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Properties

Label 15.7.29098734453...6209.1
Degree $15$
Signature $[7, 4]$
Discriminant $3^{20}\cdot 37^{2}\cdot 61^{4}\cdot 397^{4}\cdot 421^{2}$
Root discriminant $231.34$
Ramified primes $3, 37, 61, 397, 421$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T89

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-436825811, 105565329, 238047714, -62787342, -69930507, 8157366, 10317297, -534909, -1111410, -98527, 42771, 6975, -530, -144, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 144*x^13 - 530*x^12 + 6975*x^11 + 42771*x^10 - 98527*x^9 - 1111410*x^8 - 534909*x^7 + 10317297*x^6 + 8157366*x^5 - 69930507*x^4 - 62787342*x^3 + 238047714*x^2 + 105565329*x - 436825811)
 
gp: K = bnfinit(x^15 - 144*x^13 - 530*x^12 + 6975*x^11 + 42771*x^10 - 98527*x^9 - 1111410*x^8 - 534909*x^7 + 10317297*x^6 + 8157366*x^5 - 69930507*x^4 - 62787342*x^3 + 238047714*x^2 + 105565329*x - 436825811, 1)
 

Normalized defining polynomial

\( x^{15} - 144 x^{13} - 530 x^{12} + 6975 x^{11} + 42771 x^{10} - 98527 x^{9} - 1111410 x^{8} - 534909 x^{7} + 10317297 x^{6} + 8157366 x^{5} - 69930507 x^{4} - 62787342 x^{3} + 238047714 x^{2} + 105565329 x - 436825811 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(290987344534580290346324872872346209=3^{20}\cdot 37^{2}\cdot 61^{4}\cdot 397^{4}\cdot 421^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $231.34$
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:  $3, 37, 61, 397, 421$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{23825281660405272437706528605892636105796556999054515} a^{14} + \frac{456204414725892237469572366009125629036938779748877}{4765056332081054487541305721178527221159311399810903} a^{13} - \frac{8041133680741607162866857020902536829950176486919893}{23825281660405272437706528605892636105796556999054515} a^{12} - \frac{8336876538770095171931364535886765929337679136664143}{23825281660405272437706528605892636105796556999054515} a^{11} + \frac{4699919374769572241098635898142209289962944824355206}{23825281660405272437706528605892636105796556999054515} a^{10} + \frac{903979726068478315694767703207702261711993608943584}{4765056332081054487541305721178527221159311399810903} a^{9} + \frac{1832412760159653741278232388915309095202343701736022}{23825281660405272437706528605892636105796556999054515} a^{8} + \frac{4220730402385834817952037808183592818063551059299033}{23825281660405272437706528605892636105796556999054515} a^{7} + \frac{10532080422791358838405251420378525164428494236665189}{23825281660405272437706528605892636105796556999054515} a^{6} - \frac{8919904688619419873767349446956606689025409283808559}{23825281660405272437706528605892636105796556999054515} a^{5} - \frac{1564973314737072134323076002175906533889383656329752}{23825281660405272437706528605892636105796556999054515} a^{4} + \frac{11595359281756912745236522397263360934406008913946803}{23825281660405272437706528605892636105796556999054515} a^{3} - \frac{5393473330501546216466161850719414092176654502994174}{23825281660405272437706528605892636105796556999054515} a^{2} - \frac{4072761386429124658061343168243849316159852203879587}{23825281660405272437706528605892636105796556999054515} a + \frac{1180465297346507492907904986244085212189661723812258}{23825281660405272437706528605892636105796556999054515}$

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:  $10$
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:  \( 561912558547 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T89:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 466560
The 60 conjugacy class representatives for 1/2[S(3)^5]S(5) are not computed
Character table for 1/2[S(3)^5]S(5) is not computed

Intermediate fields

5.5.24217.1

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 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 $15$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $15$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
37Data not computed
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.4.3.1$x^{4} - 61$$4$$1$$3$$C_4$$[\ ]_{4}$
397Data not computed
421Data not computed