Normalized defining polynomial
\( x^{13} - 3 x^{12} + 3 x^{11} - 4 x^{10} + 4 x^{9} + 7 x^{8} - 20 x^{7} + 27 x^{6} - 35 x^{5} + 41 x^{4} + \cdots + 4 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-156408138976000\) \(\medspace = -\,2^{8}\cdot 5^{3}\cdot 137\cdot 599\cdot 59561\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.36\) | 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: | $2^{2}5^{3/4}137^{1/2}599^{1/2}59561^{1/2}\approx 935065.8145079553$ | ||
Ramified primes: | \(2\), \(5\), \(137\), \(599\), \(59561\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-24438771715}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{40}a^{11}-\frac{7}{40}a^{10}+\frac{17}{40}a^{9}-\frac{7}{20}a^{8}-\frac{9}{20}a^{7}-\frac{1}{8}a^{6}+\frac{3}{10}a^{5}+\frac{9}{40}a^{4}+\frac{1}{40}a^{3}-\frac{9}{40}a^{2}-\frac{7}{20}a+\frac{3}{20}$, $\frac{1}{1400}a^{12}-\frac{1}{280}a^{11}+\frac{363}{1400}a^{10}-\frac{19}{70}a^{9}-\frac{143}{700}a^{8}-\frac{121}{1400}a^{7}-\frac{239}{700}a^{6}+\frac{633}{1400}a^{5}+\frac{99}{1400}a^{4}+\frac{193}{1400}a^{3}-\frac{9}{175}a^{2}-\frac{13}{100}a-\frac{87}{350}$
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: | $\frac{79}{350}a^{12}-\frac{8}{35}a^{11}-\frac{64}{175}a^{10}-\frac{33}{70}a^{9}-\frac{27}{175}a^{8}+\frac{871}{350}a^{7}-\frac{137}{350}a^{6}-\frac{113}{350}a^{5}-\frac{272}{175}a^{4}+\frac{81}{175}a^{3}+\frac{577}{350}a^{2}-\frac{17}{25}a-\frac{26}{175}$, $\frac{9}{1400}a^{12}-\frac{107}{280}a^{11}+\frac{1097}{1400}a^{10}-\frac{11}{28}a^{9}+\frac{743}{700}a^{8}-\frac{669}{1400}a^{7}-\frac{1163}{350}a^{6}+\frac{6817}{1400}a^{5}-\frac{7719}{1400}a^{4}+\frac{11047}{1400}a^{3}-\frac{5119}{700}a^{2}+\frac{473}{100}a-\frac{409}{175}$, $\frac{31}{700}a^{12}-\frac{13}{280}a^{11}-\frac{209}{1400}a^{10}+\frac{41}{280}a^{9}-\frac{81}{700}a^{8}+\frac{86}{175}a^{7}-\frac{61}{1400}a^{6}-\frac{607}{700}a^{5}+\frac{1343}{1400}a^{4}-\frac{1789}{1400}a^{3}+\frac{331}{1400}a^{2}-\frac{51}{100}a+\frac{447}{700}$, $\frac{423}{1400}a^{12}-\frac{213}{280}a^{11}+\frac{599}{1400}a^{10}-\frac{149}{140}a^{9}+\frac{761}{700}a^{8}+\frac{4117}{1400}a^{7}-\frac{1461}{350}a^{6}+\frac{7359}{1400}a^{5}-\frac{13073}{1400}a^{4}+\frac{12689}{1400}a^{3}-\frac{4553}{700}a^{2}+\frac{551}{100}a-\frac{463}{175}$, $\frac{423}{1400}a^{12}-\frac{213}{280}a^{11}+\frac{599}{1400}a^{10}-\frac{149}{140}a^{9}+\frac{761}{700}a^{8}+\frac{4117}{1400}a^{7}-\frac{1461}{350}a^{6}+\frac{7359}{1400}a^{5}-\frac{13073}{1400}a^{4}+\frac{12689}{1400}a^{3}-\frac{4553}{700}a^{2}+\frac{451}{100}a-\frac{463}{175}$, $\frac{67}{175}a^{12}-\frac{137}{280}a^{11}+\frac{3}{1400}a^{10}-\frac{353}{280}a^{9}-\frac{313}{700}a^{8}+\frac{2117}{700}a^{7}-\frac{2983}{1400}a^{6}+\frac{1557}{350}a^{5}-\frac{7381}{1400}a^{4}+\frac{6043}{1400}a^{3}-\frac{3347}{1400}a^{2}+\frac{137}{100}a+\frac{221}{700}$, $\frac{201}{700}a^{12}-\frac{129}{280}a^{11}+\frac{571}{1400}a^{10}-\frac{431}{280}a^{9}+\frac{159}{700}a^{8}+\frac{597}{350}a^{7}-\frac{2981}{1400}a^{6}+\frac{4523}{700}a^{5}-\frac{12317}{1400}a^{4}+\frac{11751}{1400}a^{3}-\frac{10429}{1400}a^{2}+\frac{609}{100}a-\frac{2153}{700}$ | 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: | \( 127.407170199 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 127.407170199 \cdot 1}{2\cdot\sqrt{156408138976000}}\cr\approx \mathstrut & 0.399046520814 \end{aligned}\]
Galois group
A non-solvable group of order 6227020800 |
The 101 conjugacy class representatives for $S_{13}$ |
Character table for $S_{13}$ |
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 | R | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.13.0.1}{13} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.13.0.1}{13} }$ | ${\href{/padicField/59.13.0.1}{13} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.8.2 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(5\) | 5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.9.0.1 | $x^{9} + 2 x^{3} + x + 3$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(137\) | 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
137.9.0.1 | $x^{9} + x^{3} + 80 x^{2} + 122 x + 134$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(599\) | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
\(59561\) | $\Q_{59561}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |