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Properties

Label 15.1.40948384850...9479.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,7^{7}\cdot 17^{7}\cdot 59^{4}$
Root discriminant $27.59$
Ramified primes $7, 17, 59$
Class number $1$
Class group Trivial
Galois group 15T79

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17, 51, 159, 115, 102, 10, -242, 26, -101, 71, 45, 1, -12, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 + x^13 - 12*x^12 + x^11 + 45*x^10 + 71*x^9 - 101*x^8 + 26*x^7 - 242*x^6 + 10*x^5 + 102*x^4 + 115*x^3 + 159*x^2 + 51*x - 17)
 
gp: K = bnfinit(x^15 - x^14 + x^13 - 12*x^12 + x^11 + 45*x^10 + 71*x^9 - 101*x^8 + 26*x^7 - 242*x^6 + 10*x^5 + 102*x^4 + 115*x^3 + 159*x^2 + 51*x - 17, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} + x^{13} - 12 x^{12} + x^{11} + 45 x^{10} + 71 x^{9} - 101 x^{8} + 26 x^{7} - 242 x^{6} + 10 x^{5} + 102 x^{4} + 115 x^{3} + 159 x^{2} + 51 x - 17 \)

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:  \(-4094838485015927379479=-\,7^{7}\cdot 17^{7}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.59$
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:  $7, 17, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{11} + \frac{2}{7} a^{10} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{6108741172904027} a^{14} + \frac{212082982971207}{6108741172904027} a^{13} + \frac{329563392847069}{6108741172904027} a^{12} + \frac{737809597611519}{6108741172904027} a^{11} - \frac{2710236400652758}{6108741172904027} a^{10} + \frac{2056775464087975}{6108741172904027} a^{9} + \frac{313190190593921}{872677310414861} a^{8} + \frac{1302522447350814}{6108741172904027} a^{7} + \frac{1429448205452485}{6108741172904027} a^{6} + \frac{2630762908675442}{6108741172904027} a^{5} + \frac{2931455917753175}{6108741172904027} a^{4} - \frac{565875279103138}{6108741172904027} a^{3} - \frac{8195330604958}{124668187202123} a^{2} + \frac{1237726695769086}{6108741172904027} a - \frac{883114848543098}{6108741172904027}$

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:  \( 68592.8576024 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T79:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38880
The 45 conjugacy class representatives for 1/2[S(3)^5]D(5)
Character table for 1/2[S(3)^5]D(5) is not computed

Intermediate fields

5.1.14161.1

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
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 $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ $15$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $15$ R

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.3.0.1$x^{3} - x + 17$$1$$3$$0$$C_3$$[\ ]^{3}$
59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
59.6.4.2$x^{6} - 59 x^{3} + 6962$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$