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Properties

Label 17.1.26407329303...3424.1
Degree $17$
Signature $[1, 8]$
Discriminant $2^{30}\cdot 199^{8}$
Root discriminant $41.02$
Ramified primes $2, 199$
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group $\PSL(2,16)$ (as 17T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-40, 122, -844, 1608, -1192, 1580, -180, -1540, -1488, 612, -12, -520, -92, 116, 16, -14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 - 14*x^15 + 16*x^14 + 116*x^13 - 92*x^12 - 520*x^11 - 12*x^10 + 612*x^9 - 1488*x^8 - 1540*x^7 - 180*x^6 + 1580*x^5 - 1192*x^4 + 1608*x^3 - 844*x^2 + 122*x - 40)
 
gp: K = bnfinit(x^17 - 2*x^16 - 14*x^15 + 16*x^14 + 116*x^13 - 92*x^12 - 520*x^11 - 12*x^10 + 612*x^9 - 1488*x^8 - 1540*x^7 - 180*x^6 + 1580*x^5 - 1192*x^4 + 1608*x^3 - 844*x^2 + 122*x - 40, 1)
 

Normalized defining polynomial

\( x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40 \)

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:  \(2640732930336770744777703424=2^{30}\cdot 199^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.02$
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:  $2, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{529640928484482379991048562878} a^{16} + \frac{113709381659494316635300408134}{264820464242241189995524281439} a^{15} - \frac{61943235101129197733417044674}{264820464242241189995524281439} a^{14} + \frac{12423157068358265845474486687}{264820464242241189995524281439} a^{13} + \frac{34244214218755210466985536001}{264820464242241189995524281439} a^{12} + \frac{34348135403482203219264793005}{264820464242241189995524281439} a^{11} - \frac{9688021455469352629644571467}{264820464242241189995524281439} a^{10} - \frac{36132884583782676782893806384}{264820464242241189995524281439} a^{9} + \frac{1304692737829081305122023802}{3952544242421510298440660917} a^{8} + \frac{110058323648138546863537087508}{264820464242241189995524281439} a^{7} + \frac{71755077950124959957321195748}{264820464242241189995524281439} a^{6} + \frac{15493423858179442401252435656}{264820464242241189995524281439} a^{5} + \frac{2149771624335915152310798123}{4488482444783748982974987821} a^{4} - \frac{68870993893524926894326427714}{264820464242241189995524281439} a^{3} - \frac{18952672497162653835076964687}{264820464242241189995524281439} a^{2} + \frac{65424079293829850455509679791}{264820464242241189995524281439} a - \frac{122752095623347146298207799848}{264820464242241189995524281439}$

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

Class group and class number

$C_{15}$, which has order $15$ (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:  \( 5943625.48287 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,16)$ (as 17T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 4080
The 17 conjugacy class representatives for $\PSL(2,16)$
Character table for $\PSL(2,16)$

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $17$ $17$ $17$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $17$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $17$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $17$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $17$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
15.2e30_199e8.240.1c1$15$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.2e30_199e8.240.1c2$15$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.2e30_199e8.240.1c3$15$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.2e30_199e8.240.1c4$15$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.2e30_199e8.240.1c5$15$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.2e30_199e8.240.1c6$15$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.2e30_199e8.240.1c7$15$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.2e30_199e8.240.1c8$15$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $-1$
* 16.2e30_199e8.17t6.1c1$16$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $0$
17.2e30_199e8.51.1c1$17$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.2e30_199e8.68.1c1$17$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.2e30_199e8.68.1c2$17$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.2e30_199e8.120.1c1$17$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.2e30_199e8.120.1c2$17$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.2e30_199e8.120.1c3$17$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $1$
17.2e30_199e8.120.1c4$17$ $ 2^{30} \cdot 199^{8}$ $x^{17} - 2 x^{16} - 14 x^{15} + 16 x^{14} + 116 x^{13} - 92 x^{12} - 520 x^{11} - 12 x^{10} + 612 x^{9} - 1488 x^{8} - 1540 x^{7} - 180 x^{6} + 1580 x^{5} - 1192 x^{4} + 1608 x^{3} - 844 x^{2} + 122 x - 40$ $\PSL(2,16)$ (as 17T6) $1$ $1$

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 *.