Properties

Label 18.2.226...125.1
Degree $18$
Signature $[2, 8]$
Discriminant $2.262\times 10^{24}$
Root discriminant $22.54$
Ramified primes $3, 5$
Class number $1$
Class group trivial
Galois group $C_3^2 : C_4$ (as 18T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 9*x^15 + 30*x^14 + 12*x^13 - 87*x^11 - 39*x^10 + 133*x^9 + 177*x^8 - 48*x^7 - 261*x^6 - 156*x^5 + 129*x^4 + 240*x^3 + 135*x^2 + 30*x + 20)
 
gp: K = bnfinit(x^18 - 3*x^17 - 9*x^15 + 30*x^14 + 12*x^13 - 87*x^11 - 39*x^10 + 133*x^9 + 177*x^8 - 48*x^7 - 261*x^6 - 156*x^5 + 129*x^4 + 240*x^3 + 135*x^2 + 30*x + 20, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20, 30, 135, 240, 129, -156, -261, -48, 177, 133, -39, -87, 0, 12, 30, -9, 0, -3, 1]);
 

\( x^{18} - 3 x^{17} - 9 x^{15} + 30 x^{14} + 12 x^{13} - 87 x^{11} - 39 x^{10} + 133 x^{9} + 177 x^{8} - 48 x^{7} - 261 x^{6} - 156 x^{5} + 129 x^{4} + 240 x^{3} + 135 x^{2} + 30 x + 20 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2261987535219532470703125\)\(\medspace = 3^{32}\cdot 5^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $22.54$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{12} + \frac{1}{10} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{10} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{15} + \frac{1}{10} a^{13} + \frac{1}{10} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{10} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{100} a^{16} + \frac{1}{100} a^{15} - \frac{1}{20} a^{14} - \frac{9}{50} a^{13} - \frac{17}{100} a^{12} - \frac{6}{25} a^{11} - \frac{43}{100} a^{10} - \frac{3}{100} a^{9} - \frac{11}{25} a^{8} - \frac{13}{50} a^{7} - \frac{1}{100} a^{6} - \frac{7}{25} a^{5} - \frac{7}{50} a^{4} - \frac{1}{5} a^{3} + \frac{9}{20} a^{2} - \frac{1}{5}$, $\frac{1}{8165181308402200} a^{17} - \frac{1107977017497}{2041295327100550} a^{16} - \frac{28983675025203}{1020647663550275} a^{15} - \frac{141766698683313}{8165181308402200} a^{14} - \frac{53306222641}{56311595230360} a^{13} + \frac{1626316832449869}{8165181308402200} a^{12} + \frac{3371483577131663}{8165181308402200} a^{11} - \frac{758966312313633}{4082590654201100} a^{10} + \frac{1380348127231183}{8165181308402200} a^{9} - \frac{22305245965515}{163303626168044} a^{8} + \frac{3815880098446703}{8165181308402200} a^{7} - \frac{3266433040755179}{8165181308402200} a^{6} + \frac{361939030345039}{4082590654201100} a^{5} - \frac{255806080064867}{4082590654201100} a^{4} - \frac{143337695314541}{326607252336088} a^{3} + \frac{191678506413929}{1633036261680440} a^{2} + \frac{132720717096673}{816518130840220} a + \frac{119305148371769}{408259065420110}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 265400.055751 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{8}\cdot 265400.055751 \cdot 1}{2\sqrt{2261987535219532470703125}}\approx 0.857284257172$

Galois group

$C_3:S_3.C_2$ (as 18T10):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{5}) \), 6.2.4100625.2 x2, 6.2.4100625.1 x2, 9.1.672605015625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 6 siblings: 6.2.4100625.1, 6.2.4100625.2
Degree 9 sibling: 9.1.672605015625.1
Degree 12 siblings: Deg 12, Deg 12

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$