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Properties

Label 15.1.25814526680...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 61^{13}$
Root discriminant $424.09$
Ramified primes $2, 3, 5, 61$
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![-280715886, 519196590, -277582410, 69494445, 13641630, 2047624, -71525, 767775, 75600, -22575, 1086, 1185, 135, -45, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 45*x^13 + 135*x^12 + 1185*x^11 + 1086*x^10 - 22575*x^9 + 75600*x^8 + 767775*x^7 - 71525*x^6 + 2047624*x^5 + 13641630*x^4 + 69494445*x^3 - 277582410*x^2 + 519196590*x - 280715886)
 
gp: K = bnfinit(x^15 - 5*x^14 - 45*x^13 + 135*x^12 + 1185*x^11 + 1086*x^10 - 22575*x^9 + 75600*x^8 + 767775*x^7 - 71525*x^6 + 2047624*x^5 + 13641630*x^4 + 69494445*x^3 - 277582410*x^2 + 519196590*x - 280715886, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 45 x^{13} + 135 x^{12} + 1185 x^{11} + 1086 x^{10} - 22575 x^{9} + 75600 x^{8} + 767775 x^{7} - 71525 x^{6} + 2047624 x^{5} + 13641630 x^{4} + 69494445 x^{3} - 277582410 x^{2} + 519196590 x - 280715886 \)

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:  \(-2581452668046920778582615126630000000000=-\,2^{10}\cdot 3^{13}\cdot 5^{10}\cdot 61^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $424.09$
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:  $2, 3, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{15} a^{8} + \frac{1}{15} a^{6} - \frac{1}{3} a^{5} + \frac{2}{5} a^{4} - \frac{1}{3} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{15} a^{9} + \frac{1}{15} a^{7} + \frac{2}{5} a^{5} - \frac{1}{3} a^{4} - \frac{4}{15} a^{3} + \frac{2}{5} a$, $\frac{1}{30195} a^{10} + \frac{196}{6039} a^{9} + \frac{166}{30195} a^{8} - \frac{805}{6039} a^{7} - \frac{4069}{30195} a^{6} + \frac{1043}{6039} a^{5} - \frac{2878}{10065} a^{4} - \frac{131}{671} a^{3} + \frac{214}{3355} a^{2} + \frac{1}{671} a - \frac{28}{61}$, $\frac{1}{30195} a^{11} - \frac{1}{915} a^{9} + \frac{34}{2745} a^{8} - \frac{31}{915} a^{7} + \frac{31}{305} a^{6} - \frac{1123}{2745} a^{5} - \frac{97}{915} a^{4} - \frac{131}{915} a^{3} - \frac{94}{305} a^{2} - \frac{401}{3355} a + \frac{11}{305}$, $\frac{1}{150975} a^{12} + \frac{2}{150975} a^{11} + \frac{1}{150975} a^{10} + \frac{284}{30195} a^{9} - \frac{134}{30195} a^{8} + \frac{614}{30195} a^{7} + \frac{1367}{10065} a^{6} + \frac{1103}{10065} a^{5} - \frac{186}{3355} a^{4} - \frac{602}{10065} a^{3} + \frac{71}{1525} a^{2} + \frac{1502}{16775} a - \frac{529}{1525}$, $\frac{1}{45745425} a^{13} - \frac{17}{9149085} a^{12} - \frac{293}{45745425} a^{11} + \frac{293}{45745425} a^{10} + \frac{303542}{9149085} a^{9} - \frac{2624}{149985} a^{8} + \frac{1514972}{9149085} a^{7} + \frac{280321}{9149085} a^{6} + \frac{1909487}{9149085} a^{5} - \frac{1330088}{3049695} a^{4} + \frac{94467}{1694275} a^{3} + \frac{263597}{1016565} a^{2} - \frac{1073233}{5082825} a - \frac{88162}{462075}$, $\frac{1}{339611968928518136017311466607175} a^{14} + \frac{1051018889849294002837972}{339611968928518136017311466607175} a^{13} - \frac{116322286784275647558781973}{67922393785703627203462293321435} a^{12} - \frac{1589679653841230506452030347}{339611968928518136017311466607175} a^{11} + \frac{898099184353085335415377589}{339611968928518136017311466607175} a^{10} + \frac{1442443089088321188795090942524}{67922393785703627203462293321435} a^{9} + \frac{2028067389301266576972442669231}{67922393785703627203462293321435} a^{8} + \frac{11157242606009519774332014609149}{67922393785703627203462293321435} a^{7} - \frac{10932922020962194284739607372639}{67922393785703627203462293321435} a^{6} - \frac{21137473721065971392599942126376}{67922393785703627203462293321435} a^{5} + \frac{38148419214442310167805930717218}{113203989642839378672437155535725} a^{4} - \frac{4906109480836124716744465633776}{12578221071426597630270795059525} a^{3} - \frac{2892582376899088283340132005734}{7546932642855958578162477035715} a^{2} + \frac{1628339911451620068909470861683}{3430423928570890262801125925325} a + \frac{143588189491765968992877607319}{311856720779171842072829629575}$

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:  \( 40164614069800.03 \) (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

3.1.3660.1, 5.1.140189140125.1

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

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{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}$
$61$61.5.4.3$x^{5} - 244$$5$$1$$4$$C_5$$[\ ]_{5}$
61.10.9.2$x^{10} - 244$$10$$1$$9$$C_{10}$$[\ ]_{10}$