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Properties

Label 15.1.16084824218...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 5^{19}\cdot 7^{7}$
Root discriminant $30.23$
Ramified primes $2, 5, 7$
Class number $1$
Class group Trivial
Galois group $D_{15}$ (as 15T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-100, 500, -1100, 500, 350, -480, -475, 110, 200, 75, -36, -20, -15, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^12 - 20*x^11 - 36*x^10 + 75*x^9 + 200*x^8 + 110*x^7 - 475*x^6 - 480*x^5 + 350*x^4 + 500*x^3 - 1100*x^2 + 500*x - 100)
 
gp: K = bnfinit(x^15 - 15*x^12 - 20*x^11 - 36*x^10 + 75*x^9 + 200*x^8 + 110*x^7 - 475*x^6 - 480*x^5 + 350*x^4 + 500*x^3 - 1100*x^2 + 500*x - 100, 1)
 

Normalized defining polynomial

\( x^{15} - 15 x^{12} - 20 x^{11} - 36 x^{10} + 75 x^{9} + 200 x^{8} + 110 x^{7} - 475 x^{6} - 480 x^{5} + 350 x^{4} + 500 x^{3} - 1100 x^{2} + 500 x - 100 \)

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:  \(-16084824218750000000000=-\,2^{10}\cdot 5^{19}\cdot 7^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.23$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{2}{7}$, $\frac{1}{35} a^{10} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{6}{35} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{35} a^{11} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} - \frac{6}{35} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{2450} a^{12} + \frac{2}{245} a^{10} + \frac{33}{490} a^{9} + \frac{38}{245} a^{8} + \frac{387}{1225} a^{7} + \frac{25}{98} a^{6} + \frac{78}{245} a^{5} - \frac{32}{245} a^{4} - \frac{47}{98} a^{3} + \frac{101}{245} a^{2} - \frac{5}{49} a - \frac{1}{49}$, $\frac{1}{2450} a^{13} + \frac{2}{245} a^{11} + \frac{1}{98} a^{10} + \frac{3}{245} a^{9} - \frac{313}{1225} a^{8} - \frac{45}{98} a^{7} - \frac{27}{245} a^{6} + \frac{17}{245} a^{5} + \frac{37}{98} a^{4} - \frac{109}{245} a^{3} + \frac{23}{49} a^{2} - \frac{15}{49} a + \frac{3}{7}$, $\frac{1}{57814075900} a^{14} - \frac{266773}{28907037950} a^{13} - \frac{2473783}{14453518975} a^{12} + \frac{46691087}{11562815180} a^{11} + \frac{8900921}{825915370} a^{10} - \frac{595827459}{14453518975} a^{9} - \frac{3016907239}{57814075900} a^{8} - \frac{11753366159}{28907037950} a^{7} + \frac{1927230831}{5781407590} a^{6} - \frac{256681531}{1651830740} a^{5} + \frac{55619131}{1156281518} a^{4} + \frac{98952169}{5781407590} a^{3} + \frac{362742193}{5781407590} a^{2} - \frac{103324541}{578140759} a - \frac{39540189}{578140759}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 913440.648087 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{15}$ (as 15T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.140.1, 5.1.765625.1

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

Sibling fields

Galois closure: 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 $15$ R R $15$ $15$ $15$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$5.5.6.1$x^{5} + 10 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
5.10.13.7$x^{10} + 5 x^{4} + 10$$10$$1$$13$$D_5$$[3/2]_{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$