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Properties

Label 17.1.54481006265...0000.1
Degree $17$
Signature $[1, 8]$
Discriminant $2^{16}\cdot 3^{20}\cdot 5^{22}$
Root discriminant $56.13$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\PSL(2,16):C_4$ (as 17T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9400, 24080, -55864, 83000, -102400, 102200, -84100, 63500, -38080, 22720, -10220, 4870, -1622, 610, -140, 40, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 5*x^16 + 40*x^15 - 140*x^14 + 610*x^13 - 1622*x^12 + 4870*x^11 - 10220*x^10 + 22720*x^9 - 38080*x^8 + 63500*x^7 - 84100*x^6 + 102200*x^5 - 102400*x^4 + 83000*x^3 - 55864*x^2 + 24080*x - 9400)
 
gp: K = bnfinit(x^17 - 5*x^16 + 40*x^15 - 140*x^14 + 610*x^13 - 1622*x^12 + 4870*x^11 - 10220*x^10 + 22720*x^9 - 38080*x^8 + 63500*x^7 - 84100*x^6 + 102200*x^5 - 102400*x^4 + 83000*x^3 - 55864*x^2 + 24080*x - 9400, 1)
 

Normalized defining polynomial

\( x^{17} - 5 x^{16} + 40 x^{15} - 140 x^{14} + 610 x^{13} - 1622 x^{12} + 4870 x^{11} - 10220 x^{10} + 22720 x^{9} - 38080 x^{8} + 63500 x^{7} - 84100 x^{6} + 102200 x^{5} - 102400 x^{4} + 83000 x^{3} - 55864 x^{2} + 24080 x - 9400 \)

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

Invariants

Degree:  $17$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(544810062656250000000000000000=2^{16}\cdot 3^{20}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.13$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{18} a^{9} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{18} a^{10} - \frac{1}{6} a^{6} - \frac{1}{3} a^{3} + \frac{4}{9} a$, $\frac{1}{18} a^{11} - \frac{1}{6} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{108} a^{12} - \frac{1}{36} a^{11} + \frac{1}{54} a^{9} + \frac{1}{18} a^{7} - \frac{1}{9} a^{4} - \frac{7}{27} a^{3} + \frac{1}{9} a^{2} - \frac{1}{9} a - \frac{5}{27}$, $\frac{1}{108} a^{13} - \frac{1}{36} a^{11} + \frac{1}{54} a^{10} + \frac{1}{18} a^{8} - \frac{1}{6} a^{6} - \frac{1}{9} a^{5} + \frac{11}{27} a^{4} - \frac{1}{3} a^{2} + \frac{4}{27} a$, $\frac{1}{108} a^{14} - \frac{1}{108} a^{11} + \frac{1}{18} a^{6} + \frac{2}{27} a^{5} + \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{7}{27} a^{2} - \frac{1}{9}$, $\frac{1}{108} a^{15} - \frac{1}{36} a^{11} + \frac{1}{54} a^{9} - \frac{1}{18} a^{7} - \frac{5}{54} a^{6} - \frac{2}{9} a^{4} + \frac{4}{9} a^{2} + \frac{4}{9} a + \frac{13}{27}$, $\frac{1}{2700} a^{16} + \frac{1}{540} a^{15} - \frac{1}{270} a^{14} + \frac{1}{270} a^{13} + \frac{1}{270} a^{12} + \frac{7}{675} a^{11} - \frac{1}{54} a^{10} + \frac{1}{90} a^{9} - \frac{1}{15} a^{8} + \frac{1}{135} a^{7} + \frac{5}{54} a^{6} + \frac{1}{27} a^{5} + \frac{11}{27} a^{4} + \frac{11}{27} a^{3} + \frac{8}{27} a^{2} - \frac{241}{675} a - \frac{2}{15}$

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:  $8$
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:  \( 1736565212.05 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,16):C_4$ (as 17T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 16320
The 17 conjugacy class representatives for $\PSL(2,16):C_4$
Character table for $\PSL(2,16):C_4$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ $17$

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
2Data not computed
3Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.7.1$x^{5} + 15 x^{3} + 5$$5$$1$$7$$F_5$$[7/4]_{4}$
5.10.15.1$x^{10} - 10 x^{6} + 5$$10$$1$$15$$F_5$$[7/4]_{4}$