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Properties

Label 15.15.1073301297...9296.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{21}\cdot 3^{20}\cdot 7^{12}\cdot 13^{9}$
Root discriminant $252.38$
Ramified primes $2, 3, 7, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T52

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2913293824, 72902592, 2032974720, 124252044, -529878552, -67895478, 62607300, 11200371, -3305736, -724486, 76020, 20664, -616, -252, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 252*x^13 - 616*x^12 + 20664*x^11 + 76020*x^10 - 724486*x^9 - 3305736*x^8 + 11200371*x^7 + 62607300*x^6 - 67895478*x^5 - 529878552*x^4 + 124252044*x^3 + 2032974720*x^2 + 72902592*x - 2913293824)
 
gp: K = bnfinit(x^15 - 252*x^13 - 616*x^12 + 20664*x^11 + 76020*x^10 - 724486*x^9 - 3305736*x^8 + 11200371*x^7 + 62607300*x^6 - 67895478*x^5 - 529878552*x^4 + 124252044*x^3 + 2032974720*x^2 + 72902592*x - 2913293824, 1)
 

Normalized defining polynomial

\( x^{15} - 252 x^{13} - 616 x^{12} + 20664 x^{11} + 76020 x^{10} - 724486 x^{9} - 3305736 x^{8} + 11200371 x^{7} + 62607300 x^{6} - 67895478 x^{5} - 529878552 x^{4} + 124252044 x^{3} + 2032974720 x^{2} + 72902592 x - 2913293824 \)

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:  \(1073301297242305974749981302950199296=2^{21}\cdot 3^{20}\cdot 7^{12}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $252.38$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{14} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{14} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{28} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{28} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{28} a^{10} - \frac{1}{28} a^{6}$, $\frac{1}{56} a^{11} - \frac{1}{56} a^{9} + \frac{1}{56} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{392} a^{12} - \frac{1}{56} a^{10} - \frac{1}{56} a^{8} + \frac{1}{56} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{784} a^{13} + \frac{1}{56} a^{7} + \frac{3}{16} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{2116354438732059109122968510857014240512} a^{14} + \frac{179309925957833846664303074675970707}{529088609683014777280742127714253560128} a^{13} + \frac{84373847120583613966642328781692267}{75584087097573539611534589673464794304} a^{12} - \frac{288122246950156484440120935607266337}{37792043548786769805767294836732397152} a^{11} + \frac{556168924776102681551188575598475761}{37792043548786769805767294836732397152} a^{10} + \frac{45440216806377427371108950993085715}{75584087097573539611534589673464794304} a^{9} - \frac{226773206577656533177186169294554221}{151168174195147079223069179346929588608} a^{8} + \frac{490229555066368139881438342620850229}{18896021774393384902883647418366198576} a^{7} - \frac{4249710499195676628177846680590414187}{302336348390294158446138358693859177216} a^{6} - \frac{899371516241570062819549895304330995}{5398863364112395686538184976676056736} a^{5} + \frac{3754418613209421761433216367517983557}{21595453456449582746152739906704226944} a^{4} - \frac{379487768335642947833902956957306549}{1349715841028098921634546244169014184} a^{3} - \frac{1625732358516630314002803491921384045}{10797726728224791373076369953352113472} a^{2} - \frac{1284412476300192839460999987799514863}{2699431682056197843269092488338028368} a - \frac{73606248074983378567096864735715707}{337428960257024730408636561042253546}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 35534230107200 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T52:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3240
The 24 conjugacy class representatives for [3^4:2]F(5)
Character table for [3^4:2]F(5) is not computed

Intermediate fields

5.5.6889792.1

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 30 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$