Normalized defining polynomial
\( x^{14} - 7 x^{13} + 35 x^{12} - 119 x^{11} + 329 x^{10} - 721 x^{9} + 1337 x^{8} - 1745 x^{7} + \cdots + 24578 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-2523048836405617863000000000000\) \(\medspace = -\,2^{12}\cdot 3^{12}\cdot 5^{12}\cdot 7^{15}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(148.45\) | 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))
| |
Galois root discriminant: | $2^{6/7}3^{6/7}5^{6/7}7^{47/42}\approx 162.85791940621738$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | 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)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}-\frac{1}{7}$, $\frac{1}{35}a^{8}+\frac{1}{35}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{13}{35}a+\frac{6}{35}$, $\frac{1}{35}a^{9}-\frac{2}{35}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{3}-\frac{8}{35}a^{2}-\frac{1}{5}a+\frac{9}{35}$, $\frac{1}{35}a^{10}+\frac{2}{5}a^{5}-\frac{3}{7}a^{3}+\frac{1}{5}$, $\frac{1}{245}a^{11}+\frac{3}{245}a^{10}-\frac{1}{245}a^{9}+\frac{3}{245}a^{8}+\frac{3}{49}a^{7}-\frac{11}{35}a^{6}+\frac{3}{35}a^{5}-\frac{8}{245}a^{4}-\frac{23}{49}a^{3}+\frac{1}{245}a^{2}-\frac{17}{245}a-\frac{24}{49}$, $\frac{1}{245}a^{12}-\frac{3}{245}a^{10}-\frac{1}{245}a^{9}-\frac{1}{245}a^{8}-\frac{2}{49}a^{7}-\frac{6}{35}a^{6}-\frac{71}{245}a^{5}+\frac{8}{35}a^{4}+\frac{94}{245}a^{3}-\frac{111}{245}a^{2}-\frac{111}{245}a-\frac{46}{245}$, $\frac{1}{68\!\cdots\!05}a^{13}+\frac{62732934736882}{68\!\cdots\!05}a^{12}+\frac{786047160250039}{68\!\cdots\!05}a^{11}+\frac{38\!\cdots\!44}{68\!\cdots\!05}a^{10}-\frac{29\!\cdots\!93}{68\!\cdots\!05}a^{9}+\frac{731361722893637}{13\!\cdots\!01}a^{8}+\frac{107040996570730}{19\!\cdots\!43}a^{7}-\frac{12\!\cdots\!34}{68\!\cdots\!05}a^{6}+\frac{26\!\cdots\!72}{68\!\cdots\!05}a^{5}-\frac{14\!\cdots\!71}{68\!\cdots\!05}a^{4}+\frac{10\!\cdots\!07}{13\!\cdots\!01}a^{3}+\frac{25\!\cdots\!04}{68\!\cdots\!05}a^{2}-\frac{31\!\cdots\!83}{68\!\cdots\!05}a-\frac{195120740686254}{98\!\cdots\!15}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{7}$, which has order $7$ (assuming GRH)
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{10\!\cdots\!66}{68\!\cdots\!05}a^{13}-\frac{10\!\cdots\!36}{68\!\cdots\!05}a^{12}+\frac{50\!\cdots\!12}{68\!\cdots\!05}a^{11}-\frac{18\!\cdots\!54}{68\!\cdots\!05}a^{10}+\frac{51\!\cdots\!89}{68\!\cdots\!05}a^{9}-\frac{11\!\cdots\!34}{68\!\cdots\!05}a^{8}+\frac{42\!\cdots\!82}{13\!\cdots\!01}a^{7}-\frac{30\!\cdots\!87}{68\!\cdots\!05}a^{6}+\frac{21\!\cdots\!16}{68\!\cdots\!05}a^{5}-\frac{13\!\cdots\!52}{68\!\cdots\!05}a^{4}+\frac{63\!\cdots\!83}{68\!\cdots\!05}a^{3}-\frac{34\!\cdots\!57}{68\!\cdots\!05}a^{2}-\frac{13\!\cdots\!58}{68\!\cdots\!05}a-\frac{17\!\cdots\!53}{13\!\cdots\!01}$, $\frac{10\!\cdots\!23}{68\!\cdots\!05}a^{13}-\frac{10\!\cdots\!91}{98\!\cdots\!15}a^{12}+\frac{76\!\cdots\!90}{13\!\cdots\!01}a^{11}-\frac{13\!\cdots\!73}{68\!\cdots\!05}a^{10}+\frac{75\!\cdots\!59}{13\!\cdots\!01}a^{9}-\frac{17\!\cdots\!57}{14\!\cdots\!45}a^{8}+\frac{15\!\cdots\!47}{68\!\cdots\!05}a^{7}-\frac{42\!\cdots\!17}{13\!\cdots\!01}a^{6}+\frac{37\!\cdots\!87}{14\!\cdots\!45}a^{5}-\frac{23\!\cdots\!06}{13\!\cdots\!01}a^{4}+\frac{32\!\cdots\!66}{68\!\cdots\!05}a^{3}-\frac{56\!\cdots\!39}{68\!\cdots\!05}a^{2}-\frac{12\!\cdots\!93}{19\!\cdots\!43}a+\frac{41\!\cdots\!09}{68\!\cdots\!05}$, $\frac{20\!\cdots\!18}{68\!\cdots\!05}a^{13}-\frac{20\!\cdots\!48}{98\!\cdots\!15}a^{12}+\frac{67\!\cdots\!34}{68\!\cdots\!05}a^{11}-\frac{23\!\cdots\!89}{68\!\cdots\!05}a^{10}+\frac{60\!\cdots\!03}{68\!\cdots\!05}a^{9}-\frac{13\!\cdots\!76}{68\!\cdots\!05}a^{8}+\frac{35\!\cdots\!81}{98\!\cdots\!15}a^{7}-\frac{25\!\cdots\!57}{68\!\cdots\!05}a^{6}+\frac{41\!\cdots\!72}{98\!\cdots\!15}a^{5}-\frac{26\!\cdots\!87}{68\!\cdots\!05}a^{4}+\frac{89\!\cdots\!34}{13\!\cdots\!01}a^{3}+\frac{21\!\cdots\!01}{68\!\cdots\!05}a^{2}-\frac{42\!\cdots\!22}{68\!\cdots\!05}a+\frac{29\!\cdots\!83}{98\!\cdots\!15}$, $\frac{24\!\cdots\!24}{68\!\cdots\!05}a^{13}+\frac{93\!\cdots\!23}{13\!\cdots\!01}a^{12}-\frac{87\!\cdots\!82}{68\!\cdots\!05}a^{11}+\frac{12\!\cdots\!03}{13\!\cdots\!01}a^{10}-\frac{14\!\cdots\!83}{68\!\cdots\!05}a^{9}+\frac{17\!\cdots\!57}{68\!\cdots\!05}a^{8}-\frac{10\!\cdots\!18}{68\!\cdots\!05}a^{7}-\frac{29\!\cdots\!38}{68\!\cdots\!05}a^{6}+\frac{10\!\cdots\!36}{68\!\cdots\!05}a^{5}+\frac{13\!\cdots\!44}{68\!\cdots\!05}a^{4}-\frac{60\!\cdots\!69}{68\!\cdots\!05}a^{3}-\frac{26\!\cdots\!75}{13\!\cdots\!01}a^{2}+\frac{21\!\cdots\!00}{13\!\cdots\!01}a+\frac{12\!\cdots\!89}{68\!\cdots\!05}$, $\frac{29\!\cdots\!09}{68\!\cdots\!05}a^{13}-\frac{19\!\cdots\!70}{13\!\cdots\!01}a^{12}-\frac{55\!\cdots\!13}{98\!\cdots\!15}a^{11}+\frac{46\!\cdots\!13}{68\!\cdots\!05}a^{10}-\frac{36\!\cdots\!76}{13\!\cdots\!01}a^{9}+\frac{31\!\cdots\!76}{68\!\cdots\!05}a^{8}-\frac{28\!\cdots\!91}{19\!\cdots\!43}a^{7}-\frac{62\!\cdots\!69}{68\!\cdots\!05}a^{6}-\frac{29\!\cdots\!18}{68\!\cdots\!05}a^{5}+\frac{26\!\cdots\!17}{19\!\cdots\!43}a^{4}+\frac{25\!\cdots\!28}{68\!\cdots\!05}a^{3}-\frac{20\!\cdots\!33}{13\!\cdots\!01}a^{2}+\frac{29\!\cdots\!13}{68\!\cdots\!05}a+\frac{11\!\cdots\!63}{98\!\cdots\!15}$, $\frac{20\!\cdots\!19}{68\!\cdots\!05}a^{13}-\frac{15\!\cdots\!64}{68\!\cdots\!05}a^{12}+\frac{16\!\cdots\!87}{14\!\cdots\!45}a^{11}-\frac{28\!\cdots\!03}{68\!\cdots\!05}a^{10}+\frac{82\!\cdots\!16}{68\!\cdots\!05}a^{9}-\frac{38\!\cdots\!80}{13\!\cdots\!01}a^{8}+\frac{52\!\cdots\!58}{98\!\cdots\!15}a^{7}-\frac{51\!\cdots\!49}{68\!\cdots\!05}a^{6}+\frac{50\!\cdots\!08}{68\!\cdots\!05}a^{5}-\frac{38\!\cdots\!77}{98\!\cdots\!15}a^{4}+\frac{73\!\cdots\!36}{68\!\cdots\!05}a^{3}-\frac{25\!\cdots\!47}{68\!\cdots\!05}a^{2}-\frac{73\!\cdots\!84}{68\!\cdots\!05}a+\frac{11\!\cdots\!21}{98\!\cdots\!15}$ (assuming GRH) | 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: | \( 1175716963.759769 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 1175716963.759769 \cdot 7}{2\cdot\sqrt{2523048836405617863000000000000}}\cr\approx \mathstrut & 1.00153718611466 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 42 |
The 7 conjugacy class representatives for $F_7$ |
Character table for $F_7$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 7.1.600362847000000.35 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 42 |
Degree 7 sibling: | 7.1.600362847000000.35 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.600362847000000.35 |
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 | R | 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 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(5\) | 5.14.12.1 | $x^{14} + 28 x^{13} + 350 x^{12} + 2576 x^{11} + 12404 x^{10} + 41104 x^{9} + 96152 x^{8} + 160650 x^{7} + 192444 x^{6} + 165676 x^{5} + 106232 x^{4} + 65016 x^{3} + 59920 x^{2} + 57232 x + 27193$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 7.14.15.5 | $x^{14} + 7 x^{3} + 7 x^{2} + 7$ | $14$ | $1$ | $15$ | $F_7$ | $[7/6]_{6}$ |