Normalized defining polynomial
\( x^{13} - 2 x^{12} + 4 x^{11} - 5 x^{10} + 2 x^{9} + x^{8} - 6 x^{7} + 8 x^{6} - 3 x^{5} + 4 x^{4} + \cdots - 1 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(24777340928000\) \(\medspace = 2^{14}\cdot 5^{3}\cdot 12098311\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(10.72\) | 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: | $2^{223/96}5^{3/4}12098311^{1/2}\approx 58191.22094805985$ | ||
Ramified primes: | \(2\), \(5\), \(12098311\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{60491555}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{659}a^{12}-\frac{176}{659}a^{11}+\frac{314}{659}a^{10}+\frac{56}{659}a^{9}+\frac{143}{659}a^{8}+\frac{161}{659}a^{7}+\frac{317}{659}a^{6}+\frac{206}{659}a^{5}-\frac{261}{659}a^{4}-\frac{53}{659}a^{3}-\frac{1}{659}a+\frac{178}{659}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{122}{659}a^{12}-\frac{384}{659}a^{11}+\frac{745}{659}a^{10}-\frac{1076}{659}a^{9}+\frac{971}{659}a^{8}-\frac{128}{659}a^{7}-\frac{866}{659}a^{6}+\frac{1408}{659}a^{5}-\frac{1528}{659}a^{4}+\frac{783}{659}a^{3}-\frac{122}{659}a+\frac{628}{659}$, $\frac{1}{659}a^{12}-\frac{176}{659}a^{11}+\frac{314}{659}a^{10}-\frac{603}{659}a^{9}+\frac{802}{659}a^{8}-\frac{498}{659}a^{7}+\frac{317}{659}a^{6}+\frac{206}{659}a^{5}-\frac{261}{659}a^{4}-\frac{53}{659}a^{3}-a^{2}-\frac{660}{659}a-\frac{481}{659}$, $\frac{98}{659}a^{12}-\frac{114}{659}a^{11}+\frac{458}{659}a^{10}-\frac{443}{659}a^{9}+\frac{175}{659}a^{8}-\frac{38}{659}a^{7}-\frac{1225}{659}a^{6}+\frac{1077}{659}a^{5}-\frac{536}{659}a^{4}+\frac{737}{659}a^{3}+2a^{2}+\frac{561}{659}a+\frac{969}{659}$, $\frac{222}{659}a^{12}-\frac{191}{659}a^{11}+\frac{513}{659}a^{10}-\frac{748}{659}a^{9}+\frac{114}{659}a^{8}-\frac{503}{659}a^{7}-\frac{139}{659}a^{6}+\frac{920}{659}a^{5}+\frac{50}{659}a^{4}+\frac{1414}{659}a^{3}+a^{2}+\frac{437}{659}a+\frac{635}{659}$, $a$, $\frac{243}{659}a^{12}-\frac{592}{659}a^{11}+\frac{1176}{659}a^{10}-\frac{1549}{659}a^{9}+\frac{1140}{659}a^{8}-\frac{417}{659}a^{7}-\frac{731}{659}a^{6}+\frac{1292}{659}a^{5}-\frac{818}{659}a^{4}+\frac{1619}{659}a^{3}+\frac{416}{659}a+\frac{1078}{659}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 32.3528276422 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 32.3528276422 \cdot 1}{2\cdot\sqrt{24777340928000}}\cr\approx \mathstrut & 0.399911696012 \end{aligned}\]
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 | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.13.0.1}{13} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.13.0.1}{13} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
2.8.14.14 | $x^{8} + 2 x^{7} + 4 x^{3} + 2 x^{2} + 4 x + 6$ | $8$ | $1$ | $14$ | $C_2 \wr S_4$ | $[4/3, 4/3, 2, 7/3, 7/3, 5/2]_{3}^{2}$ | |
\(5\) | 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
5.4.3.3 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.5.0.1 | $x^{5} + 4 x + 3$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(12098311\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ |