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Properties

Label 18.14.1004220442...0000.1
Degree $18$
Signature $[14, 2]$
Discriminant $2^{12}\cdot 5^{11}\cdot 19^{2}\cdot 37^{6}\cdot 61\cdot 151^{4}\cdot 643^{4}$
Root discriminant $316.30$
Ramified primes $2, 5, 19, 37, 61, 151, 643$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T903

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2211171219071, -17608077562964, -22916864268836, 29751632848805, -7110505980691, -1394245968721, 987979666336, -180605815836, -84891381, 4245099965, -347682222, -14888454, 1293253, -224074, 43204, 1712, -404, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 404*x^16 + 1712*x^15 + 43204*x^14 - 224074*x^13 + 1293253*x^12 - 14888454*x^11 - 347682222*x^10 + 4245099965*x^9 - 84891381*x^8 - 180605815836*x^7 + 987979666336*x^6 - 1394245968721*x^5 - 7110505980691*x^4 + 29751632848805*x^3 - 22916864268836*x^2 - 17608077562964*x - 2211171219071)
 
gp: K = bnfinit(x^18 - 3*x^17 - 404*x^16 + 1712*x^15 + 43204*x^14 - 224074*x^13 + 1293253*x^12 - 14888454*x^11 - 347682222*x^10 + 4245099965*x^9 - 84891381*x^8 - 180605815836*x^7 + 987979666336*x^6 - 1394245968721*x^5 - 7110505980691*x^4 + 29751632848805*x^3 - 22916864268836*x^2 - 17608077562964*x - 2211171219071, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 404 x^{16} + 1712 x^{15} + 43204 x^{14} - 224074 x^{13} + 1293253 x^{12} - 14888454 x^{11} - 347682222 x^{10} + 4245099965 x^{9} - 84891381 x^{8} - 180605815836 x^{7} + 987979666336 x^{6} - 1394245968721 x^{5} - 7110505980691 x^{4} + 29751632848805 x^{3} - 22916864268836 x^{2} - 17608077562964 x - 2211171219071 \)

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:  \(1004220442431955142780653174367877800000000000=2^{12}\cdot 5^{11}\cdot 19^{2}\cdot 37^{6}\cdot 61\cdot 151^{4}\cdot 643^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $316.30$
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:  $2, 5, 19, 37, 61, 151, 643$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} + \frac{1}{4} a^{13} + \frac{1}{4} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{777511084084192710568975837049860607009409881675115411792628828095271213178296111999152439553889692188252877900228972} a^{17} - \frac{33193522478555961110154224210863185894914144144554788681467480279777545412213189308097921057968974496886858861899319}{777511084084192710568975837049860607009409881675115411792628828095271213178296111999152439553889692188252877900228972} a^{16} - \frac{121751060698883548320021986774975965325682105892079499273264925436312639946484209186705664995972903514955220937580831}{777511084084192710568975837049860607009409881675115411792628828095271213178296111999152439553889692188252877900228972} a^{15} + \frac{42041232969147800205371610526965621095180714637954957516567267114415345455199439986020277412949864374301055151290552}{194377771021048177642243959262465151752352470418778852948157207023817803294574027999788109888472423047063219475057243} a^{14} + \frac{81459094088237208685628980188082415998000844212658315765846840217121369276927532459335975229419050264422200530673193}{777511084084192710568975837049860607009409881675115411792628828095271213178296111999152439553889692188252877900228972} a^{13} + \frac{132406576258877351471617575262621662567792507414685005862415670974413285102553426209258472936330940219308457369194341}{777511084084192710568975837049860607009409881675115411792628828095271213178296111999152439553889692188252877900228972} a^{12} - \frac{117271718378742220259807689766696940911493314255490379347782641611472969380827722691038194521182770993718695035620773}{777511084084192710568975837049860607009409881675115411792628828095271213178296111999152439553889692188252877900228972} a^{11} - \frac{74720068617552599287920729208203125748410630384285759094684353843146018304695635010270598350226987469857516981146099}{777511084084192710568975837049860607009409881675115411792628828095271213178296111999152439553889692188252877900228972} a^{10} - \frac{11367753138579248036444053599631124980993889478060904857211289662943889245452632509392649746016407302417041890457341}{388755542042096355284487918524930303504704940837557705896314414047635606589148055999576219776944846094126438950114486} a^{9} + \frac{154194828411844030877728111427745150215550087306843501052211760607668598961844748842379133549226438904895185984309291}{388755542042096355284487918524930303504704940837557705896314414047635606589148055999576219776944846094126438950114486} a^{8} - \frac{27270036644686717698562190150946885164937805475640594942140682226113284057639484249197447635023116969464044088780999}{194377771021048177642243959262465151752352470418778852948157207023817803294574027999788109888472423047063219475057243} a^{7} - \frac{13364968145628898768094484090728321965780477833841836553547329586739937484962028531503902320114890586080851627513547}{59808544929553285428382756696143123616108452436547339368663756007328554859868931692242495350299207091404067530786844} a^{6} - \frac{9469953350404345773798468470969196849037062219050770643038618733423937495520606345420537993660056050583454649945417}{59808544929553285428382756696143123616108452436547339368663756007328554859868931692242495350299207091404067530786844} a^{5} + \frac{258223487886518132693584385242404890799019949227469686339872758334900856688763605048944180742634128373143214184118805}{777511084084192710568975837049860607009409881675115411792628828095271213178296111999152439553889692188252877900228972} a^{4} + \frac{41253288914706360982586835261001705493615189679082467277527794713890619205366756840392031991469230572326934651948025}{388755542042096355284487918524930303504704940837557705896314414047635606589148055999576219776944846094126438950114486} a^{3} + \frac{358852954288218596464798418768590320769159790757606777660081306274954687150155035003502954721349931404012121628008501}{777511084084192710568975837049860607009409881675115411792628828095271213178296111999152439553889692188252877900228972} a^{2} + \frac{123505739904038164746901145909405368059549236081307511487488420334478291952962070585111210316525301205999981490106305}{388755542042096355284487918524930303504704940837557705896314414047635606589148055999576219776944846094126438950114486} a - \frac{262594155727625412159587215280654878471521276334082861063952810291311429006661788825618106656557003026078316113273793}{777511084084192710568975837049860607009409881675115411792628828095271213178296111999152439553889692188252877900228972}$

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:  \( 35145253923200000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T903:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 559872
The 174 conjugacy class representatives for t18n903 are not computed
Character table for t18n903 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.148.1, 6.6.2738000.1

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 $18$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $18$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.12.6.1$x^{12} + 2026120 x^{6} - 69343957 x^{2} + 1026290563600$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.6.0.1$x^{6} - 4 x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
151Data not computed
643Data not computed