Normalized defining polynomial
\( x^{13} - 2x^{11} - 3x^{10} - 4x^{9} + 7x^{8} + 6x^{7} - 4x^{6} - 2x^{5} - 5x^{4} + 5x^{3} + 3x^{2} - 4x + 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[7, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-146839998625007\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.30\) | 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: | $146839998625007^{1/2}\approx 12117755.511026248$ | ||
Ramified primes: | \(146839998625007\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-146839998625007}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $9$ | 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: | $a$, $10a^{12}+9a^{11}-12a^{10}-42a^{9}-78a^{8}+a^{7}+66a^{6}+26a^{5}+a^{4}-52a^{3}-a^{2}+31a-9$, $13a^{12}+11a^{11}-17a^{10}-54a^{9}-97a^{8}+10a^{7}+90a^{6}+25a^{5}-8a^{4}-71a^{3}+3a^{2}+44a-14$, $21a^{12}+14a^{11}-32a^{10}-85a^{9}-141a^{8}+51a^{7}+159a^{6}+27a^{5}-25a^{4}-121a^{3}+21a^{2}+75a-30$, $8a^{12}+8a^{11}-10a^{10}-35a^{9}-65a^{8}-a^{7}+60a^{6}+25a^{5}-46a^{3}-4a^{2}+28a-6$, $8a^{12}+8a^{11}-10a^{10}-35a^{9}-65a^{8}-a^{7}+60a^{6}+25a^{5}-46a^{3}-4a^{2}+28a-7$, $17a^{12}+13a^{11}-23a^{10}-70a^{9}-122a^{8}+23a^{7}+119a^{6}+32a^{5}-13a^{4}-94a^{3}+9a^{2}+57a-19$, $32a^{12}+23a^{11}-47a^{10}-130a^{9}-222a^{8}+63a^{7}+236a^{6}+45a^{5}-31a^{4}-183a^{3}+27a^{2}+114a-44$, $4a^{12}-9a^{10}-14a^{9}-15a^{8}+33a^{7}+36a^{6}-10a^{5}-17a^{4}-26a^{3}+17a^{2}+19a-12$ | 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: | \( 115.283453799 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{3}\cdot 115.283453799 \cdot 1}{2\cdot\sqrt{146839998625007}}\cr\approx \mathstrut & 0.151030399887 \end{aligned}\]
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 | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.5.0.1}{5} }$ | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.13.0.1}{13} }$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.13.0.1}{13} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
---|---|---|---|---|---|---|---|
\(146839998625007\) | $\Q_{146839998625007}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |