Normalized defining polynomial
\( x^{14} - 7 x^{13} + 35 x^{12} - 119 x^{11} + 329 x^{10} - 721 x^{9} + 1337 x^{8} + 1284559 x^{7} + \cdots + 413845826258 \)
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: | \(-29898464040397228799207555080392669309147967488\) \(\medspace = -\,2^{12}\cdot 3^{12}\cdot 7^{11}\cdot 11^{12}\cdot 19^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(2087.80\) | 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: | $2^{6/7}3^{6/7}7^{5/6}11^{6/7}19^{6/7}\approx 2290.508824994667$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(11\), \(19\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{21}a^{6}+\frac{2}{21}a^{5}-\frac{1}{7}a^{4}+\frac{1}{21}a^{3}+\frac{3}{7}a^{2}+\frac{4}{21}a-\frac{2}{7}$, $\frac{1}{21}a^{7}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{2}{21}$, $\frac{1}{1197}a^{8}-\frac{4}{1197}a^{7}+\frac{2}{171}a^{6}-\frac{4}{171}a^{5}+\frac{7}{171}a^{4}-\frac{8}{171}a^{3}+\frac{8}{171}a^{2}+\frac{481}{1197}a+\frac{358}{1197}$, $\frac{1}{1197}a^{9}-\frac{2}{1197}a^{7}+\frac{4}{171}a^{6}-\frac{1}{19}a^{5}+\frac{20}{171}a^{4}-\frac{8}{57}a^{3}-\frac{164}{399}a^{2}-\frac{16}{171}a+\frac{235}{1197}$, $\frac{1}{8379}a^{10}+\frac{1}{8379}a^{9}-\frac{2}{8379}a^{8}+\frac{83}{8379}a^{7}-\frac{5}{1197}a^{6}-\frac{160}{1197}a^{5}-\frac{4}{1197}a^{4}+\frac{237}{931}a^{3}+\frac{3785}{8379}a^{2}+\frac{191}{931}a-\frac{3470}{8379}$, $\frac{1}{8379}a^{11}-\frac{1}{2793}a^{9}+\frac{1}{8379}a^{8}-\frac{181}{8379}a^{7}+\frac{1}{63}a^{6}-\frac{1}{57}a^{5}-\frac{758}{8379}a^{4}-\frac{346}{1197}a^{3}+\frac{811}{8379}a^{2}+\frac{3085}{8379}a+\frac{1727}{8379}$, $\frac{1}{25137}a^{12}+\frac{4}{25137}a^{9}+\frac{1}{2793}a^{8}-\frac{134}{8379}a^{7}+\frac{2}{513}a^{6}-\frac{358}{2793}a^{5}-\frac{118}{1197}a^{4}-\frac{1678}{3591}a^{3}-\frac{167}{931}a^{2}+\frac{1667}{8379}a-\frac{4082}{25137}$, $\frac{1}{17\!\cdots\!67}a^{13}+\frac{11\!\cdots\!29}{17\!\cdots\!67}a^{12}-\frac{26\!\cdots\!48}{59\!\cdots\!89}a^{11}-\frac{20\!\cdots\!33}{17\!\cdots\!67}a^{10}+\frac{23\!\cdots\!05}{93\!\cdots\!93}a^{9}-\frac{38\!\cdots\!56}{59\!\cdots\!89}a^{8}+\frac{22\!\cdots\!10}{93\!\cdots\!93}a^{7}-\frac{15\!\cdots\!75}{17\!\cdots\!67}a^{6}-\frac{55\!\cdots\!48}{59\!\cdots\!89}a^{5}-\frac{76\!\cdots\!21}{17\!\cdots\!67}a^{4}+\frac{82\!\cdots\!56}{17\!\cdots\!67}a^{3}+\frac{50\!\cdots\!01}{19\!\cdots\!63}a^{2}-\frac{43\!\cdots\!16}{17\!\cdots\!67}a-\frac{28\!\cdots\!43}{17\!\cdots\!67}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
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: | not computed | 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: | not computed | 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 $
Galois group
A solvable group of order 42 |
The 7 conjugacy class representatives for $F_7$ |
Character table for $F_7$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), 7.1.65354488358705521601472.1 |
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: | 7.1.65354488358705521601472.1 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.65354488358705521601472.1 |
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 | R | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(11\) | 11.7.6.1 | $x^{7} + 11$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
11.7.6.1 | $x^{7} + 11$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(19\) | 19.14.12.1 | $x^{14} + 126 x^{13} + 6818 x^{12} + 205632 x^{11} + 3742284 x^{10} + 41321448 x^{9} + 260402968 x^{8} + 775862822 x^{7} + 520808330 x^{6} + 165413472 x^{5} + 33744732 x^{4} + 71380792 x^{3} + 731029768 x^{2} + 4357641344 x + 11135742105$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |