Properties

Label 18.0.15485076856...2467.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 23^{6}$
Root discriminant $10.25$
Ramified primes $3, 23$
Class number $1$
Class group Trivial
Galois Group $C_3.S_3^2$ (as 18T57)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 21, -67, 147, -243, 334, -378, 378, -335, 252, -189, 109, -63, 36, -9, 9, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 9*x^15 + 36*x^14 - 63*x^13 + 109*x^12 - 189*x^11 + 252*x^10 - 335*x^9 + 378*x^8 - 378*x^7 + 334*x^6 - 243*x^5 + 147*x^4 - 67*x^3 + 21*x^2 - 3*x + 1)
gp: K = bnfinit(x^18 + 9*x^16 - 9*x^15 + 36*x^14 - 63*x^13 + 109*x^12 - 189*x^11 + 252*x^10 - 335*x^9 + 378*x^8 - 378*x^7 + 334*x^6 - 243*x^5 + 147*x^4 - 67*x^3 + 21*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut +\mathstrut 9 x^{16} \) \(\mathstrut -\mathstrut 9 x^{15} \) \(\mathstrut +\mathstrut 36 x^{14} \) \(\mathstrut -\mathstrut 63 x^{13} \) \(\mathstrut +\mathstrut 109 x^{12} \) \(\mathstrut -\mathstrut 189 x^{11} \) \(\mathstrut +\mathstrut 252 x^{10} \) \(\mathstrut -\mathstrut 335 x^{9} \) \(\mathstrut +\mathstrut 378 x^{8} \) \(\mathstrut -\mathstrut 378 x^{7} \) \(\mathstrut +\mathstrut 334 x^{6} \) \(\mathstrut -\mathstrut 243 x^{5} \) \(\mathstrut +\mathstrut 147 x^{4} \) \(\mathstrut -\mathstrut 67 x^{3} \) \(\mathstrut +\mathstrut 21 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-1548507685660102467=-\,3^{21}\cdot 23^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.25$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1153182335} a^{17} - \frac{394336921}{1153182335} a^{16} - \frac{85928896}{230636467} a^{15} - \frac{97238009}{1153182335} a^{14} - \frac{49031889}{230636467} a^{13} - \frac{138175543}{1153182335} a^{12} + \frac{462712092}{1153182335} a^{11} - \frac{279666761}{1153182335} a^{10} - \frac{280014797}{1153182335} a^{9} - \frac{524613583}{1153182335} a^{8} + \frac{63869246}{1153182335} a^{7} - \frac{17051817}{67834255} a^{6} - \frac{77106787}{1153182335} a^{5} - \frac{214113886}{1153182335} a^{4} + \frac{560320823}{1153182335} a^{3} - \frac{96385234}{230636467} a^{2} - \frac{142185499}{1153182335} a + \frac{564604746}{1153182335}$

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:  \( -\frac{3077572}{11195945} a^{17} - \frac{313528}{11195945} a^{16} - \frac{5796705}{2239189} a^{15} + \frac{22523788}{11195945} a^{14} - \frac{24327282}{2239189} a^{13} + \frac{172258676}{11195945} a^{12} - \frac{359270994}{11195945} a^{11} + \frac{555197617}{11195945} a^{10} - \frac{776133451}{11195945} a^{9} + \frac{1033752351}{11195945} a^{8} - \frac{1120967627}{11195945} a^{7} + \frac{68522744}{658585} a^{6} - \frac{989519121}{11195945} a^{5} + \frac{715640877}{11195945} a^{4} - \frac{427450321}{11195945} a^{3} + \frac{36791096}{2239189} a^{2} - \frac{55256652}{11195945} a + \frac{15041843}{11195945} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{7787269}{67834255} a^{17} - \frac{775114}{67834255} a^{16} + \frac{14561708}{13566851} a^{15} - \frac{75084051}{67834255} a^{14} + \frac{63477544}{13566851} a^{13} - \frac{513110897}{67834255} a^{12} + \frac{1040788863}{67834255} a^{11} - \frac{1598219739}{67834255} a^{10} + \frac{2471706147}{67834255} a^{9} - \frac{3126190612}{67834255} a^{8} + \frac{3731901739}{67834255} a^{7} - \frac{3993706766}{67834255} a^{6} + \frac{3491474512}{67834255} a^{5} - \frac{2883271559}{67834255} a^{4} + \frac{1815929392}{67834255} a^{3} - \frac{178925970}{13566851} a^{2} + \frac{312293934}{67834255} a - \frac{27417146}{67834255} \),  \( \frac{113222024}{1153182335} a^{17} + \frac{55131366}{1153182335} a^{16} + \frac{193757382}{230636467} a^{15} - \frac{543260141}{1153182335} a^{14} + \frac{622325566}{230636467} a^{13} - \frac{4879861732}{1153182335} a^{12} + \frac{7328683703}{1153182335} a^{11} - \frac{12523178434}{1153182335} a^{10} + \frac{15206422027}{1153182335} a^{9} - \frac{15405478787}{1153182335} a^{8} + \frac{17985332554}{1153182335} a^{7} - \frac{666095633}{67834255} a^{6} + \frac{5748851982}{1153182335} a^{5} - \frac{2073417299}{1153182335} a^{4} - \frac{4448509843}{1153182335} a^{3} + \frac{793439782}{230636467} a^{2} - \frac{1950402161}{1153182335} a + \frac{340926974}{1153182335} \),  \( \frac{178802317}{1153182335} a^{17} + \frac{50157738}{1153182335} a^{16} + \frac{295167334}{230636467} a^{15} - \frac{1158575403}{1153182335} a^{14} + \frac{987272678}{230636467} a^{13} - \frac{8311404001}{1153182335} a^{12} + \frac{12673519399}{1153182335} a^{11} - \frac{21722715152}{1153182335} a^{10} + \frac{25092904776}{1153182335} a^{9} - \frac{31254712571}{1153182335} a^{8} + \frac{29876349352}{1153182335} a^{7} - \frac{1475263934}{67834255} a^{6} + \frac{17684984381}{1153182335} a^{5} - \frac{6013754842}{1153182335} a^{4} + \frac{1025853856}{1153182335} a^{3} + \frac{220696501}{230636467} a^{2} - \frac{2005756463}{1153182335} a + \frac{388091067}{1153182335} \),  \( \frac{26858599}{1153182335} a^{17} + \frac{192380726}{1153182335} a^{16} + \frac{8888358}{230636467} a^{15} + \frac{1074178204}{1153182335} a^{14} - \frac{537387521}{230636467} a^{13} + \frac{3331740458}{1153182335} a^{12} - \frac{14050528177}{1153182335} a^{11} + \frac{14856338311}{1153182335} a^{10} - \frac{32487491478}{1153182335} a^{9} + \frac{41195931338}{1153182335} a^{8} - \frac{49412987666}{1153182335} a^{7} + \frac{3401221482}{67834255} a^{6} - \frac{49307725013}{1153182335} a^{5} + \frac{41730658171}{1153182335} a^{4} - \frac{23852906973}{1153182335} a^{3} + \frac{2061584994}{230636467} a^{2} - \frac{2770205536}{1153182335} a + \frac{613268979}{1153182335} \),  \( \frac{335539153}{1153182335} a^{17} + \frac{415907127}{1153182335} a^{16} + \frac{617166644}{230636467} a^{15} + \frac{449056538}{1153182335} a^{14} + \frac{1751882503}{230636467} a^{13} - \frac{9039130324}{1153182335} a^{12} + \frac{13975578746}{1153182335} a^{11} - \frac{28814422013}{1153182335} a^{10} + \frac{24996012669}{1153182335} a^{9} - \frac{34483388644}{1153182335} a^{8} + \frac{34033375843}{1153182335} a^{7} - \frac{1312022291}{67834255} a^{6} + \frac{16964605959}{1153182335} a^{5} - \frac{6132877753}{1153182335} a^{4} - \frac{2344263796}{1153182335} a^{3} + \frac{486836096}{230636467} a^{2} - \frac{3113200302}{1153182335} a + \frac{1680560288}{1153182335} \),  \( \frac{7717972}{67834255} a^{17} + \frac{13555223}{67834255} a^{16} + \frac{16262261}{13566851} a^{15} + \frac{47173707}{67834255} a^{14} + \frac{49157180}{13566851} a^{13} - \frac{167600876}{67834255} a^{12} + \frac{308768684}{67834255} a^{11} - \frac{895490907}{67834255} a^{10} + \frac{497754946}{67834255} a^{9} - \frac{1423818216}{67834255} a^{8} + \frac{1188695972}{67834255} a^{7} - \frac{1297973193}{67834255} a^{6} + \frac{1421199251}{67834255} a^{5} - \frac{920550427}{67834255} a^{4} + \frac{845243156}{67834255} a^{3} - \frac{67781282}{13566851} a^{2} + \frac{211776622}{67834255} a - \frac{901808}{67834255} \),  \( \frac{94166316}{1153182335} a^{17} + \frac{237634764}{1153182335} a^{16} + \frac{258123870}{230636467} a^{15} + \frac{1727320226}{1153182335} a^{14} + \frac{1055059070}{230636467} a^{13} + \frac{2443434597}{1153182335} a^{12} + \frac{7905724342}{1153182335} a^{11} - \frac{5121191536}{1153182335} a^{10} + \frac{4278272468}{1153182335} a^{9} - \frac{13311177608}{1153182335} a^{8} + \frac{2541094371}{1153182335} a^{7} - \frac{374200542}{67834255} a^{6} + \frac{588237258}{1153182335} a^{5} + \frac{3665657139}{1153182335} a^{4} - \frac{4964064202}{1153182335} a^{3} + \frac{1171730599}{230636467} a^{2} - \frac{3469817434}{1153182335} a + \frac{417084781}{1153182335} \),  \( \frac{305725712}{1153182335} a^{17} + \frac{412187843}{1153182335} a^{16} + \frac{616491220}{230636467} a^{15} + \frac{1183190987}{1153182335} a^{14} + \frac{2097113568}{230636467} a^{13} - \frac{5066018986}{1153182335} a^{12} + \frac{19483233154}{1153182335} a^{11} - \frac{24589697202}{1153182335} a^{10} + \frac{29036839026}{1153182335} a^{9} - \frac{40044302106}{1153182335} a^{8} + \frac{34383968397}{1153182335} a^{7} - \frac{1960080519}{67834255} a^{6} + \frac{21977459606}{1153182335} a^{5} - \frac{13295745582}{1153182335} a^{4} + \frac{3183462736}{1153182335} a^{3} - \frac{339693754}{230636467} a^{2} - \frac{1272504488}{1153182335} a + \frac{206732022}{1153182335} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 78.7017437829 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3.S_3^2$ (as 18T57):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 108
The 11 conjugacy class representatives for $C_3.S_3^2$
Character table for $C_3.S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.23.1, 6.0.14283.1, 9.1.239483061.1 x3

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

Sibling fields

Degree 9 siblings: data not computed
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$