↑

Properties

Label 15.1.78954031719...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{18}\cdot 3^{13}\cdot 5^{17}\cdot 19^{5}$
Root discriminant $98.44$
Ramified primes $2, 3, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561729, 6149115, 3088125, 666495, 792135, 356773, -96635, 55335, 2265, -1665, -1173, 525, 15, -15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 15*x^13 + 15*x^12 + 525*x^11 - 1173*x^10 - 1665*x^9 + 2265*x^8 + 55335*x^7 - 96635*x^6 + 356773*x^5 + 792135*x^4 + 666495*x^3 + 3088125*x^2 + 6149115*x + 6561729)
 
gp: K = bnfinit(x^15 - 5*x^14 - 15*x^13 + 15*x^12 + 525*x^11 - 1173*x^10 - 1665*x^9 + 2265*x^8 + 55335*x^7 - 96635*x^6 + 356773*x^5 + 792135*x^4 + 666495*x^3 + 3088125*x^2 + 6149115*x + 6561729, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 15 x^{13} + 15 x^{12} + 525 x^{11} - 1173 x^{10} - 1665 x^{9} + 2265 x^{8} + 55335 x^{7} - 96635 x^{6} + 356773 x^{5} + 792135 x^{4} + 666495 x^{3} + 3088125 x^{2} + 6149115 x + 6561729 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-789540317195400000000000000000=-\,2^{18}\cdot 3^{13}\cdot 5^{17}\cdot 19^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.44$
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:  $2, 3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{18} a^{12} + \frac{1}{18} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{7}{18} a^{3} - \frac{4}{9} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{54} a^{13} + \frac{1}{54} a^{12} - \frac{1}{18} a^{10} + \frac{7}{18} a^{9} + \frac{1}{9} a^{8} + \frac{1}{3} a^{7} + \frac{4}{9} a^{6} + \frac{7}{18} a^{5} - \frac{11}{54} a^{4} + \frac{5}{27} a^{3} - \frac{1}{18} a^{2} - \frac{1}{6} a$, $\frac{1}{47321387300024813109179683950600222964868742} a^{14} - \frac{68419430601397234172966286240565656082091}{47321387300024813109179683950600222964868742} a^{13} + \frac{39855593740843395683396292602098948158746}{7886897883337468851529947325100037160811457} a^{12} + \frac{5925834498816149905992645737861508036509}{15773795766674937703059894650200074321622914} a^{11} - \frac{3830886676315779904013279785623873151915451}{15773795766674937703059894650200074321622914} a^{10} - \frac{1930145962759601968756189404328196583523109}{7886897883337468851529947325100037160811457} a^{9} + \frac{80966547359317895705875684013727565924511}{2628965961112489617176649108366679053603819} a^{8} + \frac{3182016672851082047402159785103513453000470}{7886897883337468851529947325100037160811457} a^{7} - \frac{3290690806410145373977213596686093632260203}{15773795766674937703059894650200074321622914} a^{6} - \frac{21472371311889616388528107114873096316590601}{47321387300024813109179683950600222964868742} a^{5} - \frac{6897523867529907690499613295632054284189330}{23660693650012406554589841975300111482434371} a^{4} - \frac{1361580260682288702879531324466619839755427}{5257931922224979234353298216733358107207638} a^{3} - \frac{72172282131661957969845519118331367637873}{1752643974074993078117766072244452702402546} a^{2} - \frac{339221275768263530823030060292705473749753}{876321987037496539058883036122226351201273} a - \frac{125911705885537321747510909222595235795064}{292107329012498846352961012040742117067091}$

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:  $7$
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:  \( 5411274997.421242 \) (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

3.1.5700.1, 5.1.4050000.4

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.15.17.3$x^{15} - 5 x^{13} + 10 x^{12} + 10 x^{11} + 10 x^{9} - 5 x^{8} - 10 x^{6} + 10 x^{5} + 5 x^{3} + 10$$15$$1$$17$$F_5 \times S_3$$[5/4]_{12}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$