Normalized defining polynomial
\( x^{14} + 2800 \)
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: | \(-65712362363534280139543000000000000\) \(\medspace = -\,2^{12}\cdot 5^{12}\cdot 7^{27}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(306.88\) | 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: | $2^{6/7}5^{6/7}7^{83/42}\approx 336.68018354640174$ | ||
Ramified primes: | \(2\), \(5\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{40}a^{7}-\frac{1}{2}$, $\frac{1}{40}a^{8}-\frac{1}{2}a$, $\frac{1}{40}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{40}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{80}a^{11}-\frac{1}{4}a^{4}$, $\frac{1}{80}a^{12}-\frac{1}{4}a^{5}$, $\frac{1}{80}a^{13}-\frac{1}{4}a^{6}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{21}$, which has order $63$ (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: | $\frac{429}{5}a^{13}-\frac{18711}{80}a^{12}-\frac{7291}{8}a^{11}-\frac{8927}{5}a^{10}-\frac{83569}{40}a^{9}-\frac{1683}{8}a^{8}+\frac{23651}{4}a^{7}+16953a^{6}+\frac{113425}{4}a^{5}+25480a^{4}-17889a^{3}-\frac{256991}{2}a^{2}-\frac{597233}{2}a-423816$, $\frac{28083}{10}a^{13}-\frac{6271}{40}a^{12}-\frac{358643}{40}a^{11}-\frac{105811}{8}a^{10}+\frac{15261}{2}a^{9}+\frac{211129}{4}a^{8}+\frac{114015}{2}a^{7}-76819a^{6}-294769a^{5}-\frac{424207}{2}a^{4}+\frac{1182957}{2}a^{3}+1564493a^{2}+555023a-4011981$, $\frac{22349}{80}a^{13}-\frac{185}{4}a^{12}-\frac{61879}{80}a^{11}+\frac{6587}{40}a^{10}+2167a^{9}-\frac{24583}{20}a^{8}-\frac{31905}{4}a^{7}+\frac{10501}{4}a^{6}+25357a^{5}-\frac{11515}{4}a^{4}-\frac{132051}{2}a^{3}+27776a^{2}+217392a-111264$, $\frac{46533}{40}a^{13}+\frac{246189}{80}a^{12}+\frac{124159}{20}a^{11}+\frac{21043}{2}a^{10}+\frac{691469}{40}a^{9}+\frac{250119}{8}a^{8}+59492a^{7}+102189a^{6}+\frac{563599}{4}a^{5}+\frac{287763}{2}a^{4}+100442a^{3}+\frac{68985}{2}a^{2}-\frac{247905}{2}a-854001$, $\frac{11289}{20}a^{13}+\frac{107287}{80}a^{12}+\frac{20657}{20}a^{11}-\frac{226619}{40}a^{10}-\frac{267433}{40}a^{9}+\frac{684431}{20}a^{8}-\frac{51027}{2}a^{7}+1286a^{6}-\frac{585821}{4}a^{5}+437780a^{4}-\frac{725001}{2}a^{3}+\frac{149261}{2}a^{2}-1268981a+3540249$, $\frac{151558889}{20}a^{13}+\frac{3962927821}{80}a^{12}+\frac{9039796141}{80}a^{11}+\frac{6301554643}{40}a^{10}+\frac{1661921541}{20}a^{9}-\frac{2084364495}{8}a^{8}-\frac{1952876311}{2}a^{7}-\frac{3763871799}{2}a^{6}-\frac{8612298663}{4}a^{5}-\frac{345672355}{4}a^{4}+\frac{12905709487}{2}a^{3}+18056714928a^{2}+\frac{59440094595}{2}a+25809209019$ (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: | \( 22014727486.660755 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 22014727486.660755 \cdot 63}{2\cdot\sqrt{65712362363534280139543000000000000}}\cr\approx \mathstrut & 1.04582593120603 \end{aligned}\] (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
\(\Q(\sqrt{-7}) \), 7.1.96889010407000000.20 |
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: | 7.1.96889010407000000.20 |
Degree 21 sibling: | deg 21 |
Minimal sibling: | 7.1.96889010407000000.20 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | 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 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(5\) | 5.14.12.1 | $x^{14} + 28 x^{13} + 350 x^{12} + 2576 x^{11} + 12404 x^{10} + 41104 x^{9} + 96152 x^{8} + 160650 x^{7} + 192444 x^{6} + 165676 x^{5} + 106232 x^{4} + 65016 x^{3} + 59920 x^{2} + 57232 x + 27193$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(7\) | 7.14.27.45 | $x^{14} + 294 x^{2} + 105$ | $14$ | $1$ | $27$ | $F_7$ | $[13/6]_{6}$ |