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Properties

Label 15.3.85126572290...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{10}\cdot 3^{20}\cdot 5^{22}$
Root discriminant $72.78$
Ramified primes $2, 3, 5$
Class number $5$ (GRH)
Class group $[5]$ (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![215, -525, -75, 650, 150, -480, -105, 450, 30, -275, -3, 90, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^13 + 90*x^11 - 3*x^10 - 275*x^9 + 30*x^8 + 450*x^7 - 105*x^6 - 480*x^5 + 150*x^4 + 650*x^3 - 75*x^2 - 525*x + 215)
 
gp: K = bnfinit(x^15 - 15*x^13 + 90*x^11 - 3*x^10 - 275*x^9 + 30*x^8 + 450*x^7 - 105*x^6 - 480*x^5 + 150*x^4 + 650*x^3 - 75*x^2 - 525*x + 215, 1)
 

Normalized defining polynomial

\( x^{15} - 15 x^{13} + 90 x^{11} - 3 x^{10} - 275 x^{9} + 30 x^{8} + 450 x^{7} - 105 x^{6} - 480 x^{5} + 150 x^{4} + 650 x^{3} - 75 x^{2} - 525 x + 215 \)

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:  \(8512657229003906250000000000=2^{10}\cdot 3^{20}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.78$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{2}{9} a^{4} - \frac{4}{9} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{18} a^{10} - \frac{1}{9} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{4}{9} a + \frac{1}{6}$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{4}{9} a^{4} + \frac{7}{18} a^{3} + \frac{1}{9} a^{2} + \frac{7}{18} a - \frac{4}{9}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{7}{18} a^{4} + \frac{4}{9} a^{3} + \frac{7}{18} a^{2} + \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{54} a^{13} + \frac{1}{54} a^{12} - \frac{1}{54} a^{11} - \frac{1}{54} a^{9} - \frac{1}{54} a^{8} + \frac{5}{54} a^{7} - \frac{1}{27} a^{6} + \frac{1}{18} a^{5} + \frac{25}{54} a^{4} - \frac{10}{27} a^{3} + \frac{5}{54} a^{2} - \frac{13}{54} a - \frac{1}{27}$, $\frac{1}{54} a^{14} + \frac{1}{54} a^{12} + \frac{1}{54} a^{11} - \frac{1}{54} a^{10} - \frac{1}{18} a^{8} - \frac{7}{54} a^{7} - \frac{7}{54} a^{6} + \frac{2}{27} a^{5} + \frac{1}{3} a^{4} + \frac{1}{54} a^{3} - \frac{1}{2} a^{2} + \frac{11}{54} a + \frac{4}{27}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 80805653.0372 \) (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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$3.3.4.4$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$$[7/4]_{4}$
5.10.15.2$x^{10} - 15 x^{6} + 5$$10$$1$$15$$F_5$$[7/4]_{4}$