Normalized defining polynomial
\( x^{34} + 97 x^{32} + 4040 x^{30} + 96737 x^{28} + 1499007 x^{26} + 15996211 x^{24} + 121823703 x^{22} + \cdots + 8048569 \)
Invariants
| Degree: | $34$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 17]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-442395848806444333196713449710663325979115564010458272524910178228286521344\)
\(\medspace = -\,2^{34}\cdot 103^{32}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(156.84\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2\cdot 103^{16/17}\approx 156.8429602905213$ | ||
| Ramified primes: |
\(2\), \(103\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $34$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(412=2^{2}\cdot 103\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{412}(1,·)$, $\chi_{412}(133,·)$, $\chi_{412}(385,·)$, $\chi_{412}(9,·)$, $\chi_{412}(267,·)$, $\chi_{412}(13,·)$, $\chi_{412}(23,·)$, $\chi_{412}(409,·)$, $\chi_{412}(111,·)$, $\chi_{412}(285,·)$, $\chi_{412}(287,·)$, $\chi_{412}(219,·)$, $\chi_{412}(167,·)$, $\chi_{412}(169,·)$, $\chi_{412}(299,·)$, $\chi_{412}(175,·)$, $\chi_{412}(179,·)$, $\chi_{412}(137,·)$, $\chi_{412}(61,·)$, $\chi_{412}(215,·)$, $\chi_{412}(117,·)$, $\chi_{412}(323,·)$, $\chi_{412}(203,·)$, $\chi_{412}(207,·)$, $\chi_{412}(81,·)$, $\chi_{412}(339,·)$, $\chi_{412}(343,·)$, $\chi_{412}(79,·)$, $\chi_{412}(93,·)$, $\chi_{412}(229,·)$, $\chi_{412}(317,·)$, $\chi_{412}(373,·)$, $\chi_{412}(375,·)$, $\chi_{412}(381,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{65536}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{149}a^{26}+\frac{50}{149}a^{24}-\frac{51}{149}a^{22}-\frac{40}{149}a^{20}-\frac{8}{149}a^{18}-\frac{1}{149}a^{16}-\frac{39}{149}a^{14}+\frac{58}{149}a^{12}-\frac{13}{149}a^{10}+\frac{20}{149}a^{8}-\frac{54}{149}a^{6}+\frac{49}{149}a^{4}+\frac{44}{149}a^{2}+\frac{42}{149}$, $\frac{1}{149}a^{27}+\frac{50}{149}a^{25}-\frac{51}{149}a^{23}-\frac{40}{149}a^{21}-\frac{8}{149}a^{19}-\frac{1}{149}a^{17}-\frac{39}{149}a^{15}+\frac{58}{149}a^{13}-\frac{13}{149}a^{11}+\frac{20}{149}a^{9}-\frac{54}{149}a^{7}+\frac{49}{149}a^{5}+\frac{44}{149}a^{3}+\frac{42}{149}a$, $\frac{1}{149}a^{28}-\frac{18}{149}a^{24}-\frac{23}{149}a^{22}+\frac{55}{149}a^{20}-\frac{48}{149}a^{18}+\frac{11}{149}a^{16}+\frac{71}{149}a^{14}+\frac{67}{149}a^{12}+\frac{74}{149}a^{10}-\frac{11}{149}a^{8}+\frac{67}{149}a^{6}-\frac{22}{149}a^{4}-\frac{72}{149}a^{2}-\frac{14}{149}$, $\frac{1}{149}a^{29}-\frac{18}{149}a^{25}-\frac{23}{149}a^{23}+\frac{55}{149}a^{21}-\frac{48}{149}a^{19}+\frac{11}{149}a^{17}+\frac{71}{149}a^{15}+\frac{67}{149}a^{13}+\frac{74}{149}a^{11}-\frac{11}{149}a^{9}+\frac{67}{149}a^{7}-\frac{22}{149}a^{5}-\frac{72}{149}a^{3}-\frac{14}{149}a$, $\frac{1}{7612261}a^{30}-\frac{19974}{7612261}a^{28}+\frac{1006}{7612261}a^{26}-\frac{754918}{7612261}a^{24}-\frac{2875684}{7612261}a^{22}-\frac{54262}{7612261}a^{20}+\frac{2778503}{7612261}a^{18}+\frac{3085792}{7612261}a^{16}+\frac{2129301}{7612261}a^{14}+\frac{1364766}{7612261}a^{12}+\frac{2298267}{7612261}a^{10}+\frac{3435119}{7612261}a^{8}+\frac{1434444}{7612261}a^{6}-\frac{2516096}{7612261}a^{4}-\frac{2563819}{7612261}a^{2}-\frac{2519233}{7612261}$, $\frac{1}{7612261}a^{31}-\frac{19974}{7612261}a^{29}+\frac{1006}{7612261}a^{27}-\frac{754918}{7612261}a^{25}-\frac{2875684}{7612261}a^{23}-\frac{54262}{7612261}a^{21}+\frac{2778503}{7612261}a^{19}+\frac{3085792}{7612261}a^{17}+\frac{2129301}{7612261}a^{15}+\frac{1364766}{7612261}a^{13}+\frac{2298267}{7612261}a^{11}+\frac{3435119}{7612261}a^{9}+\frac{1434444}{7612261}a^{7}-\frac{2516096}{7612261}a^{5}-\frac{2563819}{7612261}a^{3}-\frac{2519233}{7612261}a$, $\frac{1}{41\!\cdots\!53}a^{32}+\frac{26\!\cdots\!49}{41\!\cdots\!53}a^{30}+\frac{86\!\cdots\!66}{41\!\cdots\!53}a^{28}+\frac{95\!\cdots\!22}{41\!\cdots\!53}a^{26}-\frac{61\!\cdots\!93}{41\!\cdots\!53}a^{24}-\frac{29\!\cdots\!04}{88\!\cdots\!99}a^{22}-\frac{16\!\cdots\!30}{41\!\cdots\!53}a^{20}-\frac{65\!\cdots\!96}{41\!\cdots\!53}a^{18}-\frac{41\!\cdots\!70}{41\!\cdots\!53}a^{16}+\frac{20\!\cdots\!50}{41\!\cdots\!53}a^{14}+\frac{17\!\cdots\!58}{41\!\cdots\!53}a^{12}+\frac{98\!\cdots\!01}{41\!\cdots\!53}a^{10}+\frac{89\!\cdots\!95}{41\!\cdots\!53}a^{8}-\frac{18\!\cdots\!11}{41\!\cdots\!53}a^{6}-\frac{15\!\cdots\!92}{41\!\cdots\!53}a^{4}-\frac{18\!\cdots\!70}{41\!\cdots\!53}a^{2}-\frac{41\!\cdots\!25}{41\!\cdots\!53}$, $\frac{1}{11\!\cdots\!61}a^{33}-\frac{92\!\cdots\!61}{11\!\cdots\!61}a^{31}+\frac{22\!\cdots\!19}{11\!\cdots\!61}a^{29}+\frac{20\!\cdots\!28}{11\!\cdots\!61}a^{27}-\frac{12\!\cdots\!80}{11\!\cdots\!61}a^{25}-\frac{28\!\cdots\!20}{11\!\cdots\!61}a^{23}+\frac{90\!\cdots\!75}{11\!\cdots\!61}a^{21}-\frac{38\!\cdots\!32}{11\!\cdots\!61}a^{19}-\frac{51\!\cdots\!44}{11\!\cdots\!61}a^{17}-\frac{15\!\cdots\!56}{11\!\cdots\!61}a^{15}+\frac{22\!\cdots\!74}{11\!\cdots\!61}a^{13}+\frac{47\!\cdots\!37}{11\!\cdots\!61}a^{11}-\frac{23\!\cdots\!26}{11\!\cdots\!61}a^{9}-\frac{41\!\cdots\!30}{11\!\cdots\!61}a^{7}+\frac{93\!\cdots\!72}{11\!\cdots\!61}a^{5}+\frac{27\!\cdots\!37}{11\!\cdots\!61}a^{3}+\frac{48\!\cdots\!19}{11\!\cdots\!61}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $16$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{6584604691613314584410232644810856}{175691180390737874904529991916776925161} a^{33} + \frac{618612329626957855436301613128915806}{175691180390737874904529991916776925161} a^{31} + \frac{24713817074185250208824167189890342855}{175691180390737874904529991916776925161} a^{29} + \frac{561540738274339995777521393051402837985}{175691180390737874904529991916776925161} a^{27} + \frac{8156131082120196637394360453692187207549}{175691180390737874904529991916776925161} a^{25} + \frac{80425757876536952948256112821231646257473}{175691180390737874904529991916776925161} a^{23} + \frac{556543964208175475927708689737503525208867}{175691180390737874904529991916776925161} a^{21} + \frac{2748129970372826328650487127736848762555126}{175691180390737874904529991916776925161} a^{19} + \frac{9725901127645103558088469325986608258544041}{175691180390737874904529991916776925161} a^{17} + \frac{24507991166461980786549605223032688635076258}{175691180390737874904529991916776925161} a^{15} + \frac{43170973671177914805721133534382400780860678}{175691180390737874904529991916776925161} a^{13} + \frac{51452192474828329626048270856314689082454663}{175691180390737874904529991916776925161} a^{11} + \frac{39415022973935949991658147747138208085265001}{175691180390737874904529991916776925161} a^{9} + \frac{17989903463441247259896493217492020586460844}{175691180390737874904529991916776925161} a^{7} + \frac{4396569637969126262942417169215696981937497}{175691180390737874904529991916776925161} a^{5} + \frac{490204284213911053895895068871093321232306}{175691180390737874904529991916776925161} a^{3} + \frac{16239318835584517704845353142103350429369}{175691180390737874904529991916776925161} a \)
(order $4$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
Galois group
| A cyclic group of order 34 |
| The 34 conjugacy class representatives for $C_{34}$ |
| Character table for $C_{34}$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 17.17.160470643909878751793805444097921.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $34$ | $17^{2}$ | $34$ | $34$ | $17^{2}$ | $17^{2}$ | $34$ | $34$ | $17^{2}$ | $34$ | $17^{2}$ | $17^{2}$ | $34$ | ${\href{/padicField/47.2.0.1}{2} }^{17}$ | $17^{2}$ | $34$ |
In the table, R denotes a ramified prime. 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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $34$ | $2$ | $17$ | $34$ | |||
|
\(103\)
| Deg $34$ | $17$ | $2$ | $32$ |