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Properties

Label 15.1.28900887085...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{18}\cdot 3^{13}\cdot 5^{10}\cdot 11^{13}\cdot 29^{5}$
Root discriminant $427.29$
Ramified primes $2, 3, 5, 11, 29$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4737567935, 1392333871, -271954775, -309285975, -122910997, -15645211, 2940961, 4734027, 784389, 434633, 27391, 14443, 249, 203, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + 203*x^13 + 249*x^12 + 14443*x^11 + 27391*x^10 + 434633*x^9 + 784389*x^8 + 4734027*x^7 + 2940961*x^6 - 15645211*x^5 - 122910997*x^4 - 309285975*x^3 - 271954775*x^2 + 1392333871*x + 4737567935)
 
gp: K = bnfinit(x^15 - x^14 + 203*x^13 + 249*x^12 + 14443*x^11 + 27391*x^10 + 434633*x^9 + 784389*x^8 + 4734027*x^7 + 2940961*x^6 - 15645211*x^5 - 122910997*x^4 - 309285975*x^3 - 271954775*x^2 + 1392333871*x + 4737567935, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} + 203 x^{13} + 249 x^{12} + 14443 x^{11} + 27391 x^{10} + 434633 x^{9} + 784389 x^{8} + 4734027 x^{7} + 2940961 x^{6} - 15645211 x^{5} - 122910997 x^{4} - 309285975 x^{3} - 271954775 x^{2} + 1392333871 x + 4737567935 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2890088708537260245720054366720000000000=-\,2^{18}\cdot 3^{13}\cdot 5^{10}\cdot 11^{13}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $427.29$
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:  $2, 3, 5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{10} a^{4} + \frac{3}{10} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{8} - \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{1}{2} a^{4} - \frac{2}{5} a^{3} - \frac{3}{10} a^{2} - \frac{1}{5} a$, $\frac{1}{20} a^{9} - \frac{1}{20} a^{8} - \frac{1}{5} a^{6} - \frac{1}{2} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{3}{20} a - \frac{1}{4}$, $\frac{1}{3300} a^{10} + \frac{1}{220} a^{9} + \frac{7}{330} a^{8} - \frac{17}{550} a^{7} - \frac{104}{825} a^{6} - \frac{48}{275} a^{5} - \frac{277}{825} a^{4} + \frac{69}{550} a^{3} + \frac{107}{3300} a^{2} + \frac{191}{1100} a + \frac{23}{330}$, $\frac{1}{3300} a^{11} + \frac{1}{330} a^{9} + \frac{1}{1100} a^{8} + \frac{31}{825} a^{7} + \frac{9}{550} a^{6} + \frac{68}{825} a^{5} + \frac{127}{275} a^{4} - \frac{493}{3300} a^{3} - \frac{117}{550} a^{2} + \frac{1}{66} a - \frac{13}{44}$, $\frac{1}{33000} a^{12} - \frac{1}{8250} a^{10} - \frac{31}{2750} a^{9} + \frac{1289}{33000} a^{8} - \frac{69}{2750} a^{7} + \frac{144}{1375} a^{6} + \frac{579}{2750} a^{5} - \frac{10721}{33000} a^{4} + \frac{641}{2750} a^{3} + \frac{809}{4125} a^{2} + \frac{512}{1375} a - \frac{2129}{6600}$, $\frac{1}{33000} a^{13} - \frac{1}{8250} a^{11} - \frac{1}{16500} a^{10} + \frac{239}{33000} a^{9} - \frac{166}{4125} a^{8} - \frac{107}{2750} a^{7} - \frac{634}{4125} a^{6} + \frac{7159}{33000} a^{5} + \frac{1279}{4125} a^{4} + \frac{313}{8250} a^{3} + \frac{7789}{16500} a^{2} - \frac{3287}{6600} a + \frac{13}{165}$, $\frac{1}{1883935260100296871783517397182329287495000} a^{14} - \frac{851048282424876725819575092661563889}{171266841827299715616683399743848117045000} a^{13} + \frac{2065397073605056567167029527133441251}{188393526010029687178351739718232928749500} a^{12} - \frac{42785231430612378264769545009477626449}{470983815025074217945879349295582321873750} a^{11} - \frac{134498872190703398568722207968237510399}{1883935260100296871783517397182329287495000} a^{10} + \frac{43487709387921963003544574628183683092133}{1883935260100296871783517397182329287495000} a^{9} - \frac{3909169139264853707998138286807496537317}{104663070005572048432417633176796071527500} a^{8} + \frac{2095354387867871149923365058544655848813}{78497302504179036324313224882597053645625} a^{7} + \frac{85813262801227648284715379596709649005477}{627978420033432290594505799060776429165000} a^{6} + \frac{444087875499286177415561338659103549975063}{1883935260100296871783517397182329287495000} a^{5} - \frac{18081858851716342000157300832252703478279}{188393526010029687178351739718232928749500} a^{4} + \frac{71115708055603671395173317314222198487116}{235491907512537108972939674647791160936875} a^{3} - \frac{380029917981967775661959991075017855537149}{1883935260100296871783517397182329287495000} a^{2} - \frac{522639392145509912556306218065244867765773}{1883935260100296871783517397182329287495000} a + \frac{55450777846256027384738197937281716549887}{188393526010029687178351739718232928749500}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  $7$
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:  \( 24048739853778.793 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.19140.1, 5.1.2371842000.3

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$