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Properties

Label 17.1.60446290980...3088.1
Degree $17$
Signature $[1, 8]$
Discriminant $2^{79}$
Root discriminant $25.06$
Ramified prime $2$
Class number $1$
Class group Trivial
Galois group $F_{17}$ (as 17T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![68, -2, -128, 16, 80, 40, 32, -80, -32, 64, 0, -16, 16, 8, 0, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68)
 
gp: K = bnfinit(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68, 1)
 

Normalized defining polynomial

\( x^{17} - 2 x^{16} + 8 x^{13} + 16 x^{12} - 16 x^{11} + 64 x^{9} - 32 x^{8} - 80 x^{7} + 32 x^{6} + 40 x^{5} + 80 x^{4} + 16 x^{3} - 128 x^{2} - 2 x + 68 \)

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:  \(604462909807314587353088=2^{79}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.06$
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:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23126202879577982} a^{16} + \frac{76599019108737}{11563101439788991} a^{15} - \frac{3116476919071769}{11563101439788991} a^{14} - \frac{3753338286501982}{11563101439788991} a^{13} + \frac{5417161799952214}{11563101439788991} a^{12} + \frac{4523768210181351}{11563101439788991} a^{11} + \frac{4452913895983193}{11563101439788991} a^{10} + \frac{5135972366535216}{11563101439788991} a^{9} - \frac{2929695283632081}{11563101439788991} a^{8} - \frac{5470694594022087}{11563101439788991} a^{7} + \frac{1761284834443544}{11563101439788991} a^{6} - \frac{1096278120040765}{11563101439788991} a^{5} + \frac{3361849781604128}{11563101439788991} a^{4} - \frac{4039680983478318}{11563101439788991} a^{3} + \frac{5747676031516213}{11563101439788991} a^{2} - \frac{2695941730666951}{11563101439788991} a + \frac{2408137881238597}{11563101439788991}$

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:  $8$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 600164.932841 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_{17}$ (as 17T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 272
The 17 conjugacy class representatives for $F_{17}$
Character table for $F_{17}$

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 $16{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $17$ $16{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
1.2e4.4t1.1c1$1$ $ 2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
1.2e4.4t1.1c2$1$ $ 2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
1.2e5.8t1.1c1$1$ $ 2^{5}$ $x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 2$ $C_8$ (as 8T1) $0$ $1$
1.2e5.8t1.1c2$1$ $ 2^{5}$ $x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 2$ $C_8$ (as 8T1) $0$ $1$
1.2e5.8t1.1c3$1$ $ 2^{5}$ $x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 2$ $C_8$ (as 8T1) $0$ $1$
1.2e5.8t1.1c4$1$ $ 2^{5}$ $x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} + 2$ $C_8$ (as 8T1) $0$ $1$
1.2e6.16t1.1c1$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
1.2e6.16t1.1c2$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
1.2e6.16t1.1c3$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
1.2e6.16t1.1c4$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
1.2e6.16t1.1c5$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
1.2e6.16t1.1c6$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
1.2e6.16t1.1c7$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
1.2e6.16t1.1c8$1$ $ 2^{6}$ $x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2$ $C_{16}$ (as 16T1) $0$ $-1$
* 16.2e79.17t5.1c1$16$ $ 2^{79}$ $x^{17} - 2 x^{16} + 8 x^{13} + 16 x^{12} - 16 x^{11} + 64 x^{9} - 32 x^{8} - 80 x^{7} + 32 x^{6} + 40 x^{5} + 80 x^{4} + 16 x^{3} - 128 x^{2} - 2 x + 68$ $F_{17}$ (as 17T5) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Additional information

This field is remarkable in that it is only ramified at 2, and

This polynomial was computed by Noam Elkies.