Normalized defining polynomial
\( x^{40} - x - 1 \)
Invariants
Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 19]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-12201853343608362939601549448988889059930192105219196517009951959\) \(\medspace = -\,25788481\cdot 47\!\cdots\!39\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(40.01\) | 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: | $25788481^{1/2}473151301296433975293137639591447400873676588598576105239^{1/2}\approx 1.1046199954558292e+32$ | ||
Ramified primes: | \(25788481\), \(47315\!\cdots\!05239\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-12201\!\cdots\!51959}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$
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)
| |
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: | $a$, $a^{39}-a^{19}-1$, $a^{11}+a$, $a^{36}-a^{35}+a^{34}-a^{33}+a^{32}$, $a^{9}+a^{5}$, $a^{39}-a^{38}+a^{37}-a^{5}-1$, $a^{12}+a^{10}$, $a^{16}-a^{3}$, $a^{12}-a^{9}$, $a^{15}-a^{12}+a^{9}$, $a^{39}-a^{38}+a^{37}-a^{36}+a^{35}+a^{11}-a^{3}-1$, $a^{39}-a^{38}+a^{37}-a^{36}+a^{32}-a^{31}+a^{30}-a^{29}+2a^{28}-a^{27}+a^{26}+a^{24}+a^{21}+a^{19}-a^{18}+a^{17}+a^{15}-a^{14}-a^{9}-a^{7}-a^{5}-a^{3}-a-2$, $a^{38}-a^{37}+a^{35}-a^{34}-a^{33}+a^{32}-a^{30}+a^{29}-a^{27}+a^{25}-a^{24}+a^{22}-a^{20}+a^{19}-a^{17}+a^{15}-a^{14}+a^{12}-a^{11}-a^{10}+a^{9}-a^{7}+a^{6}+a^{5}-a^{4}+a^{2}-a$, $a^{39}+a^{35}-a^{34}-a^{32}+a^{31}-a^{30}+a^{29}-a^{28}+a^{22}+a^{17}-a^{14}-a^{11}-a^{7}+a^{3}-1$, $a^{39}-a^{38}+a^{37}-a^{35}-a^{32}+a^{31}-a^{30}+a^{28}+a^{25}-a^{24}+a^{23}-a^{21}-a^{18}+a^{17}-a^{16}+a^{14}+a^{11}-a^{10}-a^{7}-a^{4}+a^{3}+a^{2}$, $2a^{39}-a^{38}+a^{37}-2a^{36}+2a^{35}-2a^{34}+a^{33}-a^{32}+a^{31}-2a^{30}+a^{29}-a^{28}+a^{27}-a^{26}+a^{25}-a^{24}+a^{23}-a^{22}+a^{21}-a^{18}+a^{17}-a^{16}-a^{12}-a^{8}-2$, $a^{39}-2a^{38}+a^{37}-2a^{36}+a^{35}+a^{33}+a^{31}-a^{25}-a^{24}-a^{23}+a^{20}+a^{19}+a^{16}-a^{12}-a^{11}-a^{10}+a^{7}+a^{6}+a^{5}+a-2$, $a^{33}-a^{32}+a^{31}-a^{30}+a^{29}-a^{24}+a^{23}+a^{17}+a^{12}+a^{10}+a^{6}+1$, $a^{39}-a^{38}+2a^{37}-2a^{36}+2a^{35}-2a^{34}+2a^{33}-a^{32}+a^{31}+a^{28}-2a^{27}+2a^{26}-2a^{25}+3a^{24}-2a^{23}+2a^{22}-2a^{21}+a^{20}-a^{19}+a^{18}-a^{16}+a^{15}-a^{14}+a^{13}-a^{12}+a^{11}-a^{10}+a^{9}-a^{8}+a^{4}-a$, $a^{39}-a^{38}+a^{37}+a^{35}-2a^{34}+a^{33}-3a^{32}+2a^{31}-2a^{30}+2a^{29}-2a^{28}-a^{26}+a^{24}+a^{23}+a^{22}-a^{20}+a^{19}-a^{18}+2a^{17}+a^{15}-a^{14}+a^{11}+2a^{10}+a^{9}-a^{7}-a^{6}+a^{3}-a^{2}-a-3$ (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: | \( 8469994147830637.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{19}\cdot 8469994147830637.0 \cdot 1}{2\cdot\sqrt{12201853343608362939601549448988889059930192105219196517009951959}}\cr\approx \mathstrut & 0.224449015254396 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 815915283247897734345611269596115894272000000000 |
The 37338 conjugacy class representatives for $S_{40}$ are not computed |
Character table for $S_{40}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.14.0.1}{14} }{,}\,{\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | $40$ | $29{,}\,{\href{/padicField/5.11.0.1}{11} }$ | $30{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | $20{,}\,17{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $19{,}\,15{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $31{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | $34{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $32{,}\,{\href{/padicField/41.8.0.1}{8} }$ | $24{,}\,{\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $18{,}\,15{,}\,{\href{/padicField/47.7.0.1}{7} }$ | $15{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(25788481\) | $\Q_{25788481}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
\(473\!\cdots\!239\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $29$ | $1$ | $29$ | $0$ | $C_{29}$ | $[\ ]^{29}$ |