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Properties

Label 15.15.1876506681...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{10}\cdot 151^{2}\cdot 3947^{2}\cdot 24181^{2}$
Root discriminant $894.45$
Ramified primes $2, 5, 13, 151, 3947, 24181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T67

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-798463067, 2758438455, -2526579387, 201820423, 632388425, -163395653, -58060643, 19828675, 2504704, -1041392, -51724, 26096, 453, -285, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 285*x^13 + 453*x^12 + 26096*x^11 - 51724*x^10 - 1041392*x^9 + 2504704*x^8 + 19828675*x^7 - 58060643*x^6 - 163395653*x^5 + 632388425*x^4 + 201820423*x^3 - 2526579387*x^2 + 2758438455*x - 798463067)
 
gp: K = bnfinit(x^15 - x^14 - 285*x^13 + 453*x^12 + 26096*x^11 - 51724*x^10 - 1041392*x^9 + 2504704*x^8 + 19828675*x^7 - 58060643*x^6 - 163395653*x^5 + 632388425*x^4 + 201820423*x^3 - 2526579387*x^2 + 2758438455*x - 798463067, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 285 x^{13} + 453 x^{12} + 26096 x^{11} - 51724 x^{10} - 1041392 x^{9} + 2504704 x^{8} + 19828675 x^{7} - 58060643 x^{6} - 163395653 x^{5} + 632388425 x^{4} + 201820423 x^{3} - 2526579387 x^{2} + 2758438455 x - 798463067 \)

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:  \(187650668178675410163314785843943833600000000=2^{24}\cdot 5^{8}\cdot 13^{10}\cdot 151^{2}\cdot 3947^{2}\cdot 24181^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $894.45$
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:  $2, 5, 13, 151, 3947, 24181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{369314152021811705248598126865621648035367939850876} a^{14} - \frac{860868441806849479296293293401438422524595374653}{184657076010905852624299063432810824017683969925438} a^{13} - \frac{73175306693502552516321117975793766776589886192571}{369314152021811705248598126865621648035367939850876} a^{12} - \frac{2231178738991803603521459464336774351336876507297}{92328538005452926312149531716405412008841984962719} a^{11} + \frac{16760861987170134604494341783050744857252802499642}{92328538005452926312149531716405412008841984962719} a^{10} - \frac{2416642608618618838152840735205178875083239608816}{92328538005452926312149531716405412008841984962719} a^{9} + \frac{8612707291741669798667899417175820237289135227075}{92328538005452926312149531716405412008841984962719} a^{8} + \frac{25718723590156081509874000762923021386971575750614}{92328538005452926312149531716405412008841984962719} a^{7} - \frac{159020744022581367126649607536331574162733083125761}{369314152021811705248598126865621648035367939850876} a^{6} - \frac{32357785467570991152005277600704897124492320499281}{184657076010905852624299063432810824017683969925438} a^{5} - \frac{165840309663946486389637630983734598374543449777683}{369314152021811705248598126865621648035367939850876} a^{4} + \frac{20733824093096679977885424527563482595302580258410}{92328538005452926312149531716405412008841984962719} a^{3} + \frac{76204016194430340135367328863104830134221297216419}{369314152021811705248598126865621648035367939850876} a^{2} - \frac{263318099769668868133526730310068947922049284385}{184657076010905852624299063432810824017683969925438} a - \frac{58457302547631289847597705105863598893510847607579}{369314152021811705248598126865621648035367939850876}$

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:  \( 442531383487000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T67:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12000
The 32 conjugacy class representatives for [1/2.F(5)^3]3
Character table for [1/2.F(5)^3]3 is not computed

Intermediate fields

3.3.169.1

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 $15$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $15$

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$$[\ ]^{3}$
2.12.24.225$x^{12} - 12 x^{11} + 16 x^{10} - 4 x^{9} - 10 x^{8} + 16 x^{7} - 8 x^{4} - 8 x^{2} + 8$$4$$3$$24$12T55$[2, 2, 3, 3]^{6}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
151.4.2.2$x^{4} - 151 x^{2} + 273612$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
151.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
3947Data not computed
24181Data not computed