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Properties

Label 15.3.54509897718...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{20}\cdot 5^{16}\cdot 17^{3}\cdot 37^{5}$
Root discriminant $82.37$
Ramified primes $2, 5, 17, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T82

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18064, -33960, -38120, -7135, 16200, 23614, 7840, -65, -4560, 300, -888, 255, -40, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 30*x^13 - 40*x^12 + 255*x^11 - 888*x^10 + 300*x^9 - 4560*x^8 - 65*x^7 + 7840*x^6 + 23614*x^5 + 16200*x^4 - 7135*x^3 - 38120*x^2 - 33960*x - 18064)
 
gp: K = bnfinit(x^15 + 30*x^13 - 40*x^12 + 255*x^11 - 888*x^10 + 300*x^9 - 4560*x^8 - 65*x^7 + 7840*x^6 + 23614*x^5 + 16200*x^4 - 7135*x^3 - 38120*x^2 - 33960*x - 18064, 1)
 

Normalized defining polynomial

\( x^{15} + 30 x^{13} - 40 x^{12} + 255 x^{11} - 888 x^{10} + 300 x^{9} - 4560 x^{8} - 65 x^{7} + 7840 x^{6} + 23614 x^{5} + 16200 x^{4} - 7135 x^{3} - 38120 x^{2} - 33960 x - 18064 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(54509897718560000000000000000=2^{20}\cdot 5^{16}\cdot 17^{3}\cdot 37^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.37$
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:  $2, 5, 17, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{3}{16} a^{5} - \frac{3}{16} a^{4} + \frac{3}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{7} + \frac{1}{16} a^{3}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{11} + \frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{5}{32} a^{4} - \frac{5}{32} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{320} a^{13} + \frac{1}{80} a^{12} - \frac{3}{320} a^{11} - \frac{1}{40} a^{10} + \frac{1}{32} a^{9} - \frac{1}{20} a^{8} + \frac{13}{160} a^{7} - \frac{1}{10} a^{6} - \frac{67}{320} a^{5} - \frac{1}{16} a^{4} - \frac{23}{320} a^{3} - \frac{9}{40} a^{2} - \frac{17}{40} a - \frac{1}{20}$, $\frac{1}{404108701332756435520} a^{14} - \frac{347269143599118323}{404108701332756435520} a^{13} + \frac{1571061465427738109}{404108701332756435520} a^{12} - \frac{10750016880925777807}{404108701332756435520} a^{11} + \frac{4302075449603654613}{202054350666378217760} a^{10} + \frac{7751244051895451177}{202054350666378217760} a^{9} + \frac{24977540812380032149}{202054350666378217760} a^{8} - \frac{2693573278647866827}{202054350666378217760} a^{7} + \frac{32835416922143780997}{404108701332756435520} a^{6} - \frac{19231747832907960311}{404108701332756435520} a^{5} - \frac{99000780059135074183}{404108701332756435520} a^{4} + \frac{80067219723061009269}{404108701332756435520} a^{3} + \frac{4990809700652666719}{12628396916648638610} a^{2} + \frac{21513711374084764477}{50513587666594554440} a - \frac{7490620881210991793}{25256793833297277220}$

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:  $8$
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:  \( 1068664983.46 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T82:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48000
The 65 conjugacy class representatives for [F(5)^3]S(3)=F(5)wrS(3) are not computed
Character table for [F(5)^3]S(3)=F(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ R ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.18.76$x^{12} + 2 x^{11} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{4} + 2 x^{2} + 2$$12$$1$$18$12T95$[4/3, 4/3, 5/3, 5/3, 2]_{3}^{2}$
$5$5.5.6.3$x^{5} + 5 x^{2} + 5$$5$$1$$6$$F_5$$[3/2]_{2}^{2}$
5.10.10.10$x^{10} + 10 x^{8} + 5 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 2$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
37Data not computed