Normalized defining polynomial
\( x^{13} - x^{12} - 2x^{11} + x^{10} + 2x^{8} + 2x^{7} - x^{6} - 2x^{5} - x^{4} + 2x^{3} + 2x^{2} - 3x + 1 \)
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: | \(4082190012125\) \(\medspace = 5^{3}\cdot 73\cdot 447363289\) | 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: | \(9.33\) | 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: | $5^{3/4}73^{1/2}447363289^{1/2}\approx 604253.4026112618$ | ||
Ramified primes: | \(5\), \(73\), \(447363289\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{163287600485}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{143}a^{12}+\frac{36}{143}a^{11}+\frac{43}{143}a^{10}+\frac{19}{143}a^{9}-\frac{12}{143}a^{8}-\frac{1}{11}a^{7}-\frac{50}{143}a^{6}+\frac{8}{143}a^{5}+\frac{8}{143}a^{4}+\frac{9}{143}a^{3}+\frac{49}{143}a^{2}-\frac{4}{13}a-\frac{58}{143}$
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)
| |
Fundamental units: | $a$, $\frac{250}{143}a^{12}-\frac{152}{143}a^{11}-\frac{547}{143}a^{10}+\frac{31}{143}a^{9}+\frac{3}{143}a^{8}+\frac{36}{11}a^{7}+\frac{656}{143}a^{6}-\frac{2}{143}a^{5}-\frac{431}{143}a^{4}-\frac{324}{143}a^{3}+\frac{381}{143}a^{2}+\frac{66}{13}a-\frac{486}{143}$, $\frac{259}{143}a^{12}+\frac{29}{143}a^{11}-\frac{589}{143}a^{10}-\frac{370}{143}a^{9}-\frac{248}{143}a^{8}+\frac{27}{11}a^{7}+\frac{1064}{143}a^{6}+\frac{785}{143}a^{5}+\frac{70}{143}a^{4}-\frac{529}{143}a^{3}-\frac{179}{143}a^{2}+\frac{43}{13}a-\frac{150}{143}$, $\frac{72}{143}a^{12}-\frac{125}{143}a^{11}-\frac{193}{143}a^{10}+\frac{224}{143}a^{9}+\frac{137}{143}a^{8}+\frac{16}{11}a^{7}+\frac{118}{143}a^{6}-\frac{425}{143}a^{5}-\frac{425}{143}a^{4}-\frac{67}{143}a^{3}+\frac{382}{143}a^{2}+\frac{24}{13}a-\frac{315}{143}$, $\frac{265}{143}a^{12}-\frac{184}{143}a^{11}-\frac{617}{143}a^{10}+\frac{30}{143}a^{9}+\frac{109}{143}a^{8}+\frac{54}{11}a^{7}+\frac{764}{143}a^{6}-\frac{25}{143}a^{5}-\frac{740}{143}a^{4}-\frac{761}{143}a^{3}+\frac{258}{143}a^{2}+\frac{58}{13}a-\frac{498}{143}$, $\frac{34}{11}a^{12}-\frac{8}{11}a^{11}-\frac{78}{11}a^{10}-\frac{25}{11}a^{9}-\frac{12}{11}a^{8}+\frac{64}{11}a^{7}+\frac{126}{11}a^{6}+\frac{52}{11}a^{5}-\frac{47}{11}a^{4}-\frac{79}{11}a^{3}+\frac{5}{11}a^{2}+7a-\frac{36}{11}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 6.75913142567 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 6.75913142567 \cdot 1}{2\cdot\sqrt{4082190012125}}\cr\approx \mathstrut & 0.205837023561 \end{aligned}\]
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 | ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(73\) | $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.10.0.1 | $x^{10} + 2 x^{6} + 15 x^{5} + 23 x^{4} + 33 x^{3} + 32 x^{2} + 69 x + 5$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(447363289\) | $\Q_{447363289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{447363289}$ | $x$ | $1$ | $1$ | $0$ | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |