Normalized defining polynomial
\( x^{13} - x^{12} - 5 x^{11} + 15 x^{9} + 9 x^{8} - 28 x^{7} - 20 x^{6} + 26 x^{5} + 20 x^{4} - 12 x^{3} + \cdots + 2 \)
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)
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Discriminant: | \(-114312593686528\) \(\medspace = -\,2^{14}\cdot 6977087017\) | 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: | \(12.06\) | 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: | $2^{223/96}6977087017^{1/2}\approx 417931.07479608984$ | ||
Ramified primes: | \(2\), \(6977087017\) | 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: | $\Q(\sqrt{-6977087017}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{113}a^{12}-\frac{40}{113}a^{11}-\frac{27}{113}a^{10}+\frac{36}{113}a^{9}-\frac{33}{113}a^{8}+\frac{53}{113}a^{7}+\frac{52}{113}a^{6}-\frac{14}{113}a^{5}+\frac{7}{113}a^{4}-\frac{27}{113}a^{3}+\frac{24}{113}a^{2}-\frac{42}{113}a-\frac{55}{113}$
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)
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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: | $\frac{210}{113}a^{12}-\frac{151}{113}a^{11}-\frac{1037}{113}a^{10}-\frac{350}{113}a^{9}+\frac{2788}{113}a^{8}+\frac{2655}{113}a^{7}-\frac{4335}{113}a^{6}-\frac{4861}{113}a^{5}+\frac{2713}{113}a^{4}+\frac{3822}{113}a^{3}-\frac{497}{113}a^{2}-\frac{1136}{113}a-\frac{137}{113}$, $\frac{188}{113}a^{12}-\frac{175}{113}a^{11}-\frac{782}{113}a^{10}-\frac{238}{113}a^{9}+\frac{2045}{113}a^{8}+\frac{1828}{113}a^{7}-\frac{2993}{113}a^{6}-\frac{2632}{113}a^{5}+\frac{977}{113}a^{4}+\frac{1591}{113}a^{3}+\frac{444}{113}a^{2}-\frac{325}{113}a-\frac{283}{113}$, $\frac{99}{113}a^{12}-\frac{118}{113}a^{11}-\frac{413}{113}a^{10}+\frac{61}{113}a^{9}+\frac{1140}{113}a^{8}+\frac{501}{113}a^{7}-\frac{2084}{113}a^{6}-\frac{595}{113}a^{5}+\frac{1597}{113}a^{4}+\frac{39}{113}a^{3}-\frac{449}{113}a^{2}+\frac{23}{113}a-\frac{21}{113}$, $\frac{119}{113}a^{12}-\frac{240}{113}a^{11}-\frac{501}{113}a^{10}+\frac{668}{113}a^{9}+\frac{1836}{113}a^{8}-\frac{812}{113}a^{7}-\frac{4660}{113}a^{6}+\frac{1046}{113}a^{5}+\frac{5918}{113}a^{4}-\frac{953}{113}a^{3}-\frac{3698}{113}a^{2}+\frac{313}{113}a+\frac{913}{113}$, $\frac{296}{113}a^{12}-\frac{427}{113}a^{11}-\frac{1212}{113}a^{10}+\frac{486}{113}a^{9}+\frac{3792}{113}a^{8}+\frac{885}{113}a^{7}-\frac{7547}{113}a^{6}-\frac{1545}{113}a^{5}+\frac{6479}{113}a^{4}+\frac{1161}{113}a^{3}-\frac{2501}{113}a^{2}-\frac{454}{113}a+\frac{331}{113}$, $\frac{121}{113}a^{12}-\frac{94}{113}a^{11}-\frac{668}{113}a^{10}-\frac{51}{113}a^{9}+\frac{1883}{113}a^{8}+\frac{1328}{113}a^{7}-\frac{3426}{113}a^{6}-\frac{2824}{113}a^{5}+\frac{3333}{113}a^{4}+\frac{2270}{113}a^{3}-\frac{1390}{113}a^{2}-\frac{788}{113}a+\frac{125}{113}$, $\frac{576}{113}a^{12}-\frac{892}{113}a^{11}-\frac{2331}{113}a^{10}+\frac{1300}{113}a^{9}+\frac{7547}{113}a^{8}+\frac{696}{113}a^{7}-\frac{15700}{113}a^{6}-\frac{1397}{113}a^{5}+\frac{14880}{113}a^{4}+\frac{720}{113}a^{3}-\frac{7081}{113}a^{2}-\frac{123}{113}a+\frac{1429}{113}$ | 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: | \( 95.7333582025 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 95.7333582025 \cdot 1}{2\cdot\sqrt{114312593686528}}\cr\approx \mathstrut & 0.350732158021 \end{aligned}\]
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 | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.13.0.1}{13} }$ | ${\href{/padicField/23.13.0.1}{13} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
2.8.14.8 | $x^{8} + 2 x^{7} + 4 x^{3} + 6 x^{2} + 4 x + 2$ | $8$ | $1$ | $14$ | $C_2 \wr S_4$ | $[4/3, 4/3, 2, 7/3, 7/3, 5/2]_{3}^{2}$ | |
\(6977087017\) | $\Q_{6977087017}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |