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Properties

Label 18.14.1450329377...5625.1
Degree $18$
Signature $[14, 2]$
Discriminant $5^{4}\cdot 13^{15}\cdot 181^{2}\cdot 389\cdot 5964373^{2}$
Root discriminant $170.29$
Ramified primes $5, 13, 181, 389, 5964373$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T857

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3469475203, -9866585215, 11033016132, -5285633443, -114785660, 1288958095, -469853154, -40903477, 65031017, -9845841, -3268795, 1093608, 31349, -47143, 2933, 937, -103, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 103*x^16 + 937*x^15 + 2933*x^14 - 47143*x^13 + 31349*x^12 + 1093608*x^11 - 3268795*x^10 - 9845841*x^9 + 65031017*x^8 - 40903477*x^7 - 469853154*x^6 + 1288958095*x^5 - 114785660*x^4 - 5285633443*x^3 + 11033016132*x^2 - 9866585215*x + 3469475203)
 
gp: K = bnfinit(x^18 - 7*x^17 - 103*x^16 + 937*x^15 + 2933*x^14 - 47143*x^13 + 31349*x^12 + 1093608*x^11 - 3268795*x^10 - 9845841*x^9 + 65031017*x^8 - 40903477*x^7 - 469853154*x^6 + 1288958095*x^5 - 114785660*x^4 - 5285633443*x^3 + 11033016132*x^2 - 9866585215*x + 3469475203, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} - 103 x^{16} + 937 x^{15} + 2933 x^{14} - 47143 x^{13} + 31349 x^{12} + 1093608 x^{11} - 3268795 x^{10} - 9845841 x^{9} + 65031017 x^{8} - 40903477 x^{7} - 469853154 x^{6} + 1288958095 x^{5} - 114785660 x^{4} - 5285633443 x^{3} + 11033016132 x^{2} - 9866585215 x + 3469475203 \)

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:  \(14503293777515810243222263717959408085625=5^{4}\cdot 13^{15}\cdot 181^{2}\cdot 389\cdot 5964373^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $170.29$
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:  $5, 13, 181, 389, 5964373$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{20} a^{16} + \frac{1}{20} a^{15} + \frac{1}{10} a^{14} - \frac{1}{4} a^{13} - \frac{3}{20} a^{12} + \frac{3}{20} a^{11} + \frac{1}{10} a^{10} + \frac{1}{4} a^{9} + \frac{1}{5} a^{8} + \frac{1}{20} a^{7} - \frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{4} a^{3} - \frac{1}{10} a^{2} + \frac{3}{10} a + \frac{1}{20}$, $\frac{1}{2387448457985035244150896907778807841100892356540} a^{17} - \frac{22883069944167317478935874717661568460856272257}{1193724228992517622075448453889403920550446178270} a^{16} + \frac{52442817000622549195966717338934404531020837388}{596862114496258811037724226944701960275223089135} a^{15} + \frac{28665085929080326204505934574747097551043630473}{477489691597007048830179381555761568220178471308} a^{14} - \frac{407520299520550672718030000756754322635340972433}{2387448457985035244150896907778807841100892356540} a^{13} + \frac{93485042449254177595144842154619399919976525632}{596862114496258811037724226944701960275223089135} a^{12} - \frac{97627956630433063232370661067912263188934585573}{2387448457985035244150896907778807841100892356540} a^{11} - \frac{45672234354959821581569792538995448300498928}{119372422899251762207544845388940392055044617827} a^{10} + \frac{162974152635404792031336194380018358302601631467}{1193724228992517622075448453889403920550446178270} a^{9} - \frac{374300577657353948077741140078407701647039211727}{1193724228992517622075448453889403920550446178270} a^{8} + \frac{157977822273892356958505609310087393570410582173}{2387448457985035244150896907778807841100892356540} a^{7} - \frac{156736265190002332562412623970645522289379502773}{1193724228992517622075448453889403920550446178270} a^{6} + \frac{297014565698684711163695681523346783981191421147}{1193724228992517622075448453889403920550446178270} a^{5} + \frac{107162809454353479919546773304590542577364186417}{477489691597007048830179381555761568220178471308} a^{4} - \frac{895131289469845696240513997852106374334728098057}{2387448457985035244150896907778807841100892356540} a^{3} + \frac{180291816882638727011016894797111622417409667931}{2387448457985035244150896907778807841100892356540} a^{2} + \frac{407578520042723300675428888037464896306415706863}{1193724228992517622075448453889403920550446178270} a + \frac{193758388881177657200701912864655614707765945893}{477489691597007048830179381555761568220178471308}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 23884669346500 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T857:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 159 conjugacy class representatives for t18n857 are not computed
Character table for t18n857 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R $18$ $18$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ $18$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.3.2.3$x^{3} - 724$$3$$1$$2$$C_3$$[\ ]_{3}$
181.3.0.1$x^{3} - x + 18$$1$$3$$0$$C_3$$[\ ]^{3}$
389Data not computed
5964373Data not computed