Normalized defining polynomial
\( x^{14} - 7 x^{13} + 11 x^{12} + 25 x^{11} - 79 x^{10} - x^{9} + 191 x^{8} - 197 x^{7} + 4 x^{6} + \cdots + 2 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-4304209926662811648\) \(\medspace = -\,2^{12}\cdot 3^{12}\cdot 7^{11}\) | 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: | \(21.43\) | 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: | $2^{6/7}3^{6/7}7^{5/6}\approx 23.509053955343234$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{339}a^{12}-\frac{2}{113}a^{11}-\frac{8}{339}a^{10}-\frac{6}{113}a^{9}+\frac{7}{339}a^{8}-\frac{33}{113}a^{7}+\frac{1}{339}a^{6}+\frac{74}{339}a^{5}-\frac{161}{339}a^{4}-\frac{121}{339}a^{3}-\frac{82}{339}a^{2}-\frac{40}{339}a-\frac{55}{113}$, $\frac{1}{8475}a^{13}+\frac{2}{2825}a^{12}+\frac{463}{2825}a^{11}-\frac{1018}{8475}a^{10}-\frac{371}{2825}a^{9}-\frac{229}{1695}a^{8}-\frac{923}{2825}a^{7}-\frac{348}{2825}a^{6}-\frac{3793}{8475}a^{5}-\frac{1036}{8475}a^{4}+\frac{129}{2825}a^{3}+\frac{3044}{8475}a^{2}+\frac{824}{8475}a+\frac{357}{2825}$
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: | $\frac{4979}{2825}a^{13}-\frac{104128}{8475}a^{12}+\frac{160318}{8475}a^{11}+\frac{128128}{2825}a^{10}-\frac{1176331}{8475}a^{9}-\frac{136}{15}a^{8}+\frac{2912347}{8475}a^{7}-\frac{2826878}{8475}a^{6}-\frac{187741}{8475}a^{5}+\frac{1565768}{8475}a^{4}-\frac{209306}{8475}a^{3}-\frac{333499}{2825}a^{2}+\frac{732263}{8475}a-\frac{160423}{8475}$, $\frac{2116}{2825}a^{13}-\frac{41687}{8475}a^{12}+\frac{52622}{8475}a^{11}+\frac{176236}{8475}a^{10}-\frac{420074}{8475}a^{9}-\frac{10844}{565}a^{8}+\frac{1094663}{8475}a^{7}-\frac{816962}{8475}a^{6}-\frac{191689}{8475}a^{5}+\frac{484472}{8475}a^{4}+\frac{7092}{2825}a^{3}-\frac{345488}{8475}a^{2}+\frac{71709}{2825}a-\frac{45067}{8475}$, $\frac{4979}{2825}a^{13}-\frac{92428}{8475}a^{12}+\frac{90118}{8475}a^{11}+\frac{440509}{8475}a^{10}-\frac{271152}{2825}a^{9}-\frac{43356}{565}a^{8}+\frac{752299}{2825}a^{7}-\frac{1131478}{8475}a^{6}-\frac{683591}{8475}a^{5}+\frac{786643}{8475}a^{4}+\frac{298819}{8475}a^{3}-\frac{618022}{8475}a^{2}+\frac{289688}{8475}a-\frac{17091}{2825}$, $\frac{5624}{8475}a^{13}-\frac{37831}{8475}a^{12}+\frac{51511}{8475}a^{11}+\frac{155543}{8475}a^{10}-\frac{405962}{8475}a^{9}-\frac{7687}{565}a^{8}+\frac{354798}{2825}a^{7}-\frac{841081}{8475}a^{6}-\frac{242882}{8475}a^{5}+\frac{564086}{8475}a^{4}+\frac{3188}{8475}a^{3}-\frac{368494}{8475}a^{2}+\frac{74267}{2825}a-\frac{37771}{8475}$, $\frac{232}{8475}a^{13}+\frac{3692}{8475}a^{12}-\frac{10184}{2825}a^{11}+\frac{12133}{2825}a^{10}+\frac{152384}{8475}a^{9}-\frac{20026}{565}a^{8}-\frac{89461}{2825}a^{7}+\frac{283514}{2825}a^{6}-\frac{67092}{2825}a^{5}-\frac{1418}{25}a^{4}+\frac{156134}{8475}a^{3}+\frac{70836}{2825}a^{2}-\frac{149432}{8475}a+\frac{9999}{2825}$, $\frac{1403}{8475}a^{13}-\frac{15307}{8475}a^{12}+\frac{16214}{2825}a^{11}+\frac{2457}{2825}a^{10}-\frac{281339}{8475}a^{9}+\frac{55123}{1695}a^{8}+\frac{191381}{2825}a^{7}-\frac{1090132}{8475}a^{6}+\frac{239521}{8475}a^{5}+\frac{170064}{2825}a^{4}-\frac{241864}{8475}a^{3}-\frac{281318}{8475}a^{2}+\frac{282947}{8475}a-\frac{54787}{8475}$ | 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: | \( 15666.333906202328 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 15666.333906202328 \cdot 1}{2\cdot\sqrt{4304209926662811648}}\cr\approx \mathstrut & 1.45965338631577 \end{aligned}\]
Galois group
A solvable group of order 42 |
The 7 conjugacy class representatives for $F_7$ |
Character table for $F_7$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 7.1.784147392.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 7 sibling: | 7.1.784147392.1 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.784147392.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(3\) | 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |