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Properties

Label 18.10.6630018602...5776.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{14}\cdot 41^{9}$
Root discriminant $66.59$
Ramified primes $2, 3, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1247, 10176, 16016, -90727, 72610, -23603, 76447, -61792, 6284, -13464, 8425, 1680, -1035, 665, -198, 0, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 3*x^16 - 198*x^14 + 665*x^13 - 1035*x^12 + 1680*x^11 + 8425*x^10 - 13464*x^9 + 6284*x^8 - 61792*x^7 + 76447*x^6 - 23603*x^5 + 72610*x^4 - 90727*x^3 + 16016*x^2 + 10176*x - 1247)
 
gp: K = bnfinit(x^18 - 3*x^17 - 3*x^16 - 198*x^14 + 665*x^13 - 1035*x^12 + 1680*x^11 + 8425*x^10 - 13464*x^9 + 6284*x^8 - 61792*x^7 + 76447*x^6 - 23603*x^5 + 72610*x^4 - 90727*x^3 + 16016*x^2 + 10176*x - 1247, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 3 x^{16} - 198 x^{14} + 665 x^{13} - 1035 x^{12} + 1680 x^{11} + 8425 x^{10} - 13464 x^{9} + 6284 x^{8} - 61792 x^{7} + 76447 x^{6} - 23603 x^{5} + 72610 x^{4} - 90727 x^{3} + 16016 x^{2} + 10176 x - 1247 \)

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:  \(663001860270502303855233859915776=2^{12}\cdot 3^{6}\cdot 7^{14}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.59$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{11} + \frac{1}{7} a^{10} - \frac{3}{7} a^{9} - \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{4} - \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{10} - \frac{1}{7} a^{9} - \frac{2}{7} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{15} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{196} a^{16} - \frac{5}{196} a^{15} + \frac{3}{98} a^{14} - \frac{1}{28} a^{13} + \frac{3}{98} a^{12} - \frac{10}{49} a^{11} - \frac{9}{196} a^{10} - \frac{13}{98} a^{9} - \frac{27}{98} a^{8} - \frac{1}{98} a^{7} + \frac{1}{14} a^{6} - \frac{39}{98} a^{5} + \frac{93}{196} a^{4} - \frac{79}{196} a^{3} + \frac{15}{196} a^{2} - \frac{1}{2} a - \frac{15}{196}$, $\frac{1}{11524998163090493788859411642915021474768} a^{17} + \frac{679950542837168565697592056668292676}{720312385193155861803713227682188842173} a^{16} + \frac{611619119900345864253226760424493371149}{11524998163090493788859411642915021474768} a^{15} + \frac{352247752191014721868818118878910627367}{11524998163090493788859411642915021474768} a^{14} + \frac{188105515542412676425934554738906461231}{11524998163090493788859411642915021474768} a^{13} + \frac{384852680786033096887971974618902098299}{5762499081545246894429705821457510737384} a^{12} + \frac{5733829526674345676545846121321604671767}{11524998163090493788859411642915021474768} a^{11} - \frac{4621288498533004655990351366727766336171}{11524998163090493788859411642915021474768} a^{10} + \frac{85294102857412594477664210438979965169}{1440624770386311723607426455364377684346} a^{9} - \frac{43930696601587417491987524964578021558}{720312385193155861803713227682188842173} a^{8} - \frac{667244498300920147620240004199967107793}{2881249540772623447214852910728755368692} a^{7} - \frac{440889843716395877131341585656946477863}{2881249540772623447214852910728755368692} a^{6} - \frac{4551827829739639571158650115450235574645}{11524998163090493788859411642915021474768} a^{5} + \frac{1294576546822438465835390174771277046831}{5762499081545246894429705821457510737384} a^{4} + \frac{1109493237440443313010842035654653743687}{2881249540772623447214852910728755368692} a^{3} + \frac{842876257927002592948086684382438398989}{11524998163090493788859411642915021474768} a^{2} - \frac{2705513170271367076808670801516607450105}{11524998163090493788859411642915021474768} a - \frac{2247656604520639208620900021173610599243}{11524998163090493788859411642915021474768}$

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:  \( 9097730080.24 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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

Sibling fields

Degree 18 siblings: 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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
41Data not computed