↑

Properties

Label 15.5.38363593816...4375.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,5^{24}\cdot 23^{5}$
Root discriminant $37.35$
Ramified primes $5, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_5^2 : C_3):C_2$ (as 15T13)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2521, -2935, 4635, 9910, 8475, -3026, -9505, -8010, -2445, 750, 871, 230, -45, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 30*x^13 - 45*x^12 + 230*x^11 + 871*x^10 + 750*x^9 - 2445*x^8 - 8010*x^7 - 9505*x^6 - 3026*x^5 + 8475*x^4 + 9910*x^3 + 4635*x^2 - 2935*x - 2521)
 
gp: K = bnfinit(x^15 - 30*x^13 - 45*x^12 + 230*x^11 + 871*x^10 + 750*x^9 - 2445*x^8 - 8010*x^7 - 9505*x^6 - 3026*x^5 + 8475*x^4 + 9910*x^3 + 4635*x^2 - 2935*x - 2521, 1)
 

Normalized defining polynomial

\( x^{15} - 30 x^{13} - 45 x^{12} + 230 x^{11} + 871 x^{10} + 750 x^{9} - 2445 x^{8} - 8010 x^{7} - 9505 x^{6} - 3026 x^{5} + 8475 x^{4} + 9910 x^{3} + 4635 x^{2} - 2935 x - 2521 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-383635938167572021484375=-\,5^{24}\cdot 23^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.35$
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:  $5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{30} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a - \frac{1}{30}$, $\frac{1}{30} a^{11} + \frac{1}{6} a^{7} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{2}{15} a + \frac{1}{6}$, $\frac{1}{60} a^{12} - \frac{1}{60} a^{10} + \frac{1}{6} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} + \frac{1}{4} a^{4} - \frac{1}{3} a^{3} - \frac{11}{60} a^{2} + \frac{1}{60}$, $\frac{1}{180} a^{13} - \frac{1}{180} a^{12} - \frac{1}{60} a^{11} + \frac{1}{180} a^{10} + \frac{5}{36} a^{9} + \frac{7}{36} a^{8} + \frac{1}{12} a^{7} - \frac{7}{36} a^{6} - \frac{5}{36} a^{5} + \frac{13}{36} a^{4} + \frac{23}{60} a^{3} - \frac{59}{180} a^{2} + \frac{1}{60} a - \frac{1}{180}$, $\frac{1}{96307068436693280820} a^{14} - \frac{227215414205666027}{96307068436693280820} a^{13} + \frac{119113079518591657}{96307068436693280820} a^{12} + \frac{187118014539066719}{19261413687338656164} a^{11} - \frac{305655324634156421}{32102356145564426940} a^{10} + \frac{658362622410563231}{19261413687338656164} a^{9} + \frac{2412826605693232739}{19261413687338656164} a^{8} + \frac{2530322955704310143}{19261413687338656164} a^{7} - \frac{1928601935281575769}{19261413687338656164} a^{6} + \frac{1028955300161781997}{6420471229112885388} a^{5} + \frac{32602088062957018009}{96307068436693280820} a^{4} - \frac{10042460400885325433}{96307068436693280820} a^{3} + \frac{16031929406045475053}{96307068436693280820} a^{2} + \frac{7469874632880884461}{19261413687338656164} a + \frac{6663449709218419792}{24076767109173320205}$

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:  $9$
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:  \( 2022515.38523 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:S_3$ (as 15T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 150
The 13 conjugacy class representatives for $(C_5^2 : C_3):C_2$
Character table for $(C_5^2 : C_3):C_2$

Intermediate fields

3.1.23.1

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 25 sibling: data not computed
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{5}$

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
$5$5.5.8.4$x^{5} - 5 x^{4} + 55$$5$$1$$8$$C_5$$[2]$
5.10.16.23$x^{10} + 20 x^{9} + 25 x^{8} + 10 x^{5} + 600 x^{4} + 25$$5$$2$$16$$D_5\times C_5$$[2, 2]^{2}$
$23$23.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
23.10.5.2$x^{10} - 279841 x^{2} + 12872686$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$