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Properties

Label 18.10.4122446872...2921.1
Degree $18$
Signature $[10, 4]$
Discriminant $41^{9}\cdot 11221481^{2}$
Root discriminant $38.88$
Ramified primes $41, 11221481$
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![-325399, -827552, 74572, 1207011, 191202, -657180, -37363, 158094, -43958, -10896, 15739, -2375, -826, 432, -164, 1, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 + x^15 - 164*x^14 + 432*x^13 - 826*x^12 - 2375*x^11 + 15739*x^10 - 10896*x^9 - 43958*x^8 + 158094*x^7 - 37363*x^6 - 657180*x^5 + 191202*x^4 + 1207011*x^3 + 74572*x^2 - 827552*x - 325399)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 + x^15 - 164*x^14 + 432*x^13 - 826*x^12 - 2375*x^11 + 15739*x^10 - 10896*x^9 - 43958*x^8 + 158094*x^7 - 37363*x^6 - 657180*x^5 + 191202*x^4 + 1207011*x^3 + 74572*x^2 - 827552*x - 325399, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} + x^{15} - 164 x^{14} + 432 x^{13} - 826 x^{12} - 2375 x^{11} + 15739 x^{10} - 10896 x^{9} - 43958 x^{8} + 158094 x^{7} - 37363 x^{6} - 657180 x^{5} + 191202 x^{4} + 1207011 x^{3} + 74572 x^{2} - 827552 x - 325399 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(41224468721177639474720732921=41^{9}\cdot 11221481^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.88$
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:  $41, 11221481$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{16270477756281147435522848410639120053774472248473} a^{17} + \frac{544497251499578758728034360722327595762064779310}{16270477756281147435522848410639120053774472248473} a^{16} - \frac{5485071218298718786702560686907359921548744690764}{16270477756281147435522848410639120053774472248473} a^{15} - \frac{4729385699454822841187602517579921297871005250351}{16270477756281147435522848410639120053774472248473} a^{14} + \frac{5023534166267757454932146000154086566794029116157}{16270477756281147435522848410639120053774472248473} a^{13} + \frac{3982600564900880511849140049204232490651351584683}{16270477756281147435522848410639120053774472248473} a^{12} - \frac{1517277234722778809553888077939044286320464665173}{16270477756281147435522848410639120053774472248473} a^{11} - \frac{2337458195224467645198901410375459078553335367813}{16270477756281147435522848410639120053774472248473} a^{10} - \frac{662344016111964170372383184804500042932725367784}{16270477756281147435522848410639120053774472248473} a^{9} + \frac{4361467239695374521293412714038924942297618451112}{16270477756281147435522848410639120053774472248473} a^{8} + \frac{4129292404743805881062231224297962632922385553052}{16270477756281147435522848410639120053774472248473} a^{7} - \frac{6485111013159948753687397390310020712248660940777}{16270477756281147435522848410639120053774472248473} a^{6} + \frac{5969215978560813345476154408328921718313551423746}{16270477756281147435522848410639120053774472248473} a^{5} + \frac{5540131944111547359503110389230324618392065495562}{16270477756281147435522848410639120053774472248473} a^{4} + \frac{7708591050777766807589107794013940893799015998096}{16270477756281147435522848410639120053774472248473} a^{3} + \frac{5913248427533566734548670388297650353237551607792}{16270477756281147435522848410639120053774472248473} a^{2} + \frac{3561188649976857021429427276050394037077842362317}{16270477756281147435522848410639120053774472248473} a + \frac{7596463424535388175325562639136171731337029612226}{16270477756281147435522848410639120053774472248473}$

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:  $13$
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:  \( 37185333.3537 \) (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

\(\Q(\sqrt{41}) \), 9.5.460080721.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
41Data not computed
11221481Data not computed