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Properties

Label 15.3.60500833661...2241.2
Degree $15$
Signature $[3, 6]$
Discriminant $3^{20}\cdot 1609^{6}$
Root discriminant $82.94$
Ramified primes $3, 1609$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $\GL(2,4):C_2$ (as 15T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-26273539, -22983852, -26735256, 23747884, 9764478, -8508177, -972656, 1286655, 16263, -99113, 417, 4203, -13, -96, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 96*x^13 - 13*x^12 + 4203*x^11 + 417*x^10 - 99113*x^9 + 16263*x^8 + 1286655*x^7 - 972656*x^6 - 8508177*x^5 + 9764478*x^4 + 23747884*x^3 - 26735256*x^2 - 22983852*x - 26273539)
 
gp: K = bnfinit(x^15 - 96*x^13 - 13*x^12 + 4203*x^11 + 417*x^10 - 99113*x^9 + 16263*x^8 + 1286655*x^7 - 972656*x^6 - 8508177*x^5 + 9764478*x^4 + 23747884*x^3 - 26735256*x^2 - 22983852*x - 26273539, 1)
 

Normalized defining polynomial

\( x^{15} - 96 x^{13} - 13 x^{12} + 4203 x^{11} + 417 x^{10} - 99113 x^{9} + 16263 x^{8} + 1286655 x^{7} - 972656 x^{6} - 8508177 x^{5} + 9764478 x^{4} + 23747884 x^{3} - 26735256 x^{2} - 22983852 x - 26273539 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(60500833661891746134479882241=3^{20}\cdot 1609^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.94$
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:  $3, 1609$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{34} a^{13} - \frac{1}{34} a^{12} + \frac{2}{17} a^{11} + \frac{3}{34} a^{10} + \frac{3}{17} a^{9} - \frac{3}{17} a^{8} - \frac{15}{34} a^{7} + \frac{5}{17} a^{6} + \frac{11}{34} a^{5} + \frac{9}{34} a^{4} - \frac{4}{17} a^{3} + \frac{8}{17} a^{2} - \frac{3}{34} a + \frac{2}{17}$, $\frac{1}{571074321003366436176601931805710945179110167189566} a^{14} + \frac{2879860734870687380557004485123718573546504180465}{285537160501683218088300965902855472589555083594783} a^{13} + \frac{2617666192718991864682678838725639300302103385954}{16796303558922542240488292111932674858209122564399} a^{12} + \frac{21369673277085359254967781987696941324792718032799}{571074321003366436176601931805710945179110167189566} a^{11} + \frac{97118611382362498012445043945043918950088373189199}{571074321003366436176601931805710945179110167189566} a^{10} - \frac{133433731952909060029456030680973831230831160871591}{285537160501683218088300965902855472589555083594783} a^{9} - \frac{280684621823041475792998119033060239321596348201821}{571074321003366436176601931805710945179110167189566} a^{8} + \frac{97826421898859902664701760593912090917886120647758}{285537160501683218088300965902855472589555083594783} a^{7} - \frac{11425956996149939679231439749136958660377560221951}{33592607117845084480976584223865349716418245128798} a^{6} + \frac{103376671567762737632884740646375594720233848459267}{571074321003366436176601931805710945179110167189566} a^{5} - \frac{36825807811307567705448335429910202072078395978463}{285537160501683218088300965902855472589555083594783} a^{4} - \frac{80798948204713401717647530901747972592818135611253}{571074321003366436176601931805710945179110167189566} a^{3} + \frac{133251937848860557157367401546787149845048143558491}{285537160501683218088300965902855472589555083594783} a^{2} + \frac{51696821245701926556807447371658671542880375713181}{285537160501683218088300965902855472589555083594783} a + \frac{162977222622164496981583781318551472059167076588039}{571074321003366436176601931805710945179110167189566}$

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:  $8$
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:  \( 84559284.0776 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_5$ (as 15T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 12 conjugacy class representatives for $\GL(2,4):C_2$
Character table for $\GL(2,4):C_2$

Intermediate fields

5.1.1609.1

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

Sibling fields

Degree 15 sibling: data not computed
Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 45 sibling: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
3Data not computed
1609Data not computed