Normalized defining polynomial
\( x^{42} - x^{35} + x^{28} - x^{21} + x^{14} - x^{7} + 1 \)
Invariants
Degree: | $42$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 21]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-118181386580595879976868414312001964434038548836769923458287039207\) \(\medspace = -\,7^{77}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(35.43\) | 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: | $7^{11/6}\approx 35.4279812935747$ | ||
Ramified primes: | \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Gal(K/\Q) }$: | $42$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(49=7^{2}\) | ||
Dirichlet character group: | $\lbrace$$\chi_{49}(1,·)$, $\chi_{49}(2,·)$, $\chi_{49}(3,·)$, $\chi_{49}(4,·)$, $\chi_{49}(5,·)$, $\chi_{49}(6,·)$, $\chi_{49}(8,·)$, $\chi_{49}(9,·)$, $\chi_{49}(10,·)$, $\chi_{49}(11,·)$, $\chi_{49}(12,·)$, $\chi_{49}(13,·)$, $\chi_{49}(15,·)$, $\chi_{49}(16,·)$, $\chi_{49}(17,·)$, $\chi_{49}(18,·)$, $\chi_{49}(19,·)$, $\chi_{49}(20,·)$, $\chi_{49}(22,·)$, $\chi_{49}(23,·)$, $\chi_{49}(24,·)$, $\chi_{49}(25,·)$, $\chi_{49}(26,·)$, $\chi_{49}(27,·)$, $\chi_{49}(29,·)$, $\chi_{49}(30,·)$, $\chi_{49}(31,·)$, $\chi_{49}(32,·)$, $\chi_{49}(33,·)$, $\chi_{49}(34,·)$, $\chi_{49}(36,·)$, $\chi_{49}(37,·)$, $\chi_{49}(38,·)$, $\chi_{49}(39,·)$, $\chi_{49}(40,·)$, $\chi_{49}(41,·)$, $\chi_{49}(43,·)$, $\chi_{49}(44,·)$, $\chi_{49}(45,·)$, $\chi_{49}(46,·)$, $\chi_{49}(47,·)$, $\chi_{49}(48,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{1048576}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{43}$, which has order $43$ (assuming GRH)
Unit group
Rank: | $20$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( a \) (order $98$) | 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: | $a^{38}+a^{24}+a^{10}$, $a^{34}-a^{13}$, $a-1$, $a^{2}+1$, $a^{4}+1$, $a^{3}-1$, $a^{2}-a+1$, $a^{6}+1$, $a^{8}+1$, $a^{33}+a^{17}$, $a^{32}-a^{15}$, $a^{31}+a^{13}$, $a^{41}+a^{40}+a^{39}+a^{38}+a^{37}+a^{36}+a^{35}+a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{13}+a^{12}+a^{10}+a^{8}-a^{5}-a^{3}-a$, $a^{9}-1$, $a^{40}+a^{30}+a^{20}+a^{10}+1$, $a^{5}-1$, $a^{27}+a^{5}$, $a^{34}-a^{19}$, $a^{29}+a^{9}$, $a^{41}-a^{34}+a^{27}+a^{24}-a^{20}+a^{13}-a^{6}+1$ (assuming GRH) | 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: | \( 1776855897760068.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{21}\cdot 1776855897760068.5 \cdot 43}{98\cdot\sqrt{118181386580595879976868414312001964434038548836769923458287039207}}\cr\approx \mathstrut & 0.131038008164406 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 42 |
The 42 conjugacy class representatives for $C_{42}$ |
Character table for $C_{42}$ is not computed |
Intermediate fields
\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 7.7.13841287201.1, 14.0.1341068619663964900807.1, \(\Q(\zeta_{49})^+\) |
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 | $21^{2}$ | $42$ | $42$ | R | $21^{2}$ | ${\href{/padicField/13.14.0.1}{14} }^{3}$ | $42$ | ${\href{/padicField/19.6.0.1}{6} }^{7}$ | $21^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{7}$ | $21^{2}$ | ${\href{/padicField/41.14.0.1}{14} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{6}$ | $42$ | $21^{2}$ | $42$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | Deg $42$ | $42$ | $1$ | $77$ |