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Properties

Label 15.15.7360671657...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 5^{14}\cdot 19^{8}\cdot 37^{5}$
Root discriminant $114.23$
Ramified primes $2, 5, 19, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $((C_5^2 : C_3):C_2):C_2$ (as 15T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-504860, 1830056, 1193096, -5986612, 1483772, 5075634, -3320427, -282922, 640304, -90469, -29158, 6294, 459, -138, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 138*x^13 + 459*x^12 + 6294*x^11 - 29158*x^10 - 90469*x^9 + 640304*x^8 - 282922*x^7 - 3320427*x^6 + 5075634*x^5 + 1483772*x^4 - 5986612*x^3 + 1193096*x^2 + 1830056*x - 504860)
 
gp: K = bnfinit(x^15 - 2*x^14 - 138*x^13 + 459*x^12 + 6294*x^11 - 29158*x^10 - 90469*x^9 + 640304*x^8 - 282922*x^7 - 3320427*x^6 + 5075634*x^5 + 1483772*x^4 - 5986612*x^3 + 1193096*x^2 + 1830056*x - 504860, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 138 x^{13} + 459 x^{12} + 6294 x^{11} - 29158 x^{10} - 90469 x^{9} + 640304 x^{8} - 282922 x^{7} - 3320427 x^{6} + 5075634 x^{5} + 1483772 x^{4} - 5986612 x^{3} + 1193096 x^{2} + 1830056 x - 504860 \)

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:  \(7360671657636832731250000000000=2^{10}\cdot 5^{14}\cdot 19^{8}\cdot 37^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $114.23$
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:  $2, 5, 19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{10} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{73733212462749131512571558627366740532} a^{14} - \frac{94825093653059612454010765401505449}{857362935613361994332227425899613262} a^{13} + \frac{7372293302655608672665005224608108161}{36866606231374565756285779313683370266} a^{12} - \frac{8186733409238110869676461679339219851}{73733212462749131512571558627366740532} a^{11} + \frac{14951613641072906270162219675705064727}{36866606231374565756285779313683370266} a^{10} + \frac{6925157382707670592128302945665435989}{36866606231374565756285779313683370266} a^{9} + \frac{32198361760469164400565454426177230041}{73733212462749131512571558627366740532} a^{8} + \frac{7357701503838232385986333923438710079}{18433303115687282878142889656841685133} a^{7} + \frac{3494420492971841028262561892558295273}{36866606231374565756285779313683370266} a^{6} - \frac{15400980398053662979624807548043546261}{73733212462749131512571558627366740532} a^{5} - \frac{17337615617808222292558133386421945787}{36866606231374565756285779313683370266} a^{4} - \frac{1439476738569442990368663428317733460}{18433303115687282878142889656841685133} a^{3} + \frac{14310980699496800113484874564656228089}{36866606231374565756285779313683370266} a^{2} - \frac{336581355151647151686428093632441836}{18433303115687282878142889656841685133} a - \frac{5990118267069146510404947018304616611}{18433303115687282878142889656841685133}$

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:  \( 276528967045 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:D_6$ (as 15T18):

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 $((C_5^2 : C_3):C_2):C_2$
Character table for $((C_5^2 : C_3):C_2):C_2$

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 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$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$$[\ ]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.12.10$x^{10} + 10 x^{8} + 20 x^{7} + 15 x^{6} - 5 x^{5} + 5 x^{4} + 5 x^{2} - 5 x + 7$$5$$2$$12$$D_{10}$$[3/2]_{2}^{2}$
$19$19.5.0.1$x^{5} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
19.10.8.3$x^{10} + 57 x^{5} + 1444$$5$$2$$8$$D_5\times C_5$$[\ ]_{5}^{10}$
37Data not computed