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Properties

Label 15.15.1005736322...9824.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 3^{20}\cdot 881^{10}$
Root discriminant $631.20$
Ramified primes $2, 3, 881$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T53

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![707492484008, 1114933543992, -2564442153288, 924353400348, 193726444848, -105940954800, -2014400102, 3590313594, -37019484, -52610105, 471300, 347247, -1282, -1005, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 1005*x^13 - 1282*x^12 + 347247*x^11 + 471300*x^10 - 52610105*x^9 - 37019484*x^8 + 3590313594*x^7 - 2014400102*x^6 - 105940954800*x^5 + 193726444848*x^4 + 924353400348*x^3 - 2564442153288*x^2 + 1114933543992*x + 707492484008)
 
gp: K = bnfinit(x^15 - 1005*x^13 - 1282*x^12 + 347247*x^11 + 471300*x^10 - 52610105*x^9 - 37019484*x^8 + 3590313594*x^7 - 2014400102*x^6 - 105940954800*x^5 + 193726444848*x^4 + 924353400348*x^3 - 2564442153288*x^2 + 1114933543992*x + 707492484008, 1)
 

Normalized defining polynomial

\( x^{15} - 1005 x^{13} - 1282 x^{12} + 347247 x^{11} + 471300 x^{10} - 52610105 x^{9} - 37019484 x^{8} + 3590313594 x^{7} - 2014400102 x^{6} - 105940954800 x^{5} + 193726444848 x^{4} + 924353400348 x^{3} - 2564442153288 x^{2} + 1114933543992 x + 707492484008 \)

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:  \(1005736322622937279939957750606609818829824=2^{10}\cdot 3^{20}\cdot 881^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $631.20$
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:  $2, 3, 881$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{3524} a^{12} - \frac{31}{881} a^{10} - \frac{401}{3524} a^{9} + \frac{133}{3524} a^{8} + \frac{423}{1762} a^{7} + \frac{143}{881} a^{6} + \frac{1017}{3524} a^{5} + \frac{181}{881} a^{4} - \frac{325}{1762} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{7048} a^{13} + \frac{757}{7048} a^{11} + \frac{60}{881} a^{10} + \frac{133}{7048} a^{9} + \frac{423}{3524} a^{8} + \frac{1453}{7048} a^{7} - \frac{813}{3524} a^{6} - \frac{350}{881} a^{5} - \frac{603}{1762} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2623818256782675630022162783865746888164452639379949022584143923309471464952} a^{14} + \frac{147012121742150178390277958282747328542033594497983359576967199196395717}{2623818256782675630022162783865746888164452639379949022584143923309471464952} a^{13} + \frac{250139445887988031202834292072650874333879128445428568508574350267059865}{2623818256782675630022162783865746888164452639379949022584143923309471464952} a^{12} + \frac{144107159448125423620024460278072126633034595685862417010729976728467658539}{2623818256782675630022162783865746888164452639379949022584143923309471464952} a^{11} + \frac{252576918015101456291447452610394318910703970226107337762050883508712818441}{2623818256782675630022162783865746888164452639379949022584143923309471464952} a^{10} + \frac{617999365867248388373039438371524106350640387583979446682751747749715487717}{2623818256782675630022162783865746888164452639379949022584143923309471464952} a^{9} - \frac{369145279531384777731588258557807608847134054865924873054424495810655614225}{2623818256782675630022162783865746888164452639379949022584143923309471464952} a^{8} - \frac{385662926583909194581112960724767857072502856209964477282742792951098195043}{2623818256782675630022162783865746888164452639379949022584143923309471464952} a^{7} + \frac{129531492463388068856588749665623863820968173553298380760548744120866924289}{1311909128391337815011081391932873444082226319689974511292071961654735732476} a^{6} + \frac{651779703332143489425084858652905717421555237075660116869552220877202350889}{1311909128391337815011081391932873444082226319689974511292071961654735732476} a^{5} + \frac{82456046444097960899627395621473493719349307857975459381164244435321025441}{655954564195668907505540695966436722041113159844987255646035980827367866238} a^{4} + \frac{133097616801730563555262091495909621770323531651433756700327623873311193114}{327977282097834453752770347983218361020556579922493627823017990413683933119} a^{3} + \frac{183554820358892888044656515360829399698021369043333573811687533287767309}{372278413277905168845369293965060568695296912511343504906944370503613999} a^{2} - \frac{232733883732047338233231338679783156815539705205482379310620645570609761}{744556826555810337690738587930121137390593825022687009813888741007227998} a - \frac{156890905534525146788214041220163601335603845467600599016029779441371363}{372278413277905168845369293965060568695296912511343504906944370503613999}$

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:  \( 76569368057100000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T53:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 4860
The 24 conjugacy class representatives for [3^4]A(5)
Character table for [3^4]A(5) is not computed

Intermediate fields

5.5.3104644.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 15 siblings: 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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{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}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.6.8.9$x^{6} + 6 x^{5} + 9$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
881Data not computed