Normalized defining polynomial
\( x^{36} - x + 1 \)
Invariants
Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(105284851424362376111761689319042885392000913033423744261\) \(\medspace = 31\cdot 43\cdot 12231367\cdot 64\!\cdots\!51\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(35.99\) | 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: | $31^{1/2}43^{1/2}12231367^{1/2}6457445387287894797679050738460435100440856551^{1/2}\approx 1.0260840678246708e+28$ | ||
Ramified primes: | \(31\), \(43\), \(12231367\), \(64574\!\cdots\!56551\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{10528\!\cdots\!44261}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $17$ | 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: | $a^{35}-1$, $a^{12}+1$, $a^{8}+a^{4}$, $a^{35}+a^{34}+a^{5}-1$, $a^{35}+a^{34}+a^{33}+a^{2}-1$, $a^{35}+a^{34}+a^{33}+a^{32}+a^{31}+a^{30}+a^{29}+a^{10}-1$, $a^{35}+a^{34}+a^{33}-a^{15}-a^{6}-1$, $a^{35}+a^{34}+a^{33}+a^{32}+a^{31}+a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+a^{24}+a^{22}+a^{20}+a^{18}+a^{16}+a^{14}+a^{12}+a^{10}-1$, $a^{18}+a^{17}-a^{15}-a^{14}-a^{13}+a^{11}+a^{10}$, $a^{34}+a^{33}+a^{32}+a^{31}-a^{29}-a^{28}+a^{26}+a^{25}+a^{24}+a^{23}-a^{21}-a^{20}+a^{16}+a^{15}-a^{13}-a^{12}+a^{8}+a^{7}-a^{5}-a^{2}+1$, $a^{35}-a^{27}+a^{25}-a^{24}-a^{23}-a^{20}-a^{19}-a^{16}-a^{13}-a^{10}-a^{6}+a^{4}-a^{3}-a^{2}+a-1$, $a^{32}-a^{28}-a^{25}-a^{23}-a^{22}-a^{20}-a^{18}-a^{17}+a^{13}-a^{12}+a^{8}-a^{6}+a^{5}-a^{4}+a^{3}-a$, $5a^{35}+5a^{34}+5a^{33}+5a^{32}+4a^{31}+4a^{30}+4a^{29}+4a^{28}+4a^{27}+3a^{26}+3a^{25}+3a^{24}+3a^{23}+2a^{22}+2a^{21}+2a^{20}+2a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{12}+a^{11}+a^{10}-6$, $a^{35}-a^{32}-a^{31}+a^{28}+2a^{26}+a^{25}+a^{24}-a^{20}-a^{19}+a^{16}+2a^{14}+a^{12}-a^{11}-a^{9}-a^{8}+a^{5}+a^{4}+a^{3}+a^{2}-a-1$, $a^{33}+2a^{32}+a^{29}+a^{28}-a^{27}-a^{26}+a^{25}+a^{24}+a^{21}-a^{19}+a^{17}-a^{16}-2a^{15}+2a^{13}-a^{11}+a^{9}-a^{7}+a^{6}-a^{4}-a^{3}+2a^{2}+a-1$, $4a^{35}+5a^{34}+4a^{33}+3a^{32}+3a^{31}+3a^{30}+2a^{29}+2a^{28}+a^{27}-a^{25}-a^{24}-2a^{23}-2a^{22}-2a^{21}-2a^{20}-3a^{19}-2a^{18}-2a^{17}-2a^{16}-a^{15}-a^{13}+a^{11}+a^{9}+2a^{8}+a^{7}+a^{6}+2a^{5}+a^{3}+2a^{2}+a-4$, $a^{35}+a^{32}+a^{31}+a^{30}+a^{29}+a^{26}+a^{23}+a^{21}+a^{20}-a^{16}+a^{14}+a^{12}-a^{10}-a^{8}-a^{7}+a^{6}+a^{5}-a^{2}-a$ (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: | \( 42496810214502.4 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 42496810214502.4 \cdot 1}{2\cdot\sqrt{105284851424362376111761689319042885392000913033423744261}}\cr\approx \mathstrut & 0.482370815184618 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 371993326789901217467999448150835200000000 |
The 17977 conjugacy class representatives for $S_{36}$ are not computed |
Character table for $S_{36}$ |
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 | $17{,}\,{\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.9.0.1}{9} }$ | $19{,}\,{\href{/padicField/3.13.0.1}{13} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $32{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | R | $32{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | R | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $34{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(31\) | 31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
31.6.0.1 | $x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(43\) | $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
43.5.0.1 | $x^{5} + 8 x + 40$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
43.6.0.1 | $x^{6} + 19 x^{3} + 28 x^{2} + 21 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
43.14.0.1 | $x^{14} + 38 x^{7} + 22 x^{6} + 24 x^{5} + 37 x^{4} + 18 x^{3} + 4 x^{2} + 19 x + 3$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(12231367\) | $\Q_{12231367}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{12231367}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(645\!\cdots\!551\) | $\Q_{64\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{64\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{64\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{64\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{64\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{64\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{64\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ |