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Properties

Label 15.15.4784085378...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 3^{20}\cdot 5^{26}\cdot 37^{8}$
Root discriminant $1109.99$
Ramified primes $2, 3, 5, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_5^2 : C_3):C_4$ (as 15T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4147330876900, 13359019608000, -12481912260000, 2393031337125, 454167461250, -149071347765, -5720024250, 3446799750, 29265150, -40279125, -51744, 251280, 0, -795, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 795*x^13 + 251280*x^11 - 51744*x^10 - 40279125*x^9 + 29265150*x^8 + 3446799750*x^7 - 5720024250*x^6 - 149071347765*x^5 + 454167461250*x^4 + 2393031337125*x^3 - 12481912260000*x^2 + 13359019608000*x + 4147330876900)
 
gp: K = bnfinit(x^15 - 795*x^13 + 251280*x^11 - 51744*x^10 - 40279125*x^9 + 29265150*x^8 + 3446799750*x^7 - 5720024250*x^6 - 149071347765*x^5 + 454167461250*x^4 + 2393031337125*x^3 - 12481912260000*x^2 + 13359019608000*x + 4147330876900, 1)
 

Normalized defining polynomial

\( x^{15} - 795 x^{13} + 251280 x^{11} - 51744 x^{10} - 40279125 x^{9} + 29265150 x^{8} + 3446799750 x^{7} - 5720024250 x^{6} - 149071347765 x^{5} + 454167461250 x^{4} + 2393031337125 x^{3} - 12481912260000 x^{2} + 13359019608000 x + 4147330876900 \)

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:  \(4784085378423726986844140625000000000000000000=2^{18}\cdot 3^{20}\cdot 5^{26}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1109.99$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{37} a^{8} - \frac{18}{37} a^{6} + \frac{13}{37} a^{4} - \frac{18}{37} a^{3}$, $\frac{1}{37} a^{9} - \frac{18}{37} a^{7} + \frac{13}{37} a^{5} - \frac{18}{37} a^{4}$, $\frac{1}{6845} a^{10} - \frac{11}{1369} a^{8} - \frac{323}{1369} a^{6} + \frac{3016}{6845} a^{5} - \frac{2}{37} a^{4} + \frac{17}{37} a^{3}$, $\frac{1}{6845} a^{11} - \frac{11}{1369} a^{9} - \frac{323}{1369} a^{7} + \frac{3016}{6845} a^{6} - \frac{2}{37} a^{5} + \frac{17}{37} a^{4}$, $\frac{1}{6845} a^{12} - \frac{3}{1369} a^{8} + \frac{3016}{6845} a^{7} - \frac{264}{1369} a^{6} - \frac{420}{1369} a^{5} - \frac{7}{37} a^{4} + \frac{4}{37} a^{3}$, $\frac{1}{3671076175} a^{13} + \frac{591}{19843655} a^{12} - \frac{20546}{734215235} a^{11} + \frac{594}{19843655} a^{10} - \frac{457569}{734215235} a^{9} + \frac{35873776}{3671076175} a^{8} + \frac{733142}{1526435} a^{7} + \frac{2882664}{19843655} a^{6} + \frac{6149984}{19843655} a^{5} - \frac{18960}{107263} a^{4} - \frac{6622}{41255} a^{3} + \frac{107}{2899} a^{2} + \frac{951}{2899} a - \frac{102}{2899}$, $\frac{1}{3198747216270959411776183274216581309451456941150} a^{14} + \frac{66431539069299779739688096369717549643}{1599373608135479705888091637108290654725728470575} a^{13} + \frac{13320545718547012205004795455433510204436897}{639749443254191882355236654843316261890291388230} a^{12} - \frac{12091337659791099193413687686600185335372646}{319874721627095941177618327421658130945145694115} a^{11} - \frac{10528908884223726829026946811321363732763472}{319874721627095941177618327421658130945145694115} a^{10} + \frac{17012083240989723985703397177599341762989234528}{1599373608135479705888091637108290654725728470575} a^{9} - \frac{32413209963135530063865280307609467483690470339}{3198747216270959411776183274216581309451456941150} a^{8} + \frac{3010806786618682449918565449617824731767134874}{8645262746678268680476171011396165701220153895} a^{7} - \frac{213080841846532638915764503589333179872012502}{8645262746678268680476171011396165701220153895} a^{6} + \frac{42054522614668129973741819890495132069366155}{1729052549335653736095234202279233140244030779} a^{5} - \frac{219459421012432716894893456212977562350465001}{467311499820446955701414649264657605471359670} a^{4} + \frac{56614391791242450964662155749770649994668307}{233655749910223477850707324632328802735679835} a^{3} + \frac{1233918759138680023383523882610384583702769}{2526008107137551111899538644673824894439782} a^{2} - \frac{288201430976824638108926392727675895204825}{1263004053568775555949769322336912447219891} a + \frac{140104077409499774335691797932908545076541}{1263004053568775555949769322336912447219891}$

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:  \( 6133343864530000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:(C_3:C_4)$ (as 15T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 300
The 8 conjugacy class representatives for $(C_5^2 : C_3):C_4$
Character table for $(C_5^2 : C_3):C_4$

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 sibling: data not computed
Degree 25 sibling: 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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$5.5.7.2$x^{5} + 10 x^{3} + 5$$5$$1$$7$$F_5$$[7/4]_{4}$
5.10.19.8$x^{10} - 10 x^{5} + 105$$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}$