Properties

Label 17.17.462...121.1
Degree $17$
Signature $[17, 0]$
Discriminant $4.629\times 10^{47}$
Root discriminant $636.58$
Ramified prime $953$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{17}$ (as 17T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 448*x^15 + 1309*x^14 + 75494*x^13 - 374314*x^12 - 5597667*x^11 + 41830550*x^10 + 136426018*x^9 - 1919481097*x^8 + 2548312782*x^7 + 27117309376*x^6 - 105628969954*x^5 + 34101907629*x^4 + 457076113374*x^3 - 817186035962*x^2 + 323264486326*x + 117028501127)
 
gp: K = bnfinit(x^17 - x^16 - 448*x^15 + 1309*x^14 + 75494*x^13 - 374314*x^12 - 5597667*x^11 + 41830550*x^10 + 136426018*x^9 - 1919481097*x^8 + 2548312782*x^7 + 27117309376*x^6 - 105628969954*x^5 + 34101907629*x^4 + 457076113374*x^3 - 817186035962*x^2 + 323264486326*x + 117028501127, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![117028501127, 323264486326, -817186035962, 457076113374, 34101907629, -105628969954, 27117309376, 2548312782, -1919481097, 136426018, 41830550, -5597667, -374314, 75494, 1309, -448, -1, 1]);
 

\(x^{17} - x^{16} - 448 x^{15} + 1309 x^{14} + 75494 x^{13} - 374314 x^{12} - 5597667 x^{11} + 41830550 x^{10} + 136426018 x^{9} - 1919481097 x^{8} + 2548312782 x^{7} + 27117309376 x^{6} - 105628969954 x^{5} + 34101907629 x^{4} + 457076113374 x^{3} - 817186035962 x^{2} + 323264486326 x + 117028501127\)  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:  $[17, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(462899178676311160288470191817767102492054133121\)\(\medspace = 953^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $636.58$
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:  $953$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $17$
This field is Galois and abelian over $\Q$.
Conductor:  \(953\)
Dirichlet character group:    $\lbrace$$\chi_{953}(256,·)$, $\chi_{953}(1,·)$, $\chi_{953}(770,·)$, $\chi_{953}(134,·)$, $\chi_{953}(16,·)$, $\chi_{953}(276,·)$, $\chi_{953}(732,·)$, $\chi_{953}(604,·)$, $\chi_{953}(417,·)$, $\chi_{953}(802,·)$, $\chi_{953}(284,·)$, $\chi_{953}(238,·)$, $\chi_{953}(882,·)$, $\chi_{953}(884,·)$, $\chi_{953}(949,·)$, $\chi_{953}(889,·)$, $\chi_{953}(443,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{109} a^{15} - \frac{29}{109} a^{14} - \frac{2}{109} a^{13} - \frac{11}{109} a^{12} + \frac{16}{109} a^{11} - \frac{27}{109} a^{10} + \frac{51}{109} a^{9} + \frac{8}{109} a^{8} - \frac{50}{109} a^{7} - \frac{37}{109} a^{6} - \frac{28}{109} a^{5} + \frac{28}{109} a^{4} - \frac{2}{109} a^{3} + \frac{41}{109} a^{2} + \frac{52}{109} a - \frac{1}{109}$, $\frac{1}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{16} - \frac{370775489065786690823086414238723888989638059417869812295596978244318634938}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{15} + \frac{43034875624085183733350666472854084663991773757520947640342016683487686374168}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{14} + \frac{11651393661818849092559244110544546456773901171982796489270121943180555316046}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{13} - \frac{56932349809385201816060465020027722982606037125415927199135434185037341281191}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{12} - \frac{52027237792250642435290381262127817070776575714972894421514218806381185770074}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{11} - \frac{8649526255971723887081930728586500907298678877541114488715547328417973375481}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{10} + \frac{45561057816752480357500442247480666434142732433290991560138549404640882721212}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{9} - \frac{52039442837110885124913468970044014330798328533242693235015391033793072842976}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{8} - \frac{5390795193929488026887231443731533730743911926727853809957885869197821218371}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{7} - \frac{32230190642935556438476733700294121950405077355734544047305371020272849389668}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{6} + \frac{38250518970714596197100910986284540606366446498273849708448476578439888016713}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{5} - \frac{19370715639306886462036223719841631497868017427033466629861897254150557994238}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{4} + \frac{58619452384842523912190176365140301756098026526619709349439266480392105900920}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{3} - \frac{6525449010304667664330414010655909074992218379193382332226982303473292535817}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a^{2} - \frac{33772976072158228296086565842456004372675935431653294303150845606604803313080}{147013901317241646584286723478269847178967463718076648691199395367954067241097} a - \frac{32041049269316688198378480255832130040158167165648228709736021637909447652935}{147013901317241646584286723478269847178967463718076648691199395367954067241097}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (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:  $16$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1792851846429171700 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{17}\cdot(2\pi)^{0}\cdot 1792851846429171700 \cdot 1}{2\sqrt{462899178676311160288470191817767102492054133121}}\approx 0.172695542633050$ (assuming GRH)

Galois group

$C_{17}$ (as 17T1):

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

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 $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$ $17$

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
953Data not computed