↑

Properties

Label 15.15.2880296731...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 5^{15}\cdot 7^{10}\cdot 13^{8}$
Root discriminant $125.11$
Ramified primes $2, 5, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_5^2 : C_4):C_3$ (as 15T19)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-228488, -1142440, 1428050, 4009525, -329550, -2226068, -11830, 511225, 4810, -57850, -192, 3365, 0, -95, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 95*x^13 + 3365*x^11 - 192*x^10 - 57850*x^9 + 4810*x^8 + 511225*x^7 - 11830*x^6 - 2226068*x^5 - 329550*x^4 + 4009525*x^3 + 1428050*x^2 - 1142440*x - 228488)
 
gp: K = bnfinit(x^15 - 95*x^13 + 3365*x^11 - 192*x^10 - 57850*x^9 + 4810*x^8 + 511225*x^7 - 11830*x^6 - 2226068*x^5 - 329550*x^4 + 4009525*x^3 + 1428050*x^2 - 1142440*x - 228488, 1)
 

Normalized defining polynomial

\( x^{15} - 95 x^{13} + 3365 x^{11} - 192 x^{10} - 57850 x^{9} + 4810 x^{8} + 511225 x^{7} - 11830 x^{6} - 2226068 x^{5} - 329550 x^{4} + 4009525 x^{3} + 1428050 x^{2} - 1142440 x - 228488 \)

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:  \(28802967316428066125000000000000=2^{12}\cdot 5^{15}\cdot 7^{10}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.11$
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:  $2, 5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13} a^{8} - \frac{4}{13} a^{6} - \frac{2}{13} a^{4} + \frac{3}{13} a^{3}$, $\frac{1}{91} a^{9} + \frac{1}{91} a^{8} - \frac{17}{91} a^{7} + \frac{9}{91} a^{6} - \frac{2}{91} a^{5} - \frac{25}{91} a^{4} + \frac{6}{13} a^{3} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{1183} a^{10} - \frac{4}{1183} a^{8} - \frac{2}{7} a^{7} + \frac{297}{1183} a^{6} - \frac{23}{1183} a^{5} - \frac{18}{91} a^{4} + \frac{2}{13} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{1183} a^{11} - \frac{4}{1183} a^{9} + \frac{2}{91} a^{8} + \frac{297}{1183} a^{7} - \frac{296}{1183} a^{6} - \frac{18}{91} a^{5} - \frac{6}{13} a^{4} + \frac{32}{91} a^{3} - \frac{2}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{1183} a^{12} - \frac{18}{1183} a^{8} - \frac{23}{1183} a^{7} - \frac{554}{1183} a^{6} - \frac{586}{1183} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{30758} a^{13} + \frac{9}{30758} a^{11} - \frac{93}{30758} a^{9} - \frac{96}{15379} a^{8} + \frac{107}{1183} a^{7} - \frac{583}{1183} a^{6} + \frac{83}{182} a^{5} + \frac{1}{13} a^{4} + \frac{5}{91} a^{3} + \frac{3}{7} a^{2} - \frac{5}{14} a + \frac{2}{7}$, $\frac{1}{450658022288473636} a^{14} + \frac{31679803485}{32189858734890974} a^{13} - \frac{9210987566447}{450658022288473636} a^{12} + \frac{91474816965613}{225329011144236818} a^{11} + \frac{6994570106197}{450658022288473636} a^{10} - \frac{286202104739821}{225329011144236818} a^{9} + \frac{459731747542069}{32189858734890974} a^{8} + \frac{2713993871059165}{17333000857248986} a^{7} - \frac{878014145184627}{2666615516499844} a^{6} - \frac{3652939564343217}{8666500428624493} a^{5} + \frac{26444790129736}{95236268446423} a^{4} - \frac{199051052153185}{1333307758249922} a^{3} + \frac{51945105656893}{205124270499988} a^{2} - \frac{519188046277}{51281067624997} a + \frac{24130435901552}{51281067624997}$

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:  \( 94158911826.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:C_{12}$ (as 15T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 300
The 14 conjugacy class representatives for $(C_5^2 : C_4):C_3$
Character table for $(C_5^2 : C_4):C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$