\frac{12323}{2181568}a^{17}+\frac{4691}{2181568}a^{16}-\frac{10269}{2181568}a^{15}+\frac{37013}{2181568}a^{14}-\frac{154669}{2181568}a^{13}+\frac{51297}{2181568}a^{12}-\frac{33651}{2181568}a^{11}-\frac{219401}{2181568}a^{10}+\frac{575009}{1090784}a^{9}-\frac{67995}{272696}a^{8}+\frac{19047}{34087}a^{7}+\frac{424331}{545392}a^{6}-\frac{634575}{545392}a^{5}+\frac{1065197}{272696}a^{4}-\frac{4412251}{545392}a^{3}+\frac{2808873}{545392}a^{2}+\frac{777745}{272696}a-\frac{332837}{272696}$ 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:  \( 522831.688605 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 522831.688605 \cdot 3}{2\cdot\sqrt{1947987996273487986556928}}\cr\approx \mathstrut & 5.45957656829 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - 6*x^14 - 16*x^12 + 37*x^10 + 166*x^8 + 372*x^6 - 88*x^4 + 36*x^2 - 8)
 
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 + 2*x^16 - 6*x^14 - 16*x^12 + 37*x^10 + 166*x^8 + 372*x^6 - 88*x^4 + 36*x^2 - 8, 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 + 2*x^16 - 6*x^14 - 16*x^12 + 37*x^10 + 166*x^8 + 372*x^6 - 88*x^4 + 36*x^2 - 8);
 
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 + 2*x^16 - 6*x^14 - 16*x^12 + 37*x^10 + 166*x^8 + 372*x^6 - 88*x^4 + 36*x^2 - 8);
 
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

$C_3^2:C_4$ (as 18T10):

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 36
The 6 conjugacy class representatives for $C_3^2 : C_4$
Character table for $C_3^2 : C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 6.2.39337984.1 x2, 6.2.802816.1 x2, 9.1.493455671296.1

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

Galois closure: data not computed
Degree 6 siblings: 6.2.802816.1, 6.2.39337984.1
Degree 9 sibling: 9.1.493455671296.1
Degree 12 siblings: 12.0.20624432955392.2, 12.0.49519263525896192.12
Minimal sibling: 6.2.802816.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }$ R ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.3.0.1}{3} }^{6}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }^{6}$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.2.3.1$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$[3]$
2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 4 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
\(7\) Copy content Toggle raw display 7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$