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Properties

Label 18.14.6970034779...0688.1
Degree $18$
Signature $[14, 2]$
Discriminant $2^{18}\cdot 23^{9}\cdot 5569^{2}\cdot 21817^{2}$
Root discriminant $75.89$
Ramified primes $2, 23, 5569, 21817$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T913

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3404825447, 0, 5773399671, 0, -2053193417, 0, 263890063, 0, -12994356, 0, 509956, 0, -67781, 0, 4445, 0, -114, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 114*x^16 + 4445*x^14 - 67781*x^12 + 509956*x^10 - 12994356*x^8 + 263890063*x^6 - 2053193417*x^4 + 5773399671*x^2 - 3404825447)
 
gp: K = bnfinit(x^18 - 114*x^16 + 4445*x^14 - 67781*x^12 + 509956*x^10 - 12994356*x^8 + 263890063*x^6 - 2053193417*x^4 + 5773399671*x^2 - 3404825447, 1)
 

Normalized defining polynomial

\( x^{18} - 114 x^{16} + 4445 x^{14} - 67781 x^{12} + 509956 x^{10} - 12994356 x^{8} + 263890063 x^{6} - 2053193417 x^{4} + 5773399671 x^{2} - 3404825447 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6970034779193566051725040347250688=2^{18}\cdot 23^{9}\cdot 5569^{2}\cdot 21817^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.89$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 23, 5569, 21817$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{23} a^{6} + \frac{1}{23} a^{4} + \frac{6}{23} a^{2}$, $\frac{1}{23} a^{7} + \frac{1}{23} a^{5} + \frac{6}{23} a^{3}$, $\frac{1}{529} a^{8} + \frac{1}{529} a^{6} - \frac{201}{529} a^{4} + \frac{4}{23} a^{2}$, $\frac{1}{529} a^{9} + \frac{1}{529} a^{7} - \frac{201}{529} a^{5} + \frac{4}{23} a^{3}$, $\frac{1}{12167} a^{10} + \frac{1}{12167} a^{8} - \frac{201}{12167} a^{6} + \frac{73}{529} a^{4} + \frac{10}{23} a^{2}$, $\frac{1}{12167} a^{11} + \frac{1}{12167} a^{9} - \frac{201}{12167} a^{7} + \frac{73}{529} a^{5} + \frac{10}{23} a^{3}$, $\frac{1}{279841} a^{12} + \frac{1}{279841} a^{10} - \frac{201}{279841} a^{8} + \frac{73}{12167} a^{6} + \frac{148}{529} a^{4} + \frac{6}{23} a^{2}$, $\frac{1}{279841} a^{13} + \frac{1}{279841} a^{11} - \frac{201}{279841} a^{9} + \frac{73}{12167} a^{7} + \frac{148}{529} a^{5} + \frac{6}{23} a^{3}$, $\frac{1}{186653947} a^{14} - \frac{45}{186653947} a^{12} + \frac{3985}{186653947} a^{10} + \frac{3833}{8115389} a^{8} + \frac{1706}{352843} a^{6} + \frac{6540}{15341} a^{4} + \frac{64}{667} a^{2} + \frac{6}{29}$, $\frac{1}{186653947} a^{15} - \frac{45}{186653947} a^{13} + \frac{3985}{186653947} a^{11} + \frac{3833}{8115389} a^{9} + \frac{1706}{352843} a^{7} + \frac{6540}{15341} a^{5} + \frac{64}{667} a^{3} + \frac{6}{29} a$, $\frac{1}{21177534656346618787} a^{16} + \frac{39282958584}{21177534656346618787} a^{14} + \frac{16562474108373}{21177534656346618787} a^{12} - \frac{34540323515092}{920762376362896469} a^{10} + \frac{25664482124208}{40033146798386803} a^{8} - \frac{34337054659841}{1740571599929861} a^{6} + \frac{34901107418211}{75677026083907} a^{4} - \frac{881322211497}{3290305481909} a^{2} - \frac{14263382049}{143056760083}$, $\frac{1}{21177534656346618787} a^{17} + \frac{39282958584}{21177534656346618787} a^{15} + \frac{16562474108373}{21177534656346618787} a^{13} - \frac{34540323515092}{920762376362896469} a^{11} + \frac{25664482124208}{40033146798386803} a^{9} - \frac{34337054659841}{1740571599929861} a^{7} + \frac{34901107418211}{75677026083907} a^{5} - \frac{881322211497}{3290305481909} a^{3} - \frac{14263382049}{143056760083} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 47961359046.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T913:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 725760
The 60 conjugacy class representatives for t18n913 are not computed
Character table for t18n913 is not computed

Intermediate fields

\(\Q(\sqrt{23}) \), 9.7.2794474079.1

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

Sibling fields

Degree 18 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.14.7.1$x^{14} - 24334 x^{8} + 148035889 x^{2} - 217908828608$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5569Data not computed
21817Data not computed