Normalized defining polynomial
\( x^{13} - 3 x^{12} + 3 x^{11} + 3 x^{10} - 13 x^{9} + 14 x^{8} + 2 x^{7} - 20 x^{6} + 18 x^{5} + x^{4} - 11 x^{3} + 6 x^{2} + x - 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(28261626739249\) \(\medspace = 2161\cdot 13078031809\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.83\) | 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))
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Galois root discriminant: | $2161^{1/2}13078031809^{1/2}\approx 5316166.545477013$ | ||
Ramified primes: | \(2161\), \(13078031809\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{28261626739249}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11}a^{12}+\frac{3}{11}a^{10}+\frac{1}{11}a^{9}+\frac{1}{11}a^{8}-\frac{5}{11}a^{7}-\frac{2}{11}a^{6}-\frac{4}{11}a^{5}-\frac{5}{11}a^{4}-\frac{3}{11}a^{3}+\frac{2}{11}a^{2}+\frac{1}{11}a+\frac{4}{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{13}{11}a^{12}-3a^{11}+\frac{28}{11}a^{10}+\frac{35}{11}a^{9}-\frac{130}{11}a^{8}+\frac{122}{11}a^{7}+\frac{29}{11}a^{6}-\frac{162}{11}a^{5}+\frac{122}{11}a^{4}+\frac{16}{11}a^{3}-\frac{62}{11}a^{2}+\frac{24}{11}a+\frac{8}{11}$, $a$, $\frac{7}{11}a^{12}-a^{11}-\frac{1}{11}a^{10}+\frac{29}{11}a^{9}-\frac{48}{11}a^{8}+\frac{9}{11}a^{7}+\frac{52}{11}a^{6}-\frac{50}{11}a^{5}-\frac{2}{11}a^{4}+\frac{23}{11}a^{3}+\frac{3}{11}a^{2}-\frac{4}{11}a+\frac{6}{11}$, $\frac{4}{11}a^{12}-a^{11}+\frac{12}{11}a^{10}+\frac{15}{11}a^{9}-\frac{62}{11}a^{8}+\frac{68}{11}a^{7}+\frac{3}{11}a^{6}-\frac{104}{11}a^{5}+\frac{112}{11}a^{4}-\frac{12}{11}a^{3}-\frac{47}{11}a^{2}+\frac{37}{11}a+\frac{5}{11}$, $\frac{5}{11}a^{12}-a^{11}+\frac{15}{11}a^{10}+\frac{5}{11}a^{9}-\frac{50}{11}a^{8}+\frac{63}{11}a^{7}-\frac{21}{11}a^{6}-\frac{64}{11}a^{5}+\frac{96}{11}a^{4}-\frac{15}{11}a^{3}-\frac{45}{11}a^{2}+\frac{38}{11}a+\frac{9}{11}$, $\frac{7}{11}a^{12}-2a^{11}+\frac{32}{11}a^{10}-\frac{4}{11}a^{9}-\frac{70}{11}a^{8}+\frac{119}{11}a^{7}-\frac{69}{11}a^{6}-\frac{61}{11}a^{5}+\frac{130}{11}a^{4}-\frac{65}{11}a^{3}-\frac{19}{11}a^{2}+\frac{40}{11}a-\frac{5}{11}$, $\frac{10}{11}a^{12}-2a^{11}+\frac{8}{11}a^{10}+\frac{43}{11}a^{9}-\frac{89}{11}a^{8}+\frac{38}{11}a^{7}+\frac{90}{11}a^{6}-\frac{117}{11}a^{5}+\frac{5}{11}a^{4}+\frac{91}{11}a^{3}-\frac{35}{11}a^{2}-\frac{23}{11}a+\frac{18}{11}$, $\frac{10}{11}a^{12}-2a^{11}+\frac{19}{11}a^{10}+\frac{21}{11}a^{9}-\frac{89}{11}a^{8}+\frac{93}{11}a^{7}-\frac{9}{11}a^{6}-\frac{95}{11}a^{5}+\frac{126}{11}a^{4}-\frac{52}{11}a^{3}-\frac{24}{11}a^{2}+\frac{54}{11}a-\frac{15}{11}$ | 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: | \( 33.0726626498 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{4}\cdot 33.0726626498 \cdot 1}{2\cdot\sqrt{28261626739249}}\cr\approx \mathstrut & 0.155135088919 \end{aligned}\]
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 | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.13.0.1}{13} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2161\) | $\Q_{2161}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(13078031809\) | $\Q_{13078031809}$ | $x$ | $1$ | $1$ | $0$ | 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |