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Properties

Label 18.2.79461810481...6409.1
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 41^{3}\cdot 97^{6}$
Root discriminant $31.22$
Ramified primes $7, 41, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-81229, 281383, -513078, 602875, -489075, 278632, -94314, 2327, 19699, -13773, 6137, -1813, 617, -272, 155, -70, 28, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 28*x^16 - 70*x^15 + 155*x^14 - 272*x^13 + 617*x^12 - 1813*x^11 + 6137*x^10 - 13773*x^9 + 19699*x^8 + 2327*x^7 - 94314*x^6 + 278632*x^5 - 489075*x^4 + 602875*x^3 - 513078*x^2 + 281383*x - 81229)
 
gp: K = bnfinit(x^18 - 7*x^17 + 28*x^16 - 70*x^15 + 155*x^14 - 272*x^13 + 617*x^12 - 1813*x^11 + 6137*x^10 - 13773*x^9 + 19699*x^8 + 2327*x^7 - 94314*x^6 + 278632*x^5 - 489075*x^4 + 602875*x^3 - 513078*x^2 + 281383*x - 81229, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 28 x^{16} - 70 x^{15} + 155 x^{14} - 272 x^{13} + 617 x^{12} - 1813 x^{11} + 6137 x^{10} - 13773 x^{9} + 19699 x^{8} + 2327 x^{7} - 94314 x^{6} + 278632 x^{5} - 489075 x^{4} + 602875 x^{3} - 513078 x^{2} + 281383 x - 81229 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(794618104817141695294816409=7^{12}\cdot 41^{3}\cdot 97^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.22$
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:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{71} a^{15} + \frac{11}{71} a^{14} - \frac{21}{71} a^{13} + \frac{33}{71} a^{12} + \frac{5}{71} a^{11} - \frac{18}{71} a^{10} + \frac{9}{71} a^{9} - \frac{30}{71} a^{8} - \frac{17}{71} a^{7} + \frac{32}{71} a^{6} + \frac{31}{71} a^{5} - \frac{29}{71} a^{4} - \frac{15}{71} a^{3} - \frac{15}{71} a^{2} - \frac{16}{71} a - \frac{22}{71}$, $\frac{1}{71} a^{16} - \frac{20}{71} a^{13} - \frac{3}{71} a^{12} - \frac{2}{71} a^{11} - \frac{6}{71} a^{10} + \frac{13}{71} a^{9} + \frac{29}{71} a^{8} + \frac{6}{71} a^{7} + \frac{34}{71} a^{6} - \frac{15}{71} a^{5} + \frac{20}{71} a^{4} + \frac{8}{71} a^{3} + \frac{7}{71} a^{2} + \frac{12}{71} a + \frac{29}{71}$, $\frac{1}{95361707464662462280630876473899186744909} a^{17} + \frac{8227046649127952069960895366249993281}{2217714127085173541410020383113934575463} a^{16} - \frac{54563259068230220291563047369046737375}{95361707464662462280630876473899186744909} a^{15} - \frac{43119191456665910490671877862884299237931}{95361707464662462280630876473899186744909} a^{14} - \frac{25773333075698686414595796999615800449718}{95361707464662462280630876473899186744909} a^{13} + \frac{8106160938204472395702734797214051028750}{95361707464662462280630876473899186744909} a^{12} - \frac{38300516695930377786961054531760041878226}{95361707464662462280630876473899186744909} a^{11} + \frac{5818264857903702459450543304763636875630}{95361707464662462280630876473899186744909} a^{10} + \frac{9128808695503427947288167258307559903822}{95361707464662462280630876473899186744909} a^{9} - \frac{490431905256825351081793775222272318165}{2217714127085173541410020383113934575463} a^{8} - \frac{31570988037429676377322348002281432449834}{95361707464662462280630876473899186744909} a^{7} - \frac{8586184767953349702324264837052360016084}{95361707464662462280630876473899186744909} a^{6} - \frac{25641167424301695941632654705720739074778}{95361707464662462280630876473899186744909} a^{5} + \frac{8810824686815834563271336752016954991246}{95361707464662462280630876473899186744909} a^{4} + \frac{11006369097396472983503619253457800059442}{95361707464662462280630876473899186744909} a^{3} + \frac{43068815169098239655666184646723546459537}{95361707464662462280630876473899186744909} a^{2} - \frac{2503744550140587328670740182974177514254}{95361707464662462280630876473899186744909} a + \frac{15634804845672516049248415776457109222139}{95361707464662462280630876473899186744909}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 506566.920161 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n767 are not computed
Character table for t18n767 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.5.467890073.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
7Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.3.0.1$x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
41.3.0.1$x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.3.4$x^{4} + 12125$$4$$1$$3$$C_4$$[\ ]_{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$