Normalized defining polynomial
\( x^{13} - 2 x^{12} - 8 x^{11} + 17 x^{10} + 21 x^{9} - 53 x^{8} - 19 x^{7} + 75 x^{6} - 48 x^{4} + \cdots - 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[9, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(737652920184769\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.92\) | 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: | $737652920184769^{1/2}\approx 27159766.57088144$ | ||
Ramified primes: | \(737652920184769\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{737652920184769}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $10$ | 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: | $8a^{12}-11a^{11}-72a^{10}+93a^{9}+235a^{8}-293a^{7}-359a^{6}+422a^{5}+290a^{4}-265a^{3}-135a^{2}+47a+21$, $7a^{12}-10a^{11}-62a^{10}+85a^{9}+196a^{8}-269a^{7}-280a^{6}+387a^{5}+198a^{4}-239a^{3}-78a^{2}+41a+11$, $a$, $a^{11}-2a^{10}-7a^{9}+15a^{8}+14a^{7}-38a^{6}-5a^{5}+37a^{4}-5a^{3}-11a^{2}+1$, $2a^{12}-4a^{11}-16a^{10}+33a^{9}+43a^{8}-98a^{7}-45a^{6}+128a^{5}+18a^{4}-69a^{3}-9a^{2}+9a+3$, $a^{12}-2a^{11}-7a^{10}+16a^{9}+13a^{8}-46a^{7}+3a^{6}+57a^{5}-28a^{4}-28a^{3}+24a^{2}+4a-5$, $5a^{12}-8a^{11}-43a^{10}+68a^{9}+129a^{8}-213a^{7}-165a^{6}+299a^{5}+92a^{4}-176a^{3}-29a^{2}+27a+5$, $a^{11}-2a^{10}-8a^{9}+17a^{8}+21a^{7}-53a^{6}-19a^{5}+74a^{4}+a^{3}-43a^{2}+2a+6$, $6a^{12}-9a^{11}-52a^{10}+75a^{9}+160a^{8}-230a^{7}-223a^{6}+317a^{5}+161a^{4}-186a^{3}-73a^{2}+30a+12$, $2a^{12}-2a^{11}-20a^{10}+18a^{9}+75a^{8}-63a^{7}-136a^{6}+105a^{5}+129a^{4}-79a^{3}-63a^{2}+18a+9$ | 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: | \( 394.327696405 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{2}\cdot 394.327696405 \cdot 1}{2\cdot\sqrt{737652920184769}}\cr\approx \mathstrut & 0.146734065565 \end{aligned}\]
Galois group
A non-solvable group of order 6227020800 |
The 101 conjugacy class representatives for $S_{13}$ are not computed |
Character table for $S_{13}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 26 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.13.0.1}{13} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.13.0.1}{13} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.13.0.1}{13} }$ | ${\href{/padicField/59.13.0.1}{13} }$ |
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 |
---|---|---|---|---|---|---|---|
\(737652920184769\) | $\Q_{737652920184769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |