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Properties

Label 15.1.17826338966...6416.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{10}\cdot 19^{7}\cdot 41^{7}$
Root discriminant $35.49$
Ramified primes $2, 19, 41$
Class number $1$
Class group Trivial
Galois group $D_{15}$ (as 15T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-608, 1520, 1416, -311, -891, 1441, -980, 95, -23, 144, -165, 131, -62, 23, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 + 23*x^13 - 62*x^12 + 131*x^11 - 165*x^10 + 144*x^9 - 23*x^8 + 95*x^7 - 980*x^6 + 1441*x^5 - 891*x^4 - 311*x^3 + 1416*x^2 + 1520*x - 608)
 
gp: K = bnfinit(x^15 - 7*x^14 + 23*x^13 - 62*x^12 + 131*x^11 - 165*x^10 + 144*x^9 - 23*x^8 + 95*x^7 - 980*x^6 + 1441*x^5 - 891*x^4 - 311*x^3 + 1416*x^2 + 1520*x - 608, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} + 23 x^{13} - 62 x^{12} + 131 x^{11} - 165 x^{10} + 144 x^{9} - 23 x^{8} + 95 x^{7} - 980 x^{6} + 1441 x^{5} - 891 x^{4} - 311 x^{3} + 1416 x^{2} + 1520 x - 608 \)

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:  \(-178263389667012351036416=-\,2^{10}\cdot 19^{7}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.49$
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:  $2, 19, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{38} a^{11} - \frac{6}{19} a^{9} - \frac{7}{38} a^{8} - \frac{8}{19} a^{7} + \frac{5}{19} a^{6} + \frac{13}{38} a^{5} + \frac{9}{19} a^{4} - \frac{9}{19} a^{3} - \frac{17}{38} a^{2}$, $\frac{1}{38} a^{12} + \frac{7}{38} a^{10} - \frac{7}{38} a^{9} - \frac{8}{19} a^{8} - \frac{9}{38} a^{7} + \frac{13}{38} a^{6} + \frac{9}{19} a^{5} + \frac{1}{38} a^{4} - \frac{17}{38} a^{3} - \frac{1}{2} a$, $\frac{1}{10108} a^{13} - \frac{8}{2527} a^{12} + \frac{111}{10108} a^{11} - \frac{269}{10108} a^{10} + \frac{1817}{5054} a^{9} - \frac{81}{1444} a^{8} + \frac{4451}{10108} a^{7} + \frac{1082}{2527} a^{6} - \frac{2225}{10108} a^{5} - \frac{657}{1444} a^{4} - \frac{1291}{5054} a^{3} + \frac{2507}{10108} a^{2} - \frac{5}{14} a - \frac{3}{133}$, $\frac{1}{48425197017369032} a^{14} - \frac{491430971475}{48425197017369032} a^{13} + \frac{588052297191879}{48425197017369032} a^{12} - \frac{249188843229447}{24212598508684516} a^{11} - \frac{10107483063024909}{48425197017369032} a^{10} - \frac{14186397256313757}{48425197017369032} a^{9} + \frac{459492914534891}{6053149627171129} a^{8} + \frac{364200952539593}{48425197017369032} a^{7} - \frac{18332076030689873}{48425197017369032} a^{6} - \frac{1564502313411567}{12106299254342258} a^{5} + \frac{21423982532635033}{48425197017369032} a^{4} + \frac{22117000135330221}{48425197017369032} a^{3} + \frac{9441941020669377}{48425197017369032} a^{2} + \frac{89212297120389}{637173644965382} a - \frac{37085505371962}{318586822482691}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1437362.27821 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{15}$ (as 15T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.3116.1, 5.1.606841.1

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

Sibling fields

Galois closure: 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 $15$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$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}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$