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Properties

Label 15.15.1177574415...6889.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 67^{2}\cdot 131^{2}\cdot 401^{7}\cdot 512779^{2}$
Root discriminant $1374.25$
Ramified primes $3, 67, 131, 401, 512779$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T79

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4061757869466624, -5077197336833280, -2538598668416640, -614578588468128, -59260129751988, 3560094069111, 1218971397600, 51799388058, -6650815716, -526988835, 13622904, 1702863, -8892, -2223, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2223*x^13 - 8892*x^12 + 1702863*x^11 + 13622904*x^10 - 526988835*x^9 - 6650815716*x^8 + 51799388058*x^7 + 1218971397600*x^6 + 3560094069111*x^5 - 59260129751988*x^4 - 614578588468128*x^3 - 2538598668416640*x^2 - 5077197336833280*x - 4061757869466624)
 
gp: K = bnfinit(x^15 - 2223*x^13 - 8892*x^12 + 1702863*x^11 + 13622904*x^10 - 526988835*x^9 - 6650815716*x^8 + 51799388058*x^7 + 1218971397600*x^6 + 3560094069111*x^5 - 59260129751988*x^4 - 614578588468128*x^3 - 2538598668416640*x^2 - 5077197336833280*x - 4061757869466624, 1)
 

Normalized defining polynomial

\( x^{15} - 2223 x^{13} - 8892 x^{12} + 1702863 x^{11} + 13622904 x^{10} - 526988835 x^{9} - 6650815716 x^{8} + 51799388058 x^{7} + 1218971397600 x^{6} + 3560094069111 x^{5} - 59260129751988 x^{4} - 614578588468128 x^{3} - 2538598668416640 x^{2} - 5077197336833280 x - 4061757869466624 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(117757441598028025329119098118152643876169986889=3^{20}\cdot 67^{2}\cdot 131^{2}\cdot 401^{7}\cdot 512779^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1374.25$
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:  $3, 67, 131, 401, 512779$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{9} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{27} a^{5}$, $\frac{1}{162} a^{6} - \frac{1}{54} a^{5} - \frac{1}{18} a^{4} - \frac{1}{18} a^{3} - \frac{1}{2} a$, $\frac{1}{162} a^{7} - \frac{1}{18} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{486} a^{8} - \frac{1}{18} a^{4} - \frac{1}{18} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{4374} a^{9} + \frac{1}{486} a^{7} + \frac{1}{486} a^{6} - \frac{1}{18} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{4374} a^{10} + \frac{1}{486} a^{7} - \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{419904} a^{11} - \frac{1}{8748} a^{10} - \frac{5}{139968} a^{9} - \frac{11}{11664} a^{8} - \frac{1}{5184} a^{7} - \frac{1}{1944} a^{6} - \frac{17}{1728} a^{5} + \frac{1}{48} a^{4} + \frac{5}{288} a^{3} + \frac{1}{9} a^{2} + \frac{5}{192} a - \frac{11}{48}$, $\frac{1}{73555621138734280704} a^{12} - \frac{5}{6718464} a^{11} - \frac{19257679995869}{2724282264397565952} a^{10} - \frac{30156863908627}{340535283049695744} a^{9} + \frac{373887104015695}{908094088132521984} a^{8} - \frac{328502482634399}{227023522033130496} a^{7} - \frac{1943712689756467}{908094088132521984} a^{6} + \frac{2984258354677}{175172470704576} a^{5} - \frac{593899750487329}{16816557187639296} a^{4} - \frac{665571568879651}{12612417890729472} a^{3} + \frac{1273320671274071}{11211038125092864} a^{2} + \frac{672803969838007}{1401379765636608} a + \frac{53115254951447}{700689882818304}$, $\frac{1}{4707559752878993965056} a^{13} - \frac{1}{588444969109874245632} a^{12} + \frac{229315175889337}{523062194764332662784} a^{11} + \frac{4661328153667133}{43588516230361055232} a^{10} - \frac{909143596509811}{174354064921444220928} a^{9} - \frac{1519}{2985984} a^{8} + \frac{42323453081648543}{19372673880160468992} a^{7} + \frac{6823508099938013}{14529505410120351744} a^{6} + \frac{9687680508138079}{1076259660008914944} a^{5} - \frac{8687824740077795}{403597372503343104} a^{4} - \frac{314418825677311313}{6457557960053489664} a^{3} - \frac{9242195201029033}{59792203333828608} a^{2} + \frac{3772116975444997}{44844152500371456} a + \frac{3478418643213143}{11211038125092864}$, $\frac{1}{903851472552766841290752} a^{14} + \frac{5}{75320956046063903440896} a^{13} + \frac{17}{11158660154972430139392} a^{12} - \frac{111132378119}{1362141132198782976} a^{11} + \frac{64795523970497255}{1239851128330270015488} a^{10} + \frac{1818542894742533}{929888346247702511616} a^{9} - \frac{7816072876480428067}{11158660154972430139392} a^{8} - \frac{478936723928473705}{464944173123851255808} a^{7} + \frac{137978410891737925}{619925564165135007744} a^{6} + \frac{1497558307127691047}{154981391041283751936} a^{5} - \frac{22117321668088339291}{413283709443423338496} a^{4} - \frac{477200662434972709}{8610077280071319552} a^{3} + \frac{603042222031272109}{4305038640035659776} a^{2} - \frac{13529933599436807}{179376610001485824} a - \frac{24938561614018993}{59792203333828608}$

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:  $14$
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:  \( 3043578245390000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T79:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38880
The 45 conjugacy class representatives for 1/2[S(3)^5]D(5)
Character table for 1/2[S(3)^5]D(5) is not computed

Intermediate fields

5.5.160801.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 45 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 $15$ $15$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.6.8.2$x^{6} + 6 x^{5} + 9 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$
3.6.8.4$x^{6} + 18 x^{2} + 63$$3$$2$$8$$C_6$$[2]^{2}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.6.0.1$x^{6} + x^{2} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
131Data not computed
401Data not computed
512779Data not computed