Normalized defining polynomial
\( x^{35} - x^{34} - 318 x^{33} + 317 x^{32} + 43270 x^{31} - 44942 x^{30} - 3310862 x^{29} + \cdots - 70566500269 \)
Invariants
| Degree: | $35$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[35, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(126\!\cdots\!761\)
\(\medspace = 11^{28}\cdot 71^{34}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(428.04\) | 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: | $11^{4/5}71^{34/35}\approx 428.03511715789494$ | ||
| Ramified primes: |
\(11\), \(71\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $35$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(781=11\cdot 71\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{781}(256,·)$, $\chi_{781}(1,·)$, $\chi_{781}(4,·)$, $\chi_{781}(654,·)$, $\chi_{781}(16,·)$, $\chi_{781}(273,·)$, $\chi_{781}(658,·)$, $\chi_{781}(533,·)$, $\chi_{781}(537,·)$, $\chi_{781}(289,·)$, $\chi_{781}(290,·)$, $\chi_{781}(554,·)$, $\chi_{781}(555,·)$, $\chi_{781}(45,·)$, $\chi_{781}(180,·)$, $\chi_{781}(311,·)$, $\chi_{781}(570,·)$, $\chi_{781}(191,·)$, $\chi_{781}(64,·)$, $\chi_{781}(713,·)$, $\chi_{781}(586,·)$, $\chi_{781}(334,·)$, $\chi_{781}(463,·)$, $\chi_{781}(720,·)$, $\chi_{781}(718,·)$, $\chi_{781}(474,·)$, $\chi_{781}(719,·)$, $\chi_{781}(735,·)$, $\chi_{781}(529,·)$, $\chi_{781}(243,·)$, $\chi_{781}(375,·)$, $\chi_{781}(379,·)$, $\chi_{781}(764,·)$, $\chi_{781}(509,·)$, $\chi_{781}(597,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{1062797297}a^{33}-\frac{234573137}{1062797297}a^{32}-\frac{124532291}{1062797297}a^{31}-\frac{338863135}{1062797297}a^{30}-\frac{347967678}{1062797297}a^{29}+\frac{460210024}{1062797297}a^{28}+\frac{504350687}{1062797297}a^{27}+\frac{266591191}{1062797297}a^{26}+\frac{345196345}{1062797297}a^{25}-\frac{274819430}{1062797297}a^{24}+\frac{531104667}{1062797297}a^{23}+\frac{222016088}{1062797297}a^{22}+\frac{266886399}{1062797297}a^{21}-\frac{128921813}{1062797297}a^{20}+\frac{428415734}{1062797297}a^{19}-\frac{50991096}{1062797297}a^{18}-\frac{351479831}{1062797297}a^{17}+\frac{430710182}{1062797297}a^{16}-\frac{22441838}{1062797297}a^{15}-\frac{105663327}{1062797297}a^{14}-\frac{258046606}{1062797297}a^{13}+\frac{100991973}{1062797297}a^{12}+\frac{336357246}{1062797297}a^{11}-\frac{485065871}{1062797297}a^{10}-\frac{336235451}{1062797297}a^{9}+\frac{164689592}{1062797297}a^{8}-\frac{270651390}{1062797297}a^{7}-\frac{341354357}{1062797297}a^{6}+\frac{173862662}{1062797297}a^{5}-\frac{37680672}{1062797297}a^{4}-\frac{6183633}{1062797297}a^{3}+\frac{363805044}{1062797297}a^{2}-\frac{65758459}{1062797297}a-\frac{446518683}{1062797297}$, $\frac{1}{19\!\cdots\!41}a^{34}-\frac{79\!\cdots\!76}{19\!\cdots\!41}a^{33}+\frac{90\!\cdots\!67}{19\!\cdots\!41}a^{32}+\frac{78\!\cdots\!22}{19\!\cdots\!41}a^{31}+\frac{35\!\cdots\!62}{19\!\cdots\!41}a^{30}+\frac{19\!\cdots\!27}{19\!\cdots\!41}a^{29}+\frac{73\!\cdots\!64}{19\!\cdots\!41}a^{28}-\frac{97\!\cdots\!87}{19\!\cdots\!41}a^{27}-\frac{25\!\cdots\!38}{19\!\cdots\!41}a^{26}-\frac{19\!\cdots\!87}{19\!\cdots\!41}a^{25}+\frac{78\!\cdots\!02}{19\!\cdots\!41}a^{24}+\frac{14\!\cdots\!47}{19\!\cdots\!41}a^{23}-\frac{83\!\cdots\!04}{19\!\cdots\!41}a^{22}-\frac{79\!\cdots\!70}{19\!\cdots\!41}a^{21}+\frac{36\!\cdots\!20}{19\!\cdots\!41}a^{20}-\frac{71\!\cdots\!67}{19\!\cdots\!41}a^{19}+\frac{88\!\cdots\!08}{19\!\cdots\!41}a^{18}-\frac{97\!\cdots\!57}{19\!\cdots\!41}a^{17}-\frac{76\!\cdots\!66}{19\!\cdots\!41}a^{16}-\frac{52\!\cdots\!93}{19\!\cdots\!41}a^{15}-\frac{76\!\cdots\!61}{19\!\cdots\!41}a^{14}+\frac{71\!\cdots\!98}{19\!\cdots\!41}a^{13}+\frac{41\!\cdots\!81}{19\!\cdots\!41}a^{12}+\frac{37\!\cdots\!01}{19\!\cdots\!41}a^{11}-\frac{72\!\cdots\!84}{19\!\cdots\!41}a^{10}+\frac{84\!\cdots\!88}{19\!\cdots\!41}a^{9}+\frac{87\!\cdots\!37}{19\!\cdots\!41}a^{8}+\frac{49\!\cdots\!49}{19\!\cdots\!41}a^{7}-\frac{63\!\cdots\!56}{19\!\cdots\!41}a^{6}+\frac{51\!\cdots\!29}{19\!\cdots\!41}a^{5}-\frac{88\!\cdots\!26}{19\!\cdots\!41}a^{4}+\frac{70\!\cdots\!63}{19\!\cdots\!41}a^{3}-\frac{97\!\cdots\!87}{19\!\cdots\!41}a^{2}-\frac{41\!\cdots\!54}{19\!\cdots\!41}a+\frac{82\!\cdots\!15}{19\!\cdots\!41}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $34$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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 35 |
| The 35 conjugacy class representatives for $C_{35}$ |
| Character table for $C_{35}$ |
Intermediate fields
| 5.5.372052421521.2, 7.7.128100283921.1 |
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 | $35$ | $35$ | ${\href{/padicField/5.5.0.1}{5} }^{7}$ | ${\href{/padicField/7.7.0.1}{7} }^{5}$ | R | $35$ | ${\href{/padicField/17.5.0.1}{5} }^{7}$ | $35$ | ${\href{/padicField/23.7.0.1}{7} }^{5}$ | $35$ | $35$ | $35$ | $35$ | $35$ | ${\href{/padicField/47.7.0.1}{7} }^{5}$ | ${\href{/padicField/53.7.0.1}{7} }^{5}$ | $35$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| Deg $35$ | $5$ | $7$ | $28$ | |||
|
\(71\)
| Deg $35$ | $35$ | $1$ | $34$ |