Normalized defining polynomial
\( x^{27} - 5x - 2 \)
Invariants
Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(45866665066876460435700187354179628500729234048268894208\) \(\medspace = 2^{27}\cdot 7\cdot 12805208083\cdot 398913576491\cdot 9557048364382464605843591\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(115.22\) | 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: | not computed | ||
Ramified primes: | \(2\), \(7\), \(12805208083\), \(398913576491\), \(9557048364382464605843591\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{68346\!\cdots\!60322}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $6a^{26}+a^{25}+8a^{24}+2a^{23}+9a^{22}+2a^{21}+9a^{20}+6a^{18}-5a^{17}+a^{16}-11a^{15}-4a^{14}-16a^{13}-7a^{12}-19a^{11}-8a^{10}-19a^{9}-4a^{8}-14a^{7}+2a^{6}-10a^{5}+8a^{4}-3a^{3}+18a^{2}+6a-1$, $38a^{26}-15a^{25}+8a^{24}+3a^{22}+3a^{21}+5a^{20}+3a^{19}+4a^{18}+5a^{17}+3a^{16}+a^{15}+2a^{14}-a^{13}-4a^{12}-4a^{11}-5a^{10}-10a^{9}-9a^{8}-8a^{7}-11a^{6}-11a^{5}-6a^{4}-8a^{3}-7a^{2}-187$, $7a^{26}-9a^{25}+11a^{24}-11a^{23}+11a^{22}-12a^{21}+12a^{20}-12a^{19}+14a^{18}-15a^{17}+14a^{16}-15a^{15}+16a^{14}-15a^{13}+17a^{12}-20a^{11}+19a^{10}-19a^{9}+22a^{8}-21a^{7}+21a^{6}-26a^{5}+26a^{4}-23a^{3}+26a^{2}-28a-9$, $a^{26}+2a^{25}-a^{24}+a^{23}+2a^{22}-3a^{21}+a^{20}-2a^{19}-a^{18}-3a^{17}+a^{16}-2a^{15}+2a^{14}+3a^{12}+3a^{11}-a^{10}+4a^{9}-3a^{7}-3a^{6}+3a^{5}-12a^{4}+3a^{3}+2a^{2}-3a-1$, $3a^{26}-20a^{25}-42a^{24}-54a^{23}-52a^{22}-41a^{21}-21a^{20}+11a^{19}+50a^{18}+78a^{17}+89a^{16}+77a^{15}+54a^{14}+14a^{13}-44a^{12}-104a^{11}-137a^{10}-140a^{9}-113a^{8}-62a^{7}+16a^{6}+116a^{5}+200a^{4}+227a^{3}+212a^{2}+151a+37$, $9a^{26}+a^{25}-4a^{24}+a^{23}+5a^{21}-7a^{20}-4a^{19}+11a^{18}-9a^{17}-3a^{16}+a^{15}-2a^{14}+7a^{13}-15a^{12}-2a^{11}+19a^{10}-17a^{9}-4a^{8}+10a^{7}+6a^{6}-3a^{5}-16a^{4}+32a^{3}+14a^{2}-40a-15$, $125a^{26}-49a^{25}+20a^{24}-9a^{23}+4a^{22}-a^{21}-a^{19}+3a^{18}-2a^{17}+a^{16}+a^{14}-4a^{13}+2a^{12}-2a^{10}+a^{9}+a^{8}-3a^{6}+5a^{5}-5a^{4}+3a-629$, $70a^{26}-3a^{25}-82a^{24}+19a^{23}+91a^{22}-38a^{21}-96a^{20}+60a^{19}+96a^{18}-84a^{17}-89a^{16}+109a^{15}+73a^{14}-134a^{13}-47a^{12}+157a^{11}+10a^{10}-176a^{9}+38a^{8}+188a^{7}-98a^{6}-192a^{5}+169a^{4}+186a^{3}-247a^{2}-164a-19$, $7a^{26}+4a^{25}+3a^{24}-10a^{23}+4a^{22}-6a^{21}+26a^{20}-41a^{19}+26a^{18}+a^{17}-6a^{16}-10a^{15}+15a^{14}-3a^{13}+16a^{12}-55a^{11}+62a^{10}-16a^{9}-16a^{8}+24a^{6}-13a^{5}+6a^{4}-48a^{3}+98a^{2}-74a-47$, $2a^{26}+2a^{25}-2a^{24}+2a^{23}+5a^{21}+a^{20}+4a^{19}-a^{18}+4a^{17}+5a^{15}+7a^{13}+3a^{12}+7a^{11}-3a^{10}+3a^{9}+a^{8}+14a^{7}+9a^{6}+12a^{5}-4a^{4}+2a^{3}+2a^{2}+23a+5$, $7a^{26}-9a^{25}+16a^{24}-16a^{23}+8a^{22}-5a^{21}-7a^{20}+22a^{19}-22a^{18}+15a^{17}-14a^{16}+3a^{15}+12a^{14}-11a^{13}+16a^{12}-31a^{11}+28a^{10}-10a^{9}+7a^{8}+18a^{7}-46a^{6}+39a^{5}-23a^{4}+24a^{3}-29a-15$, $2a^{26}+a^{25}-a^{24}+2a^{23}-2a^{21}+a^{20}-2a^{18}+5a^{17}-3a^{16}+4a^{15}-a^{14}+a^{13}-2a^{12}-a^{11}-a^{9}+4a^{8}+2a^{7}+2a^{4}-10a^{3}+10a^{2}-9a-7$, $13a^{26}-13a^{25}-a^{24}-a^{23}+5a^{22}+5a^{21}-16a^{20}+12a^{19}+2a^{18}+12a^{17}-10a^{16}-9a^{15}+14a^{14}-3a^{13}+a^{12}-28a^{11}+7a^{10}+17a^{9}-9a^{8}-17a^{6}+41a^{5}+9a^{4}-14a^{3}-13a^{2}+2a-29$, $9a^{26}+4a^{25}+4a^{24}-a^{23}-5a^{22}+13a^{21}+13a^{20}-14a^{19}-13a^{18}-2a^{17}-4a^{16}+13a^{15}+13a^{14}-17a^{13}-5a^{12}+27a^{11}+24a^{10}+13a^{9}-4a^{8}-23a^{7}-5a^{6}+21a^{5}+11a^{4}-27a^{3}-45a^{2}+4a+7$ (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: | \( 1078482016789482100 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot 1078482016789482100 \cdot 1}{2\cdot\sqrt{45866665066876460435700187354179628500729234048268894208}}\cr\approx \mathstrut & 2.41147485303619 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 10888869450418352160768000000 |
The 3010 conjugacy class representatives for $S_{27}$ are not computed |
Character table for $S_{27}$ 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 | R | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | $25{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | $19{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{3}{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | $27$ | ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
Deg $24$ | $2$ | $12$ | $24$ | ||||
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.5.0.1 | $x^{5} + x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
7.12.0.1 | $x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(12805208083\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
\(398913576491\) | $\Q_{398913576491}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(955\!\cdots\!591\) | $\Q_{95\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{95\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{95\!\cdots\!91}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |