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Properties

Label 15.7.63785837229...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{24}\cdot 5^{6}\cdot 13^{8}\cdot 37^{5}\cdot 67^{2}\cdot 3095563^{2}$
Root discriminant $970.47$
Ramified primes $2, 5, 13, 37, 67, 3095563$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T96

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2704000, 9464000, -17576000, -40593800, 75319920, -13423800, -30046744, 17570644, -2674152, -252486, 74696, 770, -72, -126, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 126*x^13 - 72*x^12 + 770*x^11 + 74696*x^10 - 252486*x^9 - 2674152*x^8 + 17570644*x^7 - 30046744*x^6 - 13423800*x^5 + 75319920*x^4 - 40593800*x^3 - 17576000*x^2 + 9464000*x + 2704000)
 
gp: K = bnfinit(x^15 - 126*x^13 - 72*x^12 + 770*x^11 + 74696*x^10 - 252486*x^9 - 2674152*x^8 + 17570644*x^7 - 30046744*x^6 - 13423800*x^5 + 75319920*x^4 - 40593800*x^3 - 17576000*x^2 + 9464000*x + 2704000, 1)
 

Normalized defining polynomial

\( x^{15} - 126 x^{13} - 72 x^{12} + 770 x^{11} + 74696 x^{10} - 252486 x^{9} - 2674152 x^{8} + 17570644 x^{7} - 30046744 x^{6} - 13423800 x^{5} + 75319920 x^{4} - 40593800 x^{3} - 17576000 x^{2} + 9464000 x + 2704000 \)

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:  \(637858372296192230986633844231104626688000000=2^{24}\cdot 5^{6}\cdot 13^{8}\cdot 37^{5}\cdot 67^{2}\cdot 3095563^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $970.47$
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:  $2, 5, 13, 37, 67, 3095563$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{52} a^{10} + \frac{1}{13} a^{8} + \frac{3}{26} a^{7} - \frac{5}{26} a^{6} + \frac{6}{13} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{260} a^{11} + \frac{1}{65} a^{9} + \frac{29}{130} a^{8} - \frac{1}{26} a^{7} - \frac{27}{130} a^{6} - \frac{1}{10} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{520} a^{12} + \frac{1}{130} a^{10} + \frac{29}{260} a^{9} - \frac{1}{52} a^{8} + \frac{19}{130} a^{7} - \frac{1}{20} a^{6} - \frac{1}{10} a^{5} + \frac{1}{5} a^{4} + \frac{3}{10} a^{3}$, $\frac{1}{5200} a^{13} - \frac{3}{2600} a^{11} + \frac{7}{1300} a^{10} - \frac{1}{104} a^{9} - \frac{11}{50} a^{8} - \frac{443}{2600} a^{7} - \frac{129}{650} a^{6} + \frac{431}{1300} a^{5} + \frac{2}{25} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{7546223283506579291296353988970917388000} a^{14} + \frac{573008612639336834519767305259661}{150924465670131585825927079779418347760} a^{13} + \frac{1242830801578929642272831181566140997}{3773111641753289645648176994485458694000} a^{12} + \frac{1822846960708959264223266976242631807}{1886555820876644822824088497242729347000} a^{11} + \frac{672014277906889699508592052039844389}{150924465670131585825927079779418347760} a^{10} - \frac{12165941167292580808812592751618912511}{1886555820876644822824088497242729347000} a^{9} + \frac{135729309178932248597522662104788529557}{3773111641753289645648176994485458694000} a^{8} - \frac{180859061867117991257741733979380513933}{1886555820876644822824088497242729347000} a^{7} - \frac{361119055193484214387664485601860306419}{1886555820876644822824088497242729347000} a^{6} + \frac{52583775201946795375251095511574274563}{235819477609580602853011062155341168375} a^{5} + \frac{4259961342909602712542483991793999609}{14511967852897267867877603824944071900} a^{4} - \frac{3452116783705095848244088159789520369}{7255983926448633933938801912472035950} a^{3} - \frac{697885425430715349015958797667486661}{2902393570579453573575520764988814380} a^{2} + \frac{506856761881029983547305009271061019}{1451196785289726786787760382494407190} a + \frac{7026884599884001595998268768044115}{145119678528972678678776038249440719}$

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:  \( 234010425716000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T96:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 65 conjugacy class representatives for [A(5)^3]S(3)=A(5)wrS(3) are not computed
Character table for [A(5)^3]S(3)=A(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

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
Degree 30 siblings: data not computed
Degree 36 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 R $15$ R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ $15$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.11.7$x^{6} + 2 x^{2} + 6$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
2.6.11.7$x^{6} + 2 x^{2} + 6$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
13.10.8.1$x^{10} - 13 x^{5} + 338$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$37$37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
37.10.5.1$x^{10} - 2738 x^{6} + 1874161 x^{2} - 11719128733$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.5.0.1$x^{5} - x + 21$$1$$5$$0$$C_5$$[\ ]^{5}$
3095563Data not computed