Normalized defining polynomial
\( x^{14} - 7 x^{13} + 25 x^{12} - 59 x^{11} + 103 x^{10} - 141 x^{9} + 159 x^{8} - 153 x^{7} + 129 x^{6} + \cdots + 1 \)
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: | \(-9095120158391\) \(\medspace = -\,71^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(8.43\) | 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: | $71^{1/2}\approx 8.426149773176359$ | ||
Ramified primes: | \(71\) | 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: | \(\Q(\sqrt{-71}) \) | ||
$\card{ \Gal(K/\Q) }$: | $14$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{1}{7}a^{11}-\frac{1}{7}a^{10}-\frac{3}{7}a^{9}+\frac{1}{7}a^{8}-\frac{3}{7}a^{7}+\frac{3}{7}a^{6}+\frac{1}{7}a^{5}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}-\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{77}a^{13}-\frac{1}{77}a^{12}-\frac{3}{77}a^{11}-\frac{2}{7}a^{10}-\frac{1}{11}a^{9}+\frac{37}{77}a^{8}-\frac{26}{77}a^{7}-\frac{12}{77}a^{6}-\frac{9}{77}a^{5}-\frac{17}{77}a^{4}+\frac{3}{7}a^{3}+\frac{6}{77}a^{2}-\frac{9}{77}a+\frac{9}{77}$
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: | $\frac{53}{77}a^{13}-\frac{306}{77}a^{12}+\frac{897}{77}a^{11}-\frac{153}{7}a^{10}+\frac{2236}{77}a^{9}-\frac{306}{11}a^{8}+\frac{1537}{77}a^{7}-\frac{779}{77}a^{6}+\frac{194}{77}a^{5}+\frac{254}{77}a^{4}-\frac{52}{7}a^{3}+\frac{571}{77}a^{2}-\frac{180}{77}a+\frac{37}{77}$, $\frac{3}{7}a^{12}-\frac{18}{7}a^{11}+\frac{53}{7}a^{10}-\frac{100}{7}a^{9}+\frac{136}{7}a^{8}-\frac{142}{7}a^{7}+\frac{121}{7}a^{6}-\frac{88}{7}a^{5}+8a^{4}-\frac{30}{7}a^{3}+\frac{4}{7}a^{2}+\frac{5}{7}a+\frac{4}{7}$, $\frac{46}{77}a^{13}-\frac{288}{77}a^{12}+\frac{929}{77}a^{11}-26a^{10}+\frac{3253}{77}a^{9}-\frac{4238}{77}a^{8}+\frac{4689}{77}a^{7}-\frac{4512}{77}a^{6}+\frac{3810}{77}a^{5}-\frac{2784}{77}a^{4}+\frac{148}{7}a^{3}-\frac{113}{11}a^{2}+\frac{389}{77}a-\frac{114}{77}$, $\frac{37}{77}a^{13}-\frac{32}{11}a^{12}+\frac{703}{77}a^{11}-\frac{134}{7}a^{10}+\frac{2304}{77}a^{9}-\frac{2822}{77}a^{8}+\frac{2833}{77}a^{7}-\frac{2391}{77}a^{6}+\frac{1713}{77}a^{5}-\frac{1014}{77}a^{4}+\frac{29}{7}a^{3}+\frac{24}{77}a^{2}-\frac{80}{77}a+\frac{58}{77}$, $\frac{2}{77}a^{13}-\frac{79}{77}a^{12}+\frac{456}{77}a^{11}-\frac{130}{7}a^{10}+\frac{427}{11}a^{9}-\frac{4623}{77}a^{8}+\frac{5569}{77}a^{7}-\frac{5568}{77}a^{6}+\frac{4756}{77}a^{5}-\frac{3576}{77}a^{4}+\frac{202}{7}a^{3}-\frac{989}{77}a^{2}+\frac{213}{77}a-\frac{59}{77}$, $\frac{27}{77}a^{13}-\frac{181}{77}a^{12}+\frac{612}{77}a^{11}-\frac{124}{7}a^{10}+\frac{325}{11}a^{9}-\frac{3005}{77}a^{8}+\frac{3302}{77}a^{7}-\frac{3096}{77}a^{6}+\frac{2529}{77}a^{5}-\frac{1768}{77}a^{4}+\frac{95}{7}a^{3}-\frac{454}{77}a^{2}+\frac{219}{77}a-\frac{65}{77}$ | 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: | \( 2.56161916129 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 2.56161916129 \cdot 1}{2\cdot\sqrt{9095120158391}}\cr\approx \mathstrut & 0.164187248733 \end{aligned}\]
Galois group
A solvable group of order 14 |
The 5 conjugacy class representatives for $D_{7}$ |
Character table for $D_{7}$ |
Intermediate fields
\(\Q(\sqrt{-71}) \), 7.1.357911.1 x7 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 7 sibling: | 7.1.357911.1 |
Minimal sibling: | 7.1.357911.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{7}$ | ${\href{/padicField/11.2.0.1}{2} }^{7}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.2.0.1}{2} }^{7}$ | ${\href{/padicField/19.7.0.1}{7} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{7}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{7}$ | ${\href{/padicField/37.7.0.1}{7} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{7}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}$ |
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 |
---|---|---|---|---|---|---|---|
\(71\) | 71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |