Normalized defining polynomial
\( x^{13} - 2 x^{12} - 4 x^{11} + 7 x^{10} + 6 x^{9} - 6 x^{8} - 6 x^{7} - 8 x^{6} + 8 x^{5} + 15 x^{4} - 7 x^{3} - 7 x^{2} + x + 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-53877019237375\) \(\medspace = -\,5^{3}\cdot 204793\cdot 2104643\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.38\) | 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: | $5^{3/4}204793^{1/2}2104643^{1/2}\approx 2195200.9241965474$ | ||
Ramified primes: | \(5\), \(204793\), \(2104643\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-2155080769495}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{851}a^{12}+\frac{162}{851}a^{11}+\frac{183}{851}a^{10}+\frac{234}{851}a^{9}+\frac{87}{851}a^{8}-\frac{205}{851}a^{7}+\frac{18}{37}a^{6}-\frac{192}{851}a^{5}+\frac{7}{851}a^{4}+\frac{312}{851}a^{3}+\frac{101}{851}a^{2}+\frac{388}{851}a-\frac{192}{851}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{173}{851}a^{12}-\frac{57}{851}a^{11}-\frac{1530}{851}a^{10}+\frac{485}{851}a^{9}+\frac{3988}{851}a^{8}-\frac{574}{851}a^{7}-\frac{142}{37}a^{6}-\frac{2580}{851}a^{5}-\frac{1342}{851}a^{4}+\frac{7171}{851}a^{3}+\frac{2155}{851}a^{2}-\frac{4360}{851}a-\frac{27}{851}$, $a$, $\frac{314}{851}a^{12}-\frac{1043}{851}a^{11}-\frac{406}{851}a^{10}+\frac{3694}{851}a^{9}-\frac{765}{851}a^{8}-\frac{3949}{851}a^{7}-\frac{9}{37}a^{6}-\frac{718}{851}a^{5}+\frac{6453}{851}a^{4}+\frac{2656}{851}a^{3}-\frac{7432}{851}a^{2}-\frac{712}{851}a+\frac{1835}{851}$, $\frac{10}{851}a^{12}-\frac{82}{851}a^{11}+\frac{128}{851}a^{10}-\frac{213}{851}a^{9}+\frac{19}{851}a^{8}+\frac{1354}{851}a^{7}-\frac{5}{37}a^{6}-\frac{1069}{851}a^{5}-\frac{781}{851}a^{4}-\frac{1135}{851}a^{3}+\frac{2712}{851}a^{2}+\frac{2178}{851}a-\frac{218}{851}$, $\frac{191}{851}a^{12}+\frac{306}{851}a^{11}-\frac{1640}{851}a^{10}-\frac{2111}{851}a^{9}+\frac{3852}{851}a^{8}+\frac{4246}{851}a^{7}-\frac{77}{37}a^{6}-\frac{4334}{851}a^{5}-\frac{6322}{851}a^{4}+\frac{2575}{851}a^{3}+\frac{8228}{851}a^{2}-\frac{1631}{851}a-\frac{1781}{851}$, $\frac{589}{851}a^{12}-\frac{745}{851}a^{11}-\frac{2843}{851}a^{10}+\frac{1666}{851}a^{9}+\frac{5289}{851}a^{8}+\frac{948}{851}a^{7}-\frac{202}{37}a^{6}-\frac{7564}{851}a^{5}-\frac{132}{851}a^{4}+\frac{8462}{851}a^{3}+\frac{4174}{851}a^{2}-\frac{3791}{851}a-\frac{2458}{851}$, $\frac{208}{851}a^{12}-\frac{344}{851}a^{11}-\frac{1082}{851}a^{10}+\frac{1867}{851}a^{9}+\frac{1076}{851}a^{8}-\frac{2643}{851}a^{7}+\frac{7}{37}a^{6}-\frac{790}{851}a^{5}+\frac{1456}{851}a^{4}+\frac{3624}{851}a^{3}-\frac{5373}{851}a^{2}-\frac{992}{851}a+\frac{1763}{851}$ | 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: | \( 39.9707034173 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 39.9707034173 \cdot 1}{2\cdot\sqrt{53877019237375}}\cr\approx \mathstrut & 0.213304169208 \end{aligned}\]
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 | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.9.0.1 | $x^{9} + 2 x^{3} + x + 3$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(204793\) | $\Q_{204793}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(2104643\) | $\Q_{2104643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2104643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |