Properties

Label 17.1.264...424.1
Degree $17$
Signature $[1, 8]$
Discriminant $2.641\times 10^{27}$
Root discriminant \(41.02\)
Ramified primes see page
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group $\PSL(2,16)$ (as 17T6)

Related objects

Downloads

Learn more

Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

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)
 
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]);
 

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

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

Invariants

Degree:  $17$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(2640732930336770744777703424\) \(\medspace = 2^{30}\cdot 199^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(41.02\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:   \(2\), \(199\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{52\!\cdots\!78}a^{16}+\frac{11\!\cdots\!34}{26\!\cdots\!39}a^{15}-\frac{61\!\cdots\!74}{26\!\cdots\!39}a^{14}+\frac{12\!\cdots\!87}{26\!\cdots\!39}a^{13}+\frac{34\!\cdots\!01}{26\!\cdots\!39}a^{12}+\frac{34\!\cdots\!05}{26\!\cdots\!39}a^{11}-\frac{96\!\cdots\!67}{26\!\cdots\!39}a^{10}-\frac{36\!\cdots\!84}{26\!\cdots\!39}a^{9}+\frac{13\!\cdots\!02}{39\!\cdots\!17}a^{8}+\frac{11\!\cdots\!08}{26\!\cdots\!39}a^{7}+\frac{71\!\cdots\!48}{26\!\cdots\!39}a^{6}+\frac{15\!\cdots\!56}{26\!\cdots\!39}a^{5}+\frac{21\!\cdots\!23}{44\!\cdots\!21}a^{4}-\frac{68\!\cdots\!14}{26\!\cdots\!39}a^{3}-\frac{18\!\cdots\!87}{26\!\cdots\!39}a^{2}+\frac{65\!\cdots\!91}{26\!\cdots\!39}a-\frac{12\!\cdots\!48}{26\!\cdots\!39}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{15}$, which has order $15$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $\frac{37\!\cdots\!53}{26\!\cdots\!39}a^{16}-\frac{76\!\cdots\!57}{26\!\cdots\!39}a^{15}-\frac{52\!\cdots\!85}{26\!\cdots\!39}a^{14}+\frac{62\!\cdots\!68}{26\!\cdots\!39}a^{13}+\frac{44\!\cdots\!02}{26\!\cdots\!39}a^{12}-\frac{36\!\cdots\!70}{26\!\cdots\!39}a^{11}-\frac{20\!\cdots\!71}{26\!\cdots\!39}a^{10}-\frac{18\!\cdots\!21}{26\!\cdots\!39}a^{9}+\frac{40\!\cdots\!19}{39\!\cdots\!17}a^{8}-\frac{50\!\cdots\!41}{26\!\cdots\!39}a^{7}-\frac{54\!\cdots\!56}{26\!\cdots\!39}a^{6}+\frac{52\!\cdots\!69}{26\!\cdots\!39}a^{5}+\frac{15\!\cdots\!82}{44\!\cdots\!21}a^{4}-\frac{10\!\cdots\!76}{26\!\cdots\!39}a^{3}+\frac{65\!\cdots\!72}{26\!\cdots\!39}a^{2}-\frac{30\!\cdots\!67}{26\!\cdots\!39}a+\frac{13\!\cdots\!99}{26\!\cdots\!39}$, $\frac{13\!\cdots\!47}{26\!\cdots\!39}a^{16}+\frac{22\!\cdots\!70}{26\!\cdots\!39}a^{15}-\frac{28\!\cdots\!18}{26\!\cdots\!39}a^{14}-\frac{44\!\cdots\!15}{26\!\cdots\!39}a^{13}+\frac{23\!\cdots\!08}{26\!\cdots\!39}a^{12}+\frac{41\!\cdots\!41}{26\!\cdots\!39}a^{11}-\frac{12\!\cdots\!67}{26\!\cdots\!39}a^{10}-\frac{23\!\cdots\!07}{26\!\cdots\!39}a^{9}+\frac{21\!\cdots\!33}{39\!\cdots\!17}a^{8}+\frac{68\!\cdots\!63}{26\!\cdots\!39}a^{7}-\frac{11\!\cdots\!01}{26\!\cdots\!39}a^{6}-\frac{77\!\cdots\!81}{26\!\cdots\!39}a^{5}+\frac{57\!\cdots\!33}{44\!\cdots\!21}a^{4}+\frac{11\!\cdots\!64}{26\!\cdots\!39}a^{3}-\frac{14\!\cdots\!42}{26\!\cdots\!39}a^{2}+\frac{34\!\cdots\!98}{26\!\cdots\!39}a-\frac{45\!\cdots\!81}{26\!\cdots\!39}$, $\frac{40\!\cdots\!05}{26\!\cdots\!39}a^{16}-\frac{57\!\cdots\!93}{26\!\cdots\!39}a^{15}-\frac{59\!\cdots\!27}{26\!\cdots\!39}a^{14}+\frac{29\!\cdots\!79}{26\!\cdots\!39}a^{13}+\frac{48\!\cdots\!64}{26\!\cdots\!39}a^{12}-\frac{87\!\cdots\!78}{26\!\cdots\!39}a^{11}-\frac{21\!\cdots\!18}{26\!\cdots\!39}a^{10}-\frac{13\!\cdots\!61}{26\!\cdots\!39}a^{9}+\frac{25\!\cdots\!19}{39\!\cdots\!17}a^{8}-\frac{49\!\cdots\!50}{26\!\cdots\!39}a^{7}-\frac{88\!\cdots\!73}{26\!\cdots\!39}a^{6}-\frac{59\!\cdots\!07}{26\!\cdots\!39}a^{5}+\frac{53\!\cdots\!49}{44\!\cdots\!21}a^{4}-\frac{21\!\cdots\!41}{26\!\cdots\!39}a^{3}+\frac{58\!\cdots\!46}{26\!\cdots\!39}a^{2}-\frac{21\!\cdots\!41}{26\!\cdots\!39}a+\frac{33\!\cdots\!37}{26\!\cdots\!39}$, $\frac{82\!\cdots\!10}{26\!\cdots\!39}a^{16}+\frac{37\!\cdots\!66}{26\!\cdots\!39}a^{15}-\frac{21\!\cdots\!05}{26\!\cdots\!39}a^{14}-\frac{65\!\cdots\!97}{26\!\cdots\!39}a^{13}+\frac{17\!\cdots\!36}{26\!\cdots\!39}a^{12}+\frac{59\!\cdots\!30}{26\!\cdots\!39}a^{11}-\frac{92\!\cdots\!13}{26\!\cdots\!39}a^{10}-\frac{31\!\cdots\!36}{26\!\cdots\!39}a^{9}+\frac{90\!\cdots\!67}{39\!\cdots\!17}a^{8}+\frac{33\!\cdots\!27}{26\!\cdots\!39}a^{7}-\frac{10\!\cdots\!35}{26\!\cdots\!39}a^{6}-\frac{12\!\cdots\!91}{26\!\cdots\!39}a^{5}+\frac{57\!\cdots\!29}{44\!\cdots\!21}a^{4}+\frac{88\!\cdots\!59}{26\!\cdots\!39}a^{3}-\frac{91\!\cdots\!03}{26\!\cdots\!39}a^{2}-\frac{26\!\cdots\!75}{26\!\cdots\!39}a-\frac{74\!\cdots\!97}{26\!\cdots\!39}$, $\frac{90\!\cdots\!48}{26\!\cdots\!39}a^{16}-\frac{13\!\cdots\!00}{26\!\cdots\!39}a^{15}-\frac{13\!\cdots\!73}{26\!\cdots\!39}a^{14}+\frac{74\!\cdots\!04}{26\!\cdots\!39}a^{13}+\frac{10\!\cdots\!86}{26\!\cdots\!39}a^{12}-\frac{26\!\cdots\!27}{26\!\cdots\!39}a^{11}-\frac{46\!\cdots\!99}{26\!\cdots\!39}a^{10}-\frac{26\!\cdots\!30}{26\!\cdots\!39}a^{9}+\frac{49\!\cdots\!46}{39\!\cdots\!17}a^{8}-\frac{11\!\cdots\!52}{26\!\cdots\!39}a^{7}-\frac{19\!\cdots\!84}{26\!\cdots\!39}a^{6}-\frac{14\!\cdots\!64}{26\!\cdots\!39}a^{5}+\frac{13\!\cdots\!04}{44\!\cdots\!21}a^{4}-\frac{72\!\cdots\!46}{26\!\cdots\!39}a^{3}+\frac{96\!\cdots\!38}{26\!\cdots\!39}a^{2}-\frac{66\!\cdots\!36}{26\!\cdots\!39}a+\frac{23\!\cdots\!39}{26\!\cdots\!39}$, $\frac{64\!\cdots\!09}{26\!\cdots\!39}a^{16}+\frac{50\!\cdots\!35}{26\!\cdots\!39}a^{15}-\frac{11\!\cdots\!98}{26\!\cdots\!39}a^{14}-\frac{16\!\cdots\!07}{26\!\cdots\!39}a^{13}+\frac{79\!\cdots\!09}{26\!\cdots\!39}a^{12}+\frac{15\!\cdots\!35}{26\!\cdots\!39}a^{11}-\frac{30\!\cdots\!09}{26\!\cdots\!39}a^{10}-\frac{91\!\cdots\!72}{26\!\cdots\!39}a^{9}-\frac{68\!\cdots\!58}{39\!\cdots\!17}a^{8}-\frac{63\!\cdots\!67}{26\!\cdots\!39}a^{7}-\frac{32\!\cdots\!10}{26\!\cdots\!39}a^{6}-\frac{47\!\cdots\!32}{26\!\cdots\!39}a^{5}-\frac{55\!\cdots\!31}{44\!\cdots\!21}a^{4}-\frac{17\!\cdots\!44}{26\!\cdots\!39}a^{3}-\frac{16\!\cdots\!78}{26\!\cdots\!39}a^{2}+\frac{37\!\cdots\!43}{26\!\cdots\!39}a-\frac{38\!\cdots\!37}{26\!\cdots\!39}$, $\frac{10\!\cdots\!00}{26\!\cdots\!39}a^{16}-\frac{18\!\cdots\!29}{26\!\cdots\!39}a^{15}-\frac{14\!\cdots\!75}{26\!\cdots\!39}a^{14}+\frac{13\!\cdots\!28}{26\!\cdots\!39}a^{13}+\frac{11\!\cdots\!38}{26\!\cdots\!39}a^{12}-\frac{69\!\cdots\!88}{26\!\cdots\!39}a^{11}-\frac{54\!\cdots\!59}{26\!\cdots\!39}a^{10}-\frac{11\!\cdots\!93}{26\!\cdots\!39}a^{9}+\frac{92\!\cdots\!53}{39\!\cdots\!17}a^{8}-\frac{13\!\cdots\!16}{26\!\cdots\!39}a^{7}-\frac{19\!\cdots\!60}{26\!\cdots\!39}a^{6}-\frac{48\!\cdots\!95}{26\!\cdots\!39}a^{5}+\frac{26\!\cdots\!03}{44\!\cdots\!21}a^{4}-\frac{11\!\cdots\!09}{26\!\cdots\!39}a^{3}+\frac{13\!\cdots\!18}{26\!\cdots\!39}a^{2}-\frac{64\!\cdots\!89}{26\!\cdots\!39}a-\frac{28\!\cdots\!39}{26\!\cdots\!39}$, $\frac{71\!\cdots\!11}{26\!\cdots\!39}a^{16}-\frac{23\!\cdots\!75}{26\!\cdots\!39}a^{15}-\frac{88\!\cdots\!81}{26\!\cdots\!39}a^{14}+\frac{25\!\cdots\!99}{26\!\cdots\!39}a^{13}+\frac{78\!\cdots\!36}{26\!\cdots\!39}a^{12}-\frac{17\!\cdots\!40}{26\!\cdots\!39}a^{11}-\frac{37\!\cdots\!44}{26\!\cdots\!39}a^{10}+\frac{51\!\cdots\!90}{26\!\cdots\!39}a^{9}+\frac{12\!\cdots\!04}{39\!\cdots\!17}a^{8}-\frac{15\!\cdots\!01}{26\!\cdots\!39}a^{7}-\frac{64\!\cdots\!59}{26\!\cdots\!39}a^{6}+\frac{23\!\cdots\!88}{26\!\cdots\!39}a^{5}+\frac{39\!\cdots\!46}{44\!\cdots\!21}a^{4}-\frac{15\!\cdots\!35}{26\!\cdots\!39}a^{3}+\frac{17\!\cdots\!71}{26\!\cdots\!39}a^{2}-\frac{16\!\cdots\!79}{26\!\cdots\!39}a+\frac{18\!\cdots\!47}{26\!\cdots\!39}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 5943625.48287 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{8}\cdot 5943625.48287 \cdot 15}{2\sqrt{2640732930336770744777703424}}\approx 4.21424132373$ (assuming GRH)

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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 ${\href{/padicField/3.5.0.1}{5} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ $17$ $17$ $17$ ${\href{/padicField/23.3.0.1}{3} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ $17$ ${\href{/padicField/31.5.0.1}{5} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ $17$ ${\href{/padicField/41.3.0.1}{3} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $17$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $17$ ${\href{/padicField/59.5.0.1}{5} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
Deg $16$$16$$1$$30$
\(199\) Copy content Toggle raw display $\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 Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
15.264...424.240.a.a$15$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.264...424.240.a.b$15$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.264...424.240.a.c$15$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.264...424.240.a.d$15$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.264...424.240.a.e$15$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.264...424.240.a.f$15$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.264...424.240.a.g$15$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.264...424.240.a.h$15$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
* 16.264...424.17t6.a.a$16$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $0$
17.264...424.51.a.a$17$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.264...424.68.a.a$17$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.264...424.68.a.b$17$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.264...424.120.a.a$17$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.264...424.120.a.b$17$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.264...424.120.a.c$17$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.264...424.120.a.d$17$ $ 2^{30} \cdot 199^{8}$ 17.1.2640732930336770744777703424.1 $\PSL(2,16)$ (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 *.