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Properties

Label 17.5.87883485894...0000.1
Degree $17$
Signature $[5, 6]$
Discriminant $2^{18}\cdot 3^{28}\cdot 5^{14}\cdot 7^{4}$
Root discriminant $75.69$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{17}$ (as 17T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2880, -10440, 10560, 980, -13660, 33731, -46312, 26723, -2266, -507, -2424, -258, 2374, -1517, 376, -17, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 8*x^16 - 17*x^15 + 376*x^14 - 1517*x^13 + 2374*x^12 - 258*x^11 - 2424*x^10 - 507*x^9 - 2266*x^8 + 26723*x^7 - 46312*x^6 + 33731*x^5 - 13660*x^4 + 980*x^3 + 10560*x^2 - 10440*x + 2880)
 
gp: K = bnfinit(x^17 - 8*x^16 - 17*x^15 + 376*x^14 - 1517*x^13 + 2374*x^12 - 258*x^11 - 2424*x^10 - 507*x^9 - 2266*x^8 + 26723*x^7 - 46312*x^6 + 33731*x^5 - 13660*x^4 + 980*x^3 + 10560*x^2 - 10440*x + 2880, 1)
 

Normalized defining polynomial

\( x^{17} - 8 x^{16} - 17 x^{15} + 376 x^{14} - 1517 x^{13} + 2374 x^{12} - 258 x^{11} - 2424 x^{10} - 507 x^{9} - 2266 x^{8} + 26723 x^{7} - 46312 x^{6} + 33731 x^{5} - 13660 x^{4} + 980 x^{3} + 10560 x^{2} - 10440 x + 2880 \)

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

Invariants

Degree:  $17$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(87883485894978177600000000000000=2^{18}\cdot 3^{28}\cdot 5^{14}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.69$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{12} a^{14} - \frac{1}{6} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{660} a^{15} + \frac{4}{165} a^{14} - \frac{6}{55} a^{13} + \frac{47}{330} a^{12} + \frac{7}{660} a^{11} - \frac{19}{110} a^{10} + \frac{31}{220} a^{9} + \frac{2}{55} a^{8} + \frac{27}{110} a^{7} + \frac{163}{330} a^{6} - \frac{271}{660} a^{5} - \frac{4}{11} a^{4} + \frac{10}{33} a^{3} - \frac{19}{66} a^{2} - \frac{5}{11} a - \frac{4}{11}$, $\frac{1}{10763356514168647525321963800} a^{16} - \frac{712558458332838183411563}{5381678257084323762660981900} a^{15} + \frac{11210673149195889744253447}{326162318611171137130968600} a^{14} + \frac{271099281471939674283499027}{2690839128542161881330490950} a^{13} + \frac{2450273953714077414924377939}{10763356514168647525321963800} a^{12} - \frac{19395049846015476233047229}{81540579652792784282742150} a^{11} + \frac{435504051866073945025230241}{1793892752361441254220327300} a^{10} + \frac{143601751971811081723180433}{1793892752361441254220327300} a^{9} + \frac{28345773967351261657939381}{1195928501574294169480218200} a^{8} + \frac{252943115071161006543174931}{5381678257084323762660981900} a^{7} + \frac{1600506806403864913201142107}{10763356514168647525321963800} a^{6} + \frac{311711887783952832898676027}{1793892752361441254220327300} a^{5} + \frac{64954419828201995727428393}{195697391166702682278581160} a^{4} - \frac{6833332392809889259350812}{269083912854216188133049095} a^{3} - \frac{2204101639827058824981094}{8154057965279278428274215} a^{2} - \frac{1133141721411510922362981}{5979642507871470847401091} a - \frac{14602925573669827019708789}{29898212539357354237005455}$

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:  $10$
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:  \( 270930416645 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{17}$ (as 17T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 177843714048000
The 156 conjugacy class representatives for $A_{17}$ are not computed
Character table for $A_{17}$ is not computed

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 R ${\href{/LocalNumberField/11.13.0.1}{13} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $17$ ${\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ $17$ ${\href{/LocalNumberField/29.13.0.1}{13} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.3.5.3$x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2}$
3.9.19.3$x^{9} + 6 x^{6} + 18 x^{2} + 6$$9$$1$$19$$S_3^2$$[2, 5/2]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.9.0.1$x^{9} + x^{2} - 6 x + 2$$1$$9$$0$$C_9$$[\ ]^{9}$