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Properties

Label 15.15.2893656810...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 5^{15}\cdot 19^{5}\cdot 73^{8}\cdot 103^{5}$
Root discriminant $1073.40$
Ramified primes $2, 5, 19, 73, 103$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 15T68

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2806200582656, 290797987840, -1090492454400, 63114118080, 139423692800, -24576836416, -4314358400, 1125484800, 43344480, -20002000, -119968, 171320, 0, -680, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 680*x^13 + 171320*x^11 - 119968*x^10 - 20002000*x^9 + 43344480*x^8 + 1125484800*x^7 - 4314358400*x^6 - 24576836416*x^5 + 139423692800*x^4 + 63114118080*x^3 - 1090492454400*x^2 + 290797987840*x + 2806200582656)
 
gp: K = bnfinit(x^15 - 680*x^13 + 171320*x^11 - 119968*x^10 - 20002000*x^9 + 43344480*x^8 + 1125484800*x^7 - 4314358400*x^6 - 24576836416*x^5 + 139423692800*x^4 + 63114118080*x^3 - 1090492454400*x^2 + 290797987840*x + 2806200582656, 1)
 

Normalized defining polynomial

\( x^{15} - 680 x^{13} + 171320 x^{11} - 119968 x^{10} - 20002000 x^{9} + 43344480 x^{8} + 1125484800 x^{7} - 4314358400 x^{6} - 24576836416 x^{5} + 139423692800 x^{4} + 63114118080 x^{3} - 1090492454400 x^{2} + 290797987840 x + 2806200582656 \)

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:  \(2893656810637015092296465010514625000000000000=2^{12}\cdot 5^{15}\cdot 19^{5}\cdot 73^{8}\cdot 103^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1073.40$
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:  $2, 5, 19, 73, 103$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{2} a$, $\frac{1}{2336} a^{8} - \frac{3}{73} a^{6} - \frac{1}{8} a^{5} + \frac{13}{146} a^{4} + \frac{21}{146} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{2336} a^{9} + \frac{25}{1168} a^{7} + \frac{13}{146} a^{5} - \frac{31}{292} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{341056} a^{10} + \frac{25}{170528} a^{8} + \frac{1997}{42632} a^{6} + \frac{991}{42632} a^{5} - \frac{45}{584} a^{4} - \frac{14}{73} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{341056} a^{11} + \frac{25}{170528} a^{9} - \frac{1335}{85264} a^{7} + \frac{991}{42632} a^{6} - \frac{45}{584} a^{5} + \frac{17}{292} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{682112} a^{12} + \frac{11}{170528} a^{8} + \frac{991}{85264} a^{7} + \frac{4759}{85264} a^{6} + \frac{3111}{42632} a^{5} - \frac{13}{292} a^{4} - \frac{21}{292} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{99588352} a^{13} + \frac{25}{49794176} a^{11} - \frac{1335}{24897088} a^{9} + \frac{991}{12448544} a^{8} + \frac{4773}{170528} a^{7} - \frac{421}{85264} a^{6} + \frac{45}{1168} a^{5} + \frac{71}{584} a^{4} + \frac{13}{584} a^{3} - \frac{1}{2} a$, $\frac{1}{4982468289271383617797586176} a^{14} - \frac{3334547439517812943}{2491234144635691808898793088} a^{13} - \frac{14040303209792549681}{2491234144635691808898793088} a^{12} + \frac{276452433012552242337}{622808536158922952224698272} a^{11} - \frac{1793257350921940428143}{1245617072317845904449396544} a^{10} - \frac{114784489809028651202603}{622808536158922952224698272} a^{9} - \frac{90278165955019153770239}{622808536158922952224698272} a^{8} + \frac{52278119811269445711741}{2132905945749736137755816} a^{7} - \frac{229852816270096983755737}{4265811891499472275511632} a^{6} + \frac{241982670229045703743381}{2132905945749736137755816} a^{5} - \frac{1441255193280637174307}{29217889667804604626792} a^{4} - \frac{51335127470635166051}{7304472416951151156698} a^{3} - \frac{23959649692338815811}{100061265985632207626} a^{2} + \frac{8201450185620543302}{50030632992816103813} a - \frac{14834946159419081449}{50030632992816103813}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 2647558800940000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T68:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12000
The 38 conjugacy class representatives for [D(5)^3:2]S(3)
Character table for [D(5)^3:2]S(3) is not computed

Intermediate fields

3.3.1957.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$5$5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.8$x^{10} + 10 x^{8} + 20 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 22$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.10.8.1$x^{10} - 73 x^{5} + 58619$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
103Data not computed