Global number field 17.17.462899178676311160288470191817767102492054133121.1

• Label: 17.17.4628991786...3121.1
• Degree: $17$
• Signature: $[17, 0]$
• Discriminant: $953^{16}$
• Root discriminant: $636.58$
• Ramified prime: $953$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{17}$ (as 17T1)

Normalized defining polynomial

\( x^{17} - x^{16} - 448 x^{15} + 1309 x^{14} + 75494 x^{13} - 374314 x^{12} - 5597667 x^{11} + 41830550 x^{10} + 136426018 x^{9} - 1919481097 x^{8} + 2548312782 x^{7} + 27117309376 x^{6} - 105628969954 x^{5} + 34101907629 x^{4} + 457076113374 x^{3} - 817186035962 x^{2} + 323264486326 x + 117028501127 \)

Invariants

Degree:		$17$
Signature:		$[17, 0]$
Discriminant:		\(462899178676311160288470191817767102492054133121=953^{16}\)
Root discriminant:		$636.58$
Ramified primes:		$953$
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}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$16$
Torsion generator:		\( -1 \) (order $2$)
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)
Regulator:		\( 1792851846429171700 \) (assuming GRH)

Galois group

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

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.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
953	Data not computed