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Properties

Label 18.10.3153544977...3125.1
Degree $18$
Signature $[10, 4]$
Discriminant $5^{9}\cdot 7^{12}\cdot 41\cdot 139^{2}\cdot 121349929^{2}$
Root discriminant $137.66$
Ramified primes $5, 7, 41, 139, 121349929$
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![1789324621, -5276020218, 6140547357, -3079626438, -39173568, 784800422, -298465159, -26267862, 43676548, -6737760, -2365365, 802216, 27317, -37506, 2288, 823, -91, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 91*x^16 + 823*x^15 + 2288*x^14 - 37506*x^13 + 27317*x^12 + 802216*x^11 - 2365365*x^10 - 6737760*x^9 + 43676548*x^8 - 26267862*x^7 - 298465159*x^6 + 784800422*x^5 - 39173568*x^4 - 3079626438*x^3 + 6140547357*x^2 - 5276020218*x + 1789324621)
 
gp: K = bnfinit(x^18 - 7*x^17 - 91*x^16 + 823*x^15 + 2288*x^14 - 37506*x^13 + 27317*x^12 + 802216*x^11 - 2365365*x^10 - 6737760*x^9 + 43676548*x^8 - 26267862*x^7 - 298465159*x^6 + 784800422*x^5 - 39173568*x^4 - 3079626438*x^3 + 6140547357*x^2 - 5276020218*x + 1789324621, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} - 91 x^{16} + 823 x^{15} + 2288 x^{14} - 37506 x^{13} + 27317 x^{12} + 802216 x^{11} - 2365365 x^{10} - 6737760 x^{9} + 43676548 x^{8} - 26267862 x^{7} - 298465159 x^{6} + 784800422 x^{5} - 39173568 x^{4} - 3079626438 x^{3} + 6140547357 x^{2} - 5276020218 x + 1789324621 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(315354497788882418993233673419533203125=5^{9}\cdot 7^{12}\cdot 41\cdot 139^{2}\cdot 121349929^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $137.66$
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:  $5, 7, 41, 139, 121349929$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{12} - \frac{1}{7} a^{11} + \frac{3}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{91} a^{16} + \frac{1}{91} a^{15} + \frac{6}{13} a^{14} - \frac{3}{91} a^{13} - \frac{3}{7} a^{12} + \frac{27}{91} a^{11} + \frac{38}{91} a^{10} + \frac{12}{91} a^{9} + \frac{3}{13} a^{8} - \frac{38}{91} a^{7} + \frac{2}{7} a^{6} - \frac{41}{91} a^{5} + \frac{31}{91} a^{4} + \frac{1}{91} a^{3} - \frac{5}{13} a^{2} + \frac{5}{13} a + \frac{5}{13}$, $\frac{1}{2128244926015877495218448451301082388834713} a^{17} + \frac{6245510119022261856061107666580137207665}{2128244926015877495218448451301082388834713} a^{16} - \frac{142083562757723924778464322736190261938055}{2128244926015877495218448451301082388834713} a^{15} + \frac{417425771847888288567721628194702661379730}{2128244926015877495218448451301082388834713} a^{14} + \frac{321770336935413654068242828586571169682593}{2128244926015877495218448451301082388834713} a^{13} - \frac{88246371319354398033803122664145413972086}{2128244926015877495218448451301082388834713} a^{12} + \frac{492941254070887915768686148332349006094517}{2128244926015877495218448451301082388834713} a^{11} + \frac{759015038069328819750647615063597709153805}{2128244926015877495218448451301082388834713} a^{10} + \frac{9545629529959717886215898836642578129206}{163711148155067499632188342407775568371901} a^{9} + \frac{48176661765865205519091290947123476814021}{163711148155067499632188342407775568371901} a^{8} - \frac{917114336709633573820308068988251879197800}{2128244926015877495218448451301082388834713} a^{7} + \frac{81780020076184285500203605839242371509573}{2128244926015877495218448451301082388834713} a^{6} + \frac{793488290013066002825380249778052331396863}{2128244926015877495218448451301082388834713} a^{5} + \frac{8202084522773986460297352778000050496187}{304034989430839642174064064471583198404959} a^{4} + \frac{783466762824395193152689321095936366817284}{2128244926015877495218448451301082388834713} a^{3} - \frac{135536448723228412179828957163753896759131}{304034989430839642174064064471583198404959} a^{2} - \frac{84230836677748837604274793233349667802876}{304034989430839642174064064471583198404959} a + \frac{150026792073135114367088579313533874804430}{304034989430839642174064064471583198404959}$

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:  $13$
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:  \( 1480622603460 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 6.6.300125.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ $18$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $18$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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
5Data not computed
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
41Data not computed
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.3.2.2$x^{3} + 556$$3$$1$$2$$C_3$$[\ ]_{3}$
121349929Data not computed