Normalized defining polynomial
\( x^{42} + 3 \)
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: | \(-1245035751069475618296892513017846796759617999278950389304336491491851278123\) \(\medspace = -\,3^{83}\cdot 7^{42}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(61.37\) | 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: | $3^{83/42}7^{47/42}\approx 77.37264242492355$ | ||
Ramified primes: | \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2}a^{21}-\frac{1}{2}$, $\frac{1}{2}a^{22}-\frac{1}{2}a$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{31}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{32}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{12}$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{13}$, $\frac{1}{2}a^{35}-\frac{1}{2}a^{14}$, $\frac{1}{2}a^{36}-\frac{1}{2}a^{15}$, $\frac{1}{2}a^{37}-\frac{1}{2}a^{16}$, $\frac{1}{2}a^{38}-\frac{1}{2}a^{17}$, $\frac{1}{2}a^{39}-\frac{1}{2}a^{18}$, $\frac{1}{2}a^{40}-\frac{1}{2}a^{19}$, $\frac{1}{2}a^{41}-\frac{1}{2}a^{20}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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)
| |
Torsion generator: | \( -\frac{1}{2} a^{21} + \frac{1}{2} \) (order $6$) | 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: | $\frac{1}{2}a^{24}+\frac{1}{2}a^{3}+1$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{21}+\frac{1}{2}a^{3}+\frac{1}{2}$, $\frac{1}{2}a^{36}+\frac{1}{2}a^{27}-\frac{1}{2}a^{21}-\frac{1}{2}a^{15}-a^{12}-\frac{1}{2}a^{6}-a^{3}+\frac{1}{2}$, $\frac{1}{2}a^{39}-\frac{1}{2}a^{36}-\frac{1}{2}a^{33}+\frac{1}{2}a^{30}+\frac{1}{2}a^{27}-\frac{1}{2}a^{24}-\frac{1}{2}a^{21}+\frac{1}{2}a^{18}+\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}+\frac{1}{2}a^{6}+\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{40}+\frac{1}{2}a^{39}-\frac{1}{2}a^{37}-\frac{1}{2}a^{36}+\frac{1}{2}a^{34}+\frac{1}{2}a^{33}-\frac{1}{2}a^{31}-\frac{1}{2}a^{30}+\frac{1}{2}a^{28}+\frac{1}{2}a^{27}-\frac{1}{2}a^{25}-\frac{1}{2}a^{24}+\frac{1}{2}a^{21}+a^{20}+\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-a^{17}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+a^{14}+\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-a^{11}-\frac{1}{2}a^{10}+\frac{1}{2}a^{9}+a^{8}+\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-a^{5}-\frac{1}{2}a^{4}+\frac{1}{2}a^{3}+a^{2}+a+\frac{1}{2}$, $\frac{1}{2}a^{41}-\frac{1}{2}a^{40}-\frac{1}{2}a^{37}+\frac{1}{2}a^{36}-\frac{1}{2}a^{35}+\frac{1}{2}a^{34}+\frac{1}{2}a^{31}-\frac{1}{2}a^{30}+\frac{1}{2}a^{29}-\frac{1}{2}a^{28}-\frac{1}{2}a^{25}+\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-a^{18}+a^{17}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+a^{12}-a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{1}{2}a^{7}-a^{6}+a^{5}-\frac{1}{2}a^{4}+\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}a+1$, $\frac{1}{2}a^{40}-\frac{1}{2}a^{39}+\frac{1}{2}a^{37}-\frac{1}{2}a^{36}+\frac{1}{2}a^{34}-\frac{1}{2}a^{33}+\frac{1}{2}a^{31}-\frac{1}{2}a^{30}+\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{25}-\frac{1}{2}a^{24}+\frac{1}{2}a^{22}-a^{20}+\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-a^{14}+\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-a^{11}+\frac{1}{2}a^{10}+\frac{1}{2}a^{9}-a^{8}+\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-a^{5}+\frac{1}{2}a^{4}+\frac{1}{2}a^{3}-a^{2}-\frac{1}{2}a+1$, $\frac{1}{2}a^{41}-\frac{1}{2}a^{39}+\frac{1}{2}a^{38}-\frac{1}{2}a^{36}+\frac{1}{2}a^{35}-\frac{1}{2}a^{33}+\frac{1}{2}a^{32}-\frac{1}{2}a^{30}+\frac{1}{2}a^{29}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{24}+\frac{1}{2}a^{23}-\frac{1}{2}a^{22}+\frac{1}{2}a^{20}-a^{19}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{11}-a^{10}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-a^{7}+\frac{1}{2}a^{6}+\frac{1}{2}a^{5}-a^{4}+\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}a+1$, $\frac{1}{2}a^{35}-\frac{1}{2}a^{14}-a^{7}-1$, $\frac{1}{2}a^{35}+\frac{1}{2}a^{28}-\frac{1}{2}a^{21}-\frac{1}{2}a^{14}+\frac{1}{2}a^{7}+\frac{1}{2}$, $\frac{1}{2}a^{39}+\frac{1}{2}a^{33}+\frac{1}{2}a^{27}+\frac{1}{2}a^{21}+\frac{1}{2}a^{18}+\frac{1}{2}a^{12}+\frac{1}{2}a^{6}-a^{3}+\frac{1}{2}$, $\frac{1}{2}a^{36}+\frac{1}{2}a^{30}+\frac{1}{2}a^{27}+\frac{1}{2}a^{21}+\frac{1}{2}a^{15}-\frac{1}{2}a^{9}+\frac{1}{2}a^{6}-a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{41}-\frac{1}{2}a^{39}+\frac{1}{2}a^{38}+\frac{1}{2}a^{37}+\frac{1}{2}a^{36}+\frac{1}{2}a^{35}+\frac{1}{2}a^{34}+\frac{1}{2}a^{33}-a^{32}-a^{31}+\frac{1}{2}a^{30}+\frac{1}{2}a^{29}+\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}+\frac{3}{2}a^{25}-\frac{1}{2}a^{24}-\frac{3}{2}a^{23}-a^{22}+\frac{1}{2}a^{20}-a^{19}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{3}{2}a^{12}-2a^{11}+a^{10}+\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{3}{2}a^{7}-\frac{1}{2}a^{6}+\frac{3}{2}a^{5}-\frac{3}{2}a^{4}-\frac{7}{2}a^{3}-\frac{3}{2}a^{2}+1$, $\frac{1}{2}a^{41}-\frac{1}{2}a^{39}+\frac{1}{2}a^{38}-\frac{1}{2}a^{37}+\frac{1}{2}a^{36}-\frac{1}{2}a^{34}+a^{33}-\frac{3}{2}a^{32}+a^{31}-\frac{1}{2}a^{28}+a^{26}-\frac{3}{2}a^{25}+\frac{3}{2}a^{24}-\frac{3}{2}a^{23}+a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-a^{14}+\frac{1}{2}a^{13}+\frac{1}{2}a^{11}-a^{10}+a^{9}-a^{8}+\frac{3}{2}a^{7}-a^{6}+\frac{3}{2}a^{4}-\frac{5}{2}a^{3}+\frac{3}{2}a^{2}+\frac{1}{2}$, $7a^{41}+4a^{40}-5a^{38}-\frac{17}{2}a^{37}-9a^{36}-7a^{35}-\frac{7}{2}a^{34}+2a^{33}+7a^{32}+\frac{19}{2}a^{31}+\frac{19}{2}a^{30}+7a^{29}+2a^{28}-4a^{27}-8a^{26}-\frac{19}{2}a^{25}-\frac{17}{2}a^{24}-\frac{11}{2}a^{23}-\frac{1}{2}a^{22}+\frac{7}{2}a^{21}+6a^{20}+8a^{19}+8a^{18}+6a^{17}+\frac{7}{2}a^{16}-5a^{14}-\frac{17}{2}a^{13}-10a^{12}-10a^{11}-\frac{15}{2}a^{10}-\frac{3}{2}a^{9}+4a^{8}+10a^{7}+15a^{6}+16a^{5}+\frac{23}{2}a^{4}+\frac{7}{2}a^{3}-\frac{13}{2}a^{2}-\frac{35}{2}a-\frac{47}{2}$, $\frac{1}{2}a^{41}+2a^{40}-4a^{39}+\frac{11}{2}a^{38}-6a^{37}+\frac{9}{2}a^{36}-a^{35}-\frac{5}{2}a^{34}+6a^{33}-\frac{17}{2}a^{32}+\frac{17}{2}a^{31}-\frac{11}{2}a^{30}+\frac{3}{2}a^{29}+4a^{28}-\frac{19}{2}a^{27}+12a^{26}-\frac{21}{2}a^{25}+6a^{24}+\frac{1}{2}a^{23}-\frac{17}{2}a^{22}+\frac{27}{2}a^{21}-\frac{27}{2}a^{20}+10a^{19}-4a^{18}-\frac{9}{2}a^{17}+11a^{16}-\frac{27}{2}a^{15}+13a^{14}-\frac{19}{2}a^{13}+2a^{12}+\frac{11}{2}a^{11}-\frac{21}{2}a^{10}+\frac{27}{2}a^{9}-\frac{25}{2}a^{8}+8a^{7}-\frac{3}{2}a^{6}-4a^{5}+\frac{17}{2}a^{4}-11a^{3}+\frac{23}{2}a^{2}-\frac{19}{2}a+\frac{13}{2}$, $\frac{5}{2}a^{41}+a^{40}+2a^{39}+\frac{3}{2}a^{38}-\frac{1}{2}a^{36}-\frac{1}{2}a^{35}-2a^{34}-\frac{3}{2}a^{33}-\frac{1}{2}a^{32}-\frac{3}{2}a^{31}+\frac{1}{2}a^{29}+\frac{3}{2}a^{28}+\frac{3}{2}a^{27}+3a^{26}+\frac{5}{2}a^{25}-\frac{1}{2}a^{24}-3a^{22}-\frac{11}{2}a^{21}-\frac{7}{2}a^{20}-2a^{19}-2a^{18}+\frac{9}{2}a^{17}+6a^{16}+\frac{9}{2}a^{15}+\frac{15}{2}a^{14}+2a^{13}-\frac{3}{2}a^{12}-\frac{9}{2}a^{11}-\frac{15}{2}a^{10}-8a^{9}-\frac{13}{2}a^{8}-\frac{3}{2}a^{7}+\frac{5}{2}a^{6}+6a^{5}+\frac{17}{2}a^{4}+\frac{17}{2}a^{3}+2a^{2}+2a-\frac{5}{2}$, $3a^{41}+\frac{1}{2}a^{40}-2a^{39}+\frac{3}{2}a^{38}+3a^{37}-\frac{1}{2}a^{36}-4a^{35}+\frac{3}{2}a^{33}-2a^{32}-\frac{9}{2}a^{31}+\frac{7}{2}a^{29}-\frac{3}{2}a^{27}+\frac{3}{2}a^{26}+\frac{11}{2}a^{25}-\frac{1}{2}a^{24}-a^{23}+\frac{1}{2}a^{22}+\frac{11}{2}a^{21}-2a^{20}-\frac{5}{2}a^{19}-2a^{18}+\frac{9}{2}a^{17}-3a^{16}-\frac{9}{2}a^{15}-3a^{14}+4a^{13}-\frac{3}{2}a^{12}-3a^{11}+\frac{3}{2}a^{10}+8a^{9}+\frac{5}{2}a^{8}-3a^{7}+\frac{5}{2}a^{6}+\frac{15}{2}a^{5}-\frac{1}{2}a^{4}-\frac{17}{2}a^{3}-a^{2}+\frac{11}{2}a-\frac{5}{2}$, $\frac{3}{2}a^{41}-2a^{39}+\frac{1}{2}a^{38}-2a^{37}-a^{36}+2a^{35}-\frac{3}{2}a^{34}+\frac{5}{2}a^{32}-3a^{31}-\frac{1}{2}a^{30}+3a^{29}-\frac{7}{2}a^{28}+\frac{1}{2}a^{27}+\frac{9}{2}a^{26}-\frac{9}{2}a^{25}+\frac{3}{2}a^{24}+4a^{23}-\frac{9}{2}a^{22}+3a^{21}+\frac{9}{2}a^{20}-5a^{19}+2a^{18}+\frac{5}{2}a^{17}-7a^{16}+2a^{15}+4a^{14}-\frac{11}{2}a^{13}+5a^{12}+\frac{5}{2}a^{11}-7a^{10}+\frac{3}{2}a^{9}-2a^{8}-\frac{13}{2}a^{7}+\frac{7}{2}a^{6}+\frac{3}{2}a^{5}-\frac{3}{2}a^{4}+\frac{11}{2}a^{3}-a^{2}-\frac{9}{2}a+2$, $2a^{41}+\frac{5}{2}a^{40}+\frac{1}{2}a^{39}-2a^{38}-\frac{5}{2}a^{37}-3a^{36}-\frac{3}{2}a^{35}+\frac{1}{2}a^{34}+\frac{1}{2}a^{33}+\frac{5}{2}a^{32}+\frac{5}{2}a^{31}+\frac{3}{2}a^{30}-a^{29}-4a^{28}-\frac{9}{2}a^{27}-a^{26}-2a^{25}+\frac{1}{2}a^{24}+\frac{5}{2}a^{23}+\frac{7}{2}a^{22}+4a^{21}-2a^{20}-\frac{9}{2}a^{19}-\frac{7}{2}a^{18}-4a^{17}-\frac{5}{2}a^{16}+\frac{3}{2}a^{14}+\frac{15}{2}a^{13}+\frac{7}{2}a^{12}-\frac{1}{2}a^{11}-\frac{7}{2}a^{10}-\frac{11}{2}a^{9}-3a^{8}-4a^{7}-\frac{7}{2}a^{6}+5a^{5}+9a^{4}+\frac{9}{2}a^{3}+\frac{3}{2}a^{2}-\frac{7}{2}a-2$ (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: | \( 2625406827600082000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{21}\cdot 2625406827600082000000 \cdot 1}{6\cdot\sqrt{1245035751069475618296892513017846796759617999278950389304336491491851278123}}\cr\approx \mathstrut & 0.716525555178365 \end{aligned}\] (assuming GRH)
Galois group
$S_3\times F_7$ (as 42T45):
A solvable group of order 252 |
The 21 conjugacy class representatives for $S_3\times F_7$ |
Character table for $S_3\times F_7$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 6.0.177147.2, 7.1.600362847.1, 14.0.1081306644173836227.1, 21.1.20371841277193344052610353606982956821.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 21 sibling: | data not computed |
Degree 42 siblings: | data not computed |
Minimal sibling: | 21.1.20371841277193344052610353606982956821.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{6}{,}\,{\href{/padicField/2.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{6}{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{6}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{6}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.14.0.1}{14} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{14}$ | ${\href{/padicField/41.2.0.1}{2} }^{21}$ | $21^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{6}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $42$ | $42$ | $1$ | $83$ | |||
\(7\) | Deg $21$ | $7$ | $3$ | $21$ | |||
Deg $21$ | $7$ | $3$ | $21$ |