Normalized defining polynomial
\( x^{42} - 2x^{21} + 2 \)
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)
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Discriminant: | \(-157423697967496898655574527484105282356823164787085832078904297275852324864\) \(\medspace = -\,2^{62}\cdot 3^{42}\cdot 7^{42}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(58.42\) | 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: | $2^{31/21}3^{7/6}7^{47/42}\approx 88.45533386347503$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $20$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -a^{21} + 1 \) (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: | $a^{28}-a^{21}-a^{7}+1$, $a^{21}+a^{7}-1$, $a^{24}-a^{21}-a^{3}+1$, $a-1$, $a^{24}-a^{3}-1$, $a^{22}-a+1$, $a^{23}-a^{2}+1$, $a^{21}+a^{6}-1$, $a^{24}+a^{6}-a^{3}-1$, $a^{27}-a^{24}+a^{21}-a^{6}+a^{3}-1$, $a^{33}-a^{27}+a^{24}-a^{21}+a^{15}-2a^{12}+a^{6}-a^{3}+1$, $a^{40}+a^{39}-a^{35}-a^{34}+a^{30}+a^{29}-a^{25}-a^{24}+a^{20}-a^{18}-a^{17}-a^{16}-a^{15}+a^{13}+a^{12}+a^{11}+a^{10}-a^{8}-a^{7}-a^{6}-a^{5}+a^{3}+a^{2}+a+1$, $a^{40}+a^{39}+a^{38}-a^{36}-a^{35}+a^{32}-a^{30}-a^{29}-a^{28}+a^{26}+a^{25}-a^{22}+a^{20}-a^{18}-a^{17}-a^{16}+a^{15}+2a^{14}+a^{13}+a^{12}-a^{11}-a^{10}+a^{8}+a^{7}+a^{6}-a^{5}-2a^{4}-a^{3}-a^{2}+a+1$, $a^{39}+a^{38}-a^{36}+a^{35}-a^{32}-a^{30}-a^{27}-a^{26}+a^{24}+a^{20}-a^{18}-2a^{17}-a^{14}-a^{12}+2a^{9}-a^{8}+a^{6}+2a^{5}-1$, $2a^{36}-a^{35}-a^{34}-2a^{31}+2a^{30}+a^{27}+2a^{26}-a^{23}-a^{22}-3a^{20}+a^{14}+2a^{13}+a^{12}+2a^{10}-4a^{9}-a^{8}-a^{6}-4a^{5}+2a^{4}+a^{2}+2a+1$, $a^{41}-4a^{40}+3a^{39}-a^{37}-a^{36}+2a^{35}+a^{34}-2a^{33}+3a^{32}-a^{31}+a^{30}-3a^{29}+4a^{28}-4a^{27}+3a^{26}-a^{25}+2a^{24}-5a^{23}+4a^{22}+a^{21}-4a^{20}+5a^{19}-4a^{18}+2a^{17}-a^{16}+5a^{15}-6a^{14}+a^{12}-4a^{10}+4a^{8}-4a^{7}+2a^{6}-2a^{5}+3a^{4}-6a^{3}+6a^{2}-6a+1$, $8a^{41}-a^{40}+9a^{39}-3a^{38}+6a^{37}-7a^{36}+a^{35}-9a^{34}+3a^{33}-2a^{32}+8a^{31}+a^{30}+7a^{29}+a^{28}+a^{27}-3a^{26}-4a^{25}-4a^{24}-4a^{23}+6a^{22}+2a^{21}-7a^{20}+2a^{19}-10a^{18}-a^{17}-8a^{16}+2a^{15}+8a^{13}+6a^{12}+2a^{11}-a^{10}-6a^{9}-4a^{8}-11a^{7}+a^{6}-7a^{5}+9a^{4}+16a^{2}-7a+7$, $a^{41}+2a^{40}-2a^{39}+5a^{38}-3a^{37}-a^{36}-3a^{33}+3a^{32}+3a^{29}-a^{28}-a^{27}+a^{26}-2a^{25}-3a^{24}+3a^{23}-3a^{22}+2a^{21}+a^{20}-4a^{19}+3a^{18}-5a^{17}+a^{15}+a^{14}-2a^{13}+6a^{12}-2a^{11}-a^{10}+3a^{9}-5a^{8}-2a^{7}+2a^{6}-a^{5}-2a^{4}+8a^{3}-3a^{2}+4a-1$, $4a^{41}+a^{40}-a^{39}+a^{38}+3a^{37}-3a^{36}-3a^{35}+a^{34}+3a^{33}+3a^{32}+a^{31}-2a^{30}-2a^{29}+a^{28}+2a^{27}-a^{25}+4a^{24}-5a^{22}-3a^{21}-2a^{20}+4a^{19}-6a^{17}-4a^{16}+5a^{15}+3a^{14}-a^{13}-3a^{12}-3a^{11}-3a^{10}+a^{9}+2a^{6}+2a^{5}-4a^{4}-9a^{3}+a^{2}+8a+1$, $2a^{41}-2a^{40}-3a^{39}-6a^{38}-6a^{37}-a^{36}+2a^{35}-3a^{33}-5a^{32}-5a^{31}+a^{30}+8a^{29}+9a^{28}+10a^{27}+2a^{26}-5a^{25}-6a^{24}+2a^{23}+6a^{22}+8a^{21}-a^{20}-5a^{19}-8a^{18}-3a^{17}+6a^{16}+7a^{15}+9a^{14}+4a^{13}+a^{12}+a^{11}+3a^{10}+2a^{9}+2a^{8}+a^{7}-8a^{6}-7a^{5}-5a^{4}-3a^{3}-3a^{2}-4a-7$ (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: | \( 507894019142814700000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{21}\cdot 507894019142814700000 \cdot 1}{4\cdot\sqrt{157423697967496898655574527484105282356823164787085832078904297275852324864}}\cr\approx \mathstrut & 0.584729881095975 \end{aligned}\] (assuming GRH)
Galois group
$D_6\times F_7$ (as 42T95):
A solvable group of order 504 |
The 42 conjugacy class representatives for $D_6\times F_7$ |
Character table for $D_6\times F_7$ is not computed |
Intermediate fields
\(\Q(\sqrt{-1}) \), 3.1.108.1, 6.0.186624.1, 7.1.52706752.1, 14.0.711168436835713024.1, 21.1.6126396525391100008339733920874496.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 42 siblings: | deg 42, deg 42, deg 42 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{6}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{6}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{6}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{7}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.14.0.1}{14} }^{2}{,}\,{\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{14}$ | ${\href{/padicField/41.2.0.1}{2} }^{20}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{4}{,}\,{\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 $42$ | $42$ | $1$ | $62$ | |||
\(3\) | 3.6.6.3 | $x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ |
Deg $18$ | $3$ | $6$ | $18$ | ||||
Deg $18$ | $3$ | $6$ | $18$ | ||||
\(7\) | Deg $42$ | $7$ | $6$ | $42$ |