Normalized defining polynomial
\( x^{40} - x + 1 \)
Invariants
Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(11976663048684220554521970551011110940069807894780803482990048041\) \(\medspace = 89\cdot 2963\cdot 2969\cdot 36703533640127\cdot 206798756592464370511\cdot 2015339302026829599091\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.99\) | 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: | $89^{1/2}2963^{1/2}2969^{1/2}36703533640127^{1/2}206798756592464370511^{1/2}2015339302026829599091^{1/2}\approx 1.0943794154078476e+32$ | ||
Ramified primes: | \(89\), \(2963\), \(2969\), \(36703533640127\), \(206798756592464370511\), \(2015339302026829599091\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{11976\!\cdots\!48041}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $19$ | 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: | $a$, $a^{13}-1$, $a^{32}+a^{31}+a^{30}$, $a^{9}+a$, $a^{39}+a^{38}+a^{37}+a^{36}+a^{35}+a^{34}+a^{33}+a^{12}-1$, $a^{39}+a^{38}+a^{37}+a^{36}+a^{35}+a^{34}+a^{33}+a^{30}+a^{29}-1$, $a^{39}+a^{38}+a^{37}+a^{36}-a^{4}$, $a^{39}+a^{38}+a^{37}+a^{36}-a^{16}+a^{6}-1$, $a^{12}+a^{7}+a^{2}$, $a^{39}+a^{38}+a^{37}+a^{36}+a^{35}+a^{34}+a^{33}+a^{32}+a^{31}+a^{30}+a^{29}+a^{27}+a^{25}+a^{23}+a^{21}+a^{19}+a^{17}+a^{15}+a^{13}+a^{11}-1$, $a^{38}+a^{37}+2a^{36}+2a^{35}+2a^{34}+2a^{33}+a^{32}+a^{31}-a^{29}-a^{28}-a^{27}-a^{26}-a^{25}-a^{24}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}-a^{11}-a^{10}-a^{9}-a^{8}-a^{6}+a^{2}+1$, $a^{39}+a^{38}+a^{37}+a^{36}+a^{34}-a^{32}+a^{28}+a^{26}+a^{25}-a^{21}-a^{19}-a^{18}+a^{14}+a^{12}+a^{11}-a^{10}-a^{7}-a^{5}+a^{3}-a^{2}$, $a^{34}-a^{31}-a^{30}-a^{29}-a^{20}+a^{17}+a^{16}+a^{15}+a^{6}-a^{3}-a^{2}$, $a^{39}+a^{38}+a^{37}+a^{36}+a^{31}-a^{29}-a^{28}-a^{27}-a^{22}+a^{18}-a^{14}-a^{9}+a^{5}$, $a^{39}+a^{38}+a^{37}+a^{36}+a^{35}+a^{34}+a^{33}+a^{32}+a^{28}+a^{14}-a^{11}-a^{6}-1$, $a^{39}+a^{38}-a^{37}+a^{33}-a^{29}-a^{27}-a^{26}+a^{25}+a^{22}-a^{21}+a^{19}-a^{18}+a^{16}+a^{14}+a^{13}-a^{7}+a^{6}+a^{5}-a^{4}+a^{3}-2a$, $a^{38}+a^{37}-a^{36}-2a^{35}-a^{34}+a^{32}+a^{31}-a^{30}-a^{29}-a^{28}-a^{27}+a^{26}-a^{22}-a^{21}+a^{18}+a^{17}-a^{16}-a^{15}-a^{14}+2a^{12}+a^{11}-a^{10}-a^{9}-2a^{8}+a^{7}+2a^{6}+a^{5}-2a^{3}-a^{2}+a+2$, $a^{38}-a^{36}+a^{34}-a^{33}-2a^{32}-a^{31}-a^{29}-a^{28}-a^{27}-a^{24}-2a^{23}+a^{21}+2a^{18}+2a^{17}+a^{16}+a^{13}+a^{12}-a^{11}-a^{10}+a^{9}+a^{8}-a^{7}-a^{6}-a^{3}-a^{2}+1$, $2a^{39}+a^{37}+2a^{36}+a^{35}-a^{34}+a^{32}-2a^{30}-a^{29}-2a^{27}-3a^{26}-a^{25}-2a^{23}-2a^{22}+a^{20}-a^{19}-a^{18}+a^{17}+a^{16}-a^{15}+2a^{13}+a^{12}-a^{11}+3a^{9}+a^{8}-a^{7}+2a^{5}-a^{3}+a-3$ (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: | \( 9483736502678452.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 9483736502678452.0 \cdot 1}{2\cdot\sqrt{11976663048684220554521970551011110940069807894780803482990048041}}\cr\approx \mathstrut & 0.398454682469241 \end{aligned}\] (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 $S_{40}$ is not computed |
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 | ${\href{/padicField/2.14.0.1}{14} }{,}\,{\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.13.0.1}{13} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $39{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/7.11.0.1}{11} }$ | $29{,}\,{\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | $35{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $36{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | $29{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $18{,}\,15{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $38{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(89\) | 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
89.2.1.1 | $x^{2} + 89$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $36$ | $1$ | $36$ | $0$ | $C_{36}$ | $[\ ]^{36}$ | ||
\(2963\) | $\Q_{2963}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2963}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | ||
\(2969\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $32$ | $1$ | $32$ | $0$ | 32T33 | $[\ ]^{32}$ | ||
\(36703533640127\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
\(206\!\cdots\!511\) | $\Q_{20\!\cdots\!11}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
\(201\!\cdots\!091\) | $\Q_{20\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |