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Properties

Label 15.15.1196021344...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 3^{20}\cdot 5^{28}\cdot 37^{8}$
Root discriminant $1375.68$
Ramified primes $2, 3, 5, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T37

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5078976310000, 8995972800000, -4216862250000, -59390642500, 448279050000, -62996451600, -13150614000, 2895435000, 108891000, -43382500, -229440, 295200, 0, -900, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 900*x^13 + 295200*x^11 - 229440*x^10 - 43382500*x^9 + 108891000*x^8 + 2895435000*x^7 - 13150614000*x^6 - 62996451600*x^5 + 448279050000*x^4 - 59390642500*x^3 - 4216862250000*x^2 + 8995972800000*x - 5078976310000)
 
gp: K = bnfinit(x^15 - 900*x^13 + 295200*x^11 - 229440*x^10 - 43382500*x^9 + 108891000*x^8 + 2895435000*x^7 - 13150614000*x^6 - 62996451600*x^5 + 448279050000*x^4 - 59390642500*x^3 - 4216862250000*x^2 + 8995972800000*x - 5078976310000, 1)
 

Normalized defining polynomial

\( x^{15} - 900 x^{13} + 295200 x^{11} - 229440 x^{10} - 43382500 x^{9} + 108891000 x^{8} + 2895435000 x^{7} - 13150614000 x^{6} - 62996451600 x^{5} + 448279050000 x^{4} - 59390642500 x^{3} - 4216862250000 x^{2} + 8995972800000 x - 5078976310000 \)

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:  \(119602134460593174671103515625000000000000000000=2^{18}\cdot 3^{20}\cdot 5^{28}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1375.68$
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:  $2, 3, 5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{10} a^{6}$, $\frac{1}{10} a^{7}$, $\frac{1}{3700} a^{8} - \frac{8}{185} a^{6} - \frac{1}{10} a^{5} - \frac{8}{37} a^{4} - \frac{76}{185} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{3700} a^{9} - \frac{8}{185} a^{7} - \frac{3}{185} a^{5} - \frac{76}{185} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{136900} a^{10} - \frac{3}{34225} a^{8} + \frac{253}{13690} a^{6} - \frac{104}{1369} a^{5} + \frac{31}{74} a^{4} - \frac{84}{185} a^{3}$, $\frac{1}{136900} a^{11} - \frac{3}{34225} a^{9} + \frac{253}{13690} a^{7} + \frac{329}{13690} a^{6} + \frac{7}{370} a^{5} - \frac{84}{185} a^{4}$, $\frac{1}{273800} a^{12} + \frac{9}{136900} a^{8} - \frac{52}{1369} a^{7} + \frac{113}{27380} a^{6} + \frac{47}{2738} a^{5} + \frac{16}{37} a^{4} - \frac{29}{370} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2380691000} a^{13} - \frac{1}{2573720} a^{12} + \frac{613}{238069100} a^{11} + \frac{11}{3217150} a^{10} + \frac{8763}{238069100} a^{9} - \frac{26163}{238069100} a^{8} + \frac{651}{1286860} a^{7} - \frac{26481}{1286860} a^{6} - \frac{29362}{321715} a^{5} + \frac{7291}{17390} a^{4} + \frac{3284}{8695} a^{3} - \frac{33}{94} a^{2} + \frac{17}{47} a - \frac{16}{47}$, $\frac{1}{22560825674707011600569799771377891409000} a^{14} + \frac{133544045697347107553023264013}{902433026988280464022791990855115656360} a^{13} - \frac{4787521799433778190891306718850933}{4512165134941402320113959954275578281800} a^{12} - \frac{3497379018974427881858635014495823}{2256082567470701160056979977137789140900} a^{11} + \frac{3492770461737213157000293697394627}{1128041283735350580028489988568894570450} a^{10} + \frac{113467622334279201852140864644180309}{1128041283735350580028489988568894570450} a^{9} - \frac{4642205410706040060013939771221779}{37601376124511686000949666285629819015} a^{8} + \frac{1158013230488265029304762599763721}{239118449122490848972652885759172140} a^{7} + \frac{133651956979823651640111854970819193}{4065013635082344432535099057905926380} a^{6} - \frac{20719915671296431059811829535796825}{406501363508234443253509905790592638} a^{5} - \frac{773841912663243130894014886084361}{10986523338060390358202970426772774} a^{4} + \frac{1657081632440856640223443787213745}{10986523338060390358202970426772774} a^{3} + \frac{14377571936465552944564200351979}{890799189572464083097538142711306} a^{2} + \frac{201707311517380004021602847862352}{445399594786232041548769071355653} a - \frac{29173069931089375882862011037887}{445399594786232041548769071355653}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 25933506913100000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T37:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1500
The 19 conjugacy class representatives for 1/2[5^3:4]S(3)
Character table for 1/2[5^3:4]S(3)

Intermediate fields

3.3.1620.1

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

Sibling fields

Degree 15 siblings: data not computed
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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ R $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $15$

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$$[\ ]_{3}^{2}$
2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.12.16.30$x^{12} + 93 x^{11} + 351 x^{10} + 3 x^{9} + 126 x^{8} - 297 x^{7} + 171 x^{6} + 243 x^{5} - 324 x^{4} - 54 x^{3} + 162 x^{2} - 243 x + 324$$3$$4$$16$$C_3 : C_4$$[2]^{4}$
$5$5.5.9.4$x^{5} + 30$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.14$x^{10} - 10 x^{5} + 80$$10$$1$$19$$C_5^2 : C_4$$[7/4, 9/4]_{4}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.10.8.1$x^{10} - 37 x^{5} + 6845$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$