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Properties

Label 15.3.79118042886...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{10}\cdot 3^{12}\cdot 5^{28}\cdot 2081^{5}$
Root discriminant $984.51$
Ramified primes $2, 3, 5, 2081$
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![75838029, 5689440, -18298740, -4599000, -5408340, -4659937, 1130400, 1139420, -92070, -106120, 2595, 4840, -30, -110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 110*x^13 - 30*x^12 + 4840*x^11 + 2595*x^10 - 106120*x^9 - 92070*x^8 + 1139420*x^7 + 1130400*x^6 - 4659937*x^5 - 5408340*x^4 - 4599000*x^3 - 18298740*x^2 + 5689440*x + 75838029)
 
gp: K = bnfinit(x^15 - 110*x^13 - 30*x^12 + 4840*x^11 + 2595*x^10 - 106120*x^9 - 92070*x^8 + 1139420*x^7 + 1130400*x^6 - 4659937*x^5 - 5408340*x^4 - 4599000*x^3 - 18298740*x^2 + 5689440*x + 75838029, 1)
 

Normalized defining polynomial

\( x^{15} - 110 x^{13} - 30 x^{12} + 4840 x^{11} + 2595 x^{10} - 106120 x^{9} - 92070 x^{8} + 1139420 x^{7} + 1130400 x^{6} - 4659937 x^{5} - 5408340 x^{4} - 4599000 x^{3} - 18298740 x^{2} + 5689440 x + 75838029 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(791180428866728914712562561035156250000000000=2^{10}\cdot 3^{12}\cdot 5^{28}\cdot 2081^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $984.51$
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:  $2, 3, 5, 2081$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{25} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{3}{25} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a + \frac{12}{25}$, $\frac{1}{25} a^{11} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{3}{25} a^{6} + \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{12}{25} a - \frac{1}{5}$, $\frac{1}{25} a^{12} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{3}{25} a^{7} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{12}{25} a^{2} - \frac{1}{5} a$, $\frac{1}{25} a^{13} + \frac{2}{5} a^{9} + \frac{3}{25} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{12}{25} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{1079984650716365722761660590963466864095125} a^{14} - \frac{7397113384209424899727307955122126505067}{1079984650716365722761660590963466864095125} a^{13} - \frac{10739652271922306112715304758395373688916}{1079984650716365722761660590963466864095125} a^{12} - \frac{1041254250554401231785688650314074308403}{1079984650716365722761660590963466864095125} a^{11} - \frac{8132912794491896187687748533707554622624}{1079984650716365722761660590963466864095125} a^{10} - \frac{399339305480284281026420894531323282701392}{1079984650716365722761660590963466864095125} a^{9} + \frac{208867260047307825927761931439458421400219}{1079984650716365722761660590963466864095125} a^{8} + \frac{162949206866112458689807613666916339089822}{1079984650716365722761660590963466864095125} a^{7} - \frac{77790052442515922823703236180587389054089}{1079984650716365722761660590963466864095125} a^{6} - \frac{113020190739694300035266985074529415434307}{1079984650716365722761660590963466864095125} a^{5} + \frac{273544035817438797822152722881886531782322}{1079984650716365722761660590963466864095125} a^{4} + \frac{393504073462944205908071488112536599178111}{1079984650716365722761660590963466864095125} a^{3} + \frac{27248200675502939938847339966134349781648}{1079984650716365722761660590963466864095125} a^{2} - \frac{398702889676929541761486118294911398185096}{1079984650716365722761660590963466864095125} a - \frac{501894121013715727220297754979439229164158}{1079984650716365722761660590963466864095125}$

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

Class group and class number

$C_{15}$, which has order $15$ (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:  $8$
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:  \( 16082642412972486 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.3.41620.1, 5.1.158203125.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$5$5.5.9.5$x^{5} + 105$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.4$x^{10} + 105$$10$$1$$19$$F_5$$[9/4]_{4}$
2081Data not computed