Properties

Label 18.0.10839257382...2103.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 1511^{2}$
Root discriminant $11.42$
Ramified primes $7, 1511$
Class number $1$
Class group Trivial
Galois Group 18T286

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 16, -27, 43, -79, 130, -155, 119, -37, -35, 61, -47, 21, -2, -6, 6, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 6*x^15 - 2*x^14 + 21*x^13 - 47*x^12 + 61*x^11 - 35*x^10 - 37*x^9 + 119*x^8 - 155*x^7 + 130*x^6 - 79*x^5 + 43*x^4 - 27*x^3 + 16*x^2 - 6*x + 1)
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 - 6*x^15 - 2*x^14 + 21*x^13 - 47*x^12 + 61*x^11 - 35*x^10 - 37*x^9 + 119*x^8 - 155*x^7 + 130*x^6 - 79*x^5 + 43*x^4 - 27*x^3 + 16*x^2 - 6*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 3 x^{17} \) \(\mathstrut +\mathstrut 6 x^{16} \) \(\mathstrut -\mathstrut 6 x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut 21 x^{13} \) \(\mathstrut -\mathstrut 47 x^{12} \) \(\mathstrut +\mathstrut 61 x^{11} \) \(\mathstrut -\mathstrut 35 x^{10} \) \(\mathstrut -\mathstrut 37 x^{9} \) \(\mathstrut +\mathstrut 119 x^{8} \) \(\mathstrut -\mathstrut 155 x^{7} \) \(\mathstrut +\mathstrut 130 x^{6} \) \(\mathstrut -\mathstrut 79 x^{5} \) \(\mathstrut +\mathstrut 43 x^{4} \) \(\mathstrut -\mathstrut 27 x^{3} \) \(\mathstrut +\mathstrut 16 x^{2} \) \(\mathstrut -\mathstrut 6 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 9]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-10839257382142572103=-\,7^{15}\cdot 1511^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.42$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 1511$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{491107} a^{17} - \frac{145852}{491107} a^{16} + \frac{68649}{491107} a^{15} - \frac{189598}{491107} a^{14} - \frac{83149}{491107} a^{13} - \frac{197736}{491107} a^{12} - \frac{169651}{491107} a^{11} - \frac{15221}{491107} a^{10} + \frac{163954}{491107} a^{9} - \frac{36046}{491107} a^{8} - \frac{27262}{491107} a^{7} + \frac{133011}{491107} a^{6} + \frac{187505}{491107} a^{5} - \frac{123529}{491107} a^{4} - \frac{170238}{491107} a^{3} + \frac{145436}{491107} a^{2} + \frac{198396}{491107} a + \frac{166230}{491107}$

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

Class group and class number

Trivial Abelian group, order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $8$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( \frac{1827668}{491107} a^{17} - \frac{7940104}{491107} a^{16} + \frac{14788489}{491107} a^{15} - \frac{18215865}{491107} a^{14} - \frac{3071987}{491107} a^{13} + \frac{51134640}{491107} a^{12} - \frac{121691777}{491107} a^{11} + \frac{165815553}{491107} a^{10} - \frac{103500318}{491107} a^{9} - \frac{92409222}{491107} a^{8} + \frac{321050754}{491107} a^{7} - \frac{420859779}{491107} a^{6} + \frac{333920965}{491107} a^{5} - \frac{186384313}{491107} a^{4} + \frac{91676340}{491107} a^{3} - \frac{66293305}{491107} a^{2} + \frac{40804457}{491107} a - \frac{11567231}{491107} \) (order $14$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{1909610}{491107} a^{17} - \frac{3835880}{491107} a^{16} + \frac{7027557}{491107} a^{15} - \frac{3352026}{491107} a^{14} - \frac{9233218}{491107} a^{13} + \frac{31700299}{491107} a^{12} - \frac{55168725}{491107} a^{11} + \frac{52051327}{491107} a^{10} + \frac{1101049}{491107} a^{9} - \frac{83733130}{491107} a^{8} + \frac{141448531}{491107} a^{7} - \frac{129092510}{491107} a^{6} + \frac{78797540}{491107} a^{5} - \frac{37585833}{491107} a^{4} + \frac{25137927}{491107} a^{3} - \frac{18050569}{491107} a^{2} + \frac{6276978}{491107} a - \frac{396862}{491107} \),  \( \frac{1242572}{491107} a^{17} - \frac{3797311}{491107} a^{16} + \frac{7334789}{491107} a^{15} - \frac{7102584}{491107} a^{14} - \frac{3457424}{491107} a^{13} + \frac{27117100}{491107} a^{12} - \frac{58273211}{491107} a^{11} + \frac{73008208}{491107} a^{10} - \frac{36626826}{491107} a^{9} - \frac{54813682}{491107} a^{8} + \frac{151841738}{491107} a^{7} - \frac{185079506}{491107} a^{6} + \frac{143338699}{491107} a^{5} - \frac{79216393}{491107} a^{4} + \frac{41816748}{491107} a^{3} - \frac{28390032}{491107} a^{2} + \frac{16540253}{491107} a - \frac{4405105}{491107} \),  \( \frac{766488}{491107} a^{17} + \frac{807490}{491107} a^{16} - \frac{1515910}{491107} a^{15} + \frac{6738258}{491107} a^{14} - \frac{6666392}{491107} a^{13} + \frac{1206744}{491107} a^{12} + \frac{15080089}{491107} a^{11} - \frac{41720051}{491107} a^{10} + \frac{55880627}{491107} a^{9} - \frac{25666406}{491107} a^{8} - \frac{44083743}{491107} a^{7} + \frac{108021243}{491107} a^{6} - \frac{106245794}{491107} a^{5} + \frac{64795144}{491107} a^{4} - \frac{25756236}{491107} a^{3} + \frac{18215118}{491107} a^{2} - \frac{15406977}{491107} a + \frac{5411230}{491107} \),  \( \frac{937932}{491107} a^{17} + \frac{3016749}{491107} a^{16} - \frac{5364865}{491107} a^{15} + \frac{15037681}{491107} a^{14} - \frac{10138408}{491107} a^{13} - \frac{8149970}{491107} a^{12} + \frac{49623610}{491107} a^{11} - \frac{102423845}{491107} a^{10} + \frac{112778363}{491107} a^{9} - \frac{22990807}{491107} a^{8} - \frac{137926189}{491107} a^{7} + \frac{262595394}{491107} a^{6} - \frac{240592791}{491107} a^{5} + \frac{144055763}{491107} a^{4} - \frac{60419495}{491107} a^{3} + \frac{44379876}{491107} a^{2} - \frac{33353825}{491107} a + \frac{11010317}{491107} \),  \( \frac{839680}{491107} a^{17} - \frac{7548054}{491107} a^{16} + \frac{14241405}{491107} a^{15} - \frac{22083372}{491107} a^{14} + \frac{4094298}{491107} a^{13} + \frac{42689710}{491107} a^{12} - \frac{118938726}{491107} a^{11} + \frac{182982199}{491107} a^{10} - \frac{144078299}{491107} a^{9} - \frac{53220426}{491107} a^{8} + \frac{317378446}{491107} a^{7} - \frac{466938003}{491107} a^{6} + \frac{392108656}{491107} a^{5} - \frac{225503791}{491107} a^{4} + \frac{105676441}{491107} a^{3} - \frac{75087725}{491107} a^{2} + \frac{50349617}{491107} a - \frac{15193922}{491107} \),  \( \frac{3959084}{491107} a^{17} - \frac{12915499}{491107} a^{16} + \frac{24259140}{491107} a^{15} - \frac{24505004}{491107} a^{14} - \frac{11728914}{491107} a^{13} + \frac{90463177}{491107} a^{12} - \frac{195531720}{491107} a^{11} + \frac{245620371}{491107} a^{10} - \frac{123027082}{491107} a^{9} - \frac{186943822}{491107} a^{8} + \frac{512717518}{491107} a^{7} - \frac{620795749}{491107} a^{6} + \frac{472480187}{491107} a^{5} - \frac{257488159}{491107} a^{4} + \frac{132954879}{491107} a^{3} - \frac{93951893}{491107} a^{2} + \frac{54228481}{491107} a - \frac{13957380}{491107} \),  \( \frac{2097275}{491107} a^{17} - \frac{7722778}{491107} a^{16} + \frac{14689923}{491107} a^{15} - \frac{16327328}{491107} a^{14} - \frac{4536522}{491107} a^{13} + \frac{52245273}{491107} a^{12} - \frac{118591847}{491107} a^{11} + \frac{156486465}{491107} a^{10} - \frac{90162100}{491107} a^{9} - \frac{99022219}{491107} a^{8} + \frac{311601049}{491107} a^{7} - \frac{396731999}{491107} a^{6} + \frac{311891533}{491107} a^{5} - \frac{173477429}{491107} a^{4} + \frac{88269024}{491107} a^{3} - \frac{62274784}{491107} a^{2} + \frac{37398175}{491107} a - \frac{10273799}{491107} \),  \( \frac{1193413}{491107} a^{17} - \frac{4021043}{491107} a^{16} + \frac{7505902}{491107} a^{15} - \frac{7865362}{491107} a^{14} - \frac{3419294}{491107} a^{13} + \frac{27631380}{491107} a^{12} - \frac{60834311}{491107} a^{11} + \frac{77230242}{491107} a^{10} - \frac{40322284}{491107} a^{9} - \frac{55724638}{491107} a^{8} + \frac{158658891}{491107} a^{7} - \frac{194990797}{491107} a^{6} + \frac{150734292}{491107} a^{5} - \frac{84594514}{491107} a^{4} + \frac{44538845}{491107} a^{3} - \frac{30798185}{491107} a^{2} + \frac{16976309}{491107} a - \frac{4085195}{491107} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 524.185517787 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T286:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 1296
The 34 conjugacy class representatives for t18n286
Character table for t18n286 is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.3.177767639.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
1511Data not computed