Normalized defining polynomial
\( x^{21} - 2847 x^{14} + 1401686 x^{7} + 62748517 \)
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[3, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-362095812116921075757469201554122678840602303=-\,7^{29}\cdot 13^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $132.39$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $7, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{13} a^{7}$, $\frac{1}{13} a^{8}$, $\frac{1}{13} a^{9}$, $\frac{1}{13} a^{10}$, $\frac{1}{1183} a^{11} + \frac{2}{91} a^{10} + \frac{2}{91} a^{9} - \frac{3}{91} a^{8} + \frac{3}{91} a^{7} - \frac{24}{91} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{1183} a^{12} - \frac{1}{91} a^{10} + \frac{1}{91} a^{9} - \frac{3}{91} a^{8} - \frac{1}{91} a^{7} - \frac{24}{91} a^{5} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{1183} a^{13} - \frac{1}{91} a^{10} + \frac{2}{91} a^{9} + \frac{2}{91} a^{8} - \frac{3}{91} a^{7} - \frac{24}{91} a^{6} + \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{27113177} a^{14} + \frac{37416}{2085629} a^{7} + \frac{3001}{12341}$, $\frac{1}{27113177} a^{15} + \frac{37416}{2085629} a^{8} + \frac{3001}{12341} a$, $\frac{1}{352471301} a^{16} + \frac{197849}{27113177} a^{9} + \frac{52365}{160433} a^{2}$, $\frac{1}{352471301} a^{17} + \frac{197849}{27113177} a^{10} + \frac{52365}{160433} a^{3}$, $\frac{1}{4582126913} a^{18} - \frac{100098}{352471301} a^{11} - \frac{2}{91} a^{10} - \frac{2}{91} a^{9} + \frac{3}{91} a^{8} - \frac{3}{91} a^{7} - \frac{841476}{2085629} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{4582126913} a^{19} - \frac{100098}{352471301} a^{12} + \frac{1}{91} a^{10} - \frac{1}{91} a^{9} + \frac{3}{91} a^{8} + \frac{1}{91} a^{7} - \frac{841476}{2085629} a^{5} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{59567649869} a^{20} + \frac{1091690}{4582126913} a^{13} + \frac{3}{91} a^{10} + \frac{1}{91} a^{9} + \frac{1}{91} a^{8} + \frac{2}{91} a^{7} + \frac{5300816}{27113177} a^{6} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a + \frac{1}{7}$
Class group and class number
$C_{2}\times C_{14}\times C_{14}$, which has order $392$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 238332776520.71344 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times D_7$ (as 21T3):
| A solvable group of order 42 |
| The 15 conjugacy class representatives for $C_3\times D_7$ |
| Character table for $C_3\times D_7$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 7.1.194779153450063.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $21$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | R | $21$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | $21$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{21}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | $21$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| $13$ | 13.7.6.1 | $x^{7} - 13$ | $7$ | $1$ | $6$ | $D_{7}$ | $[\ ]_{7}^{2}$ |
| 13.7.6.1 | $x^{7} - 13$ | $7$ | $1$ | $6$ | $D_{7}$ | $[\ ]_{7}^{2}$ | |
| 13.7.6.1 | $x^{7} - 13$ | $7$ | $1$ | $6$ | $D_{7}$ | $[\ ]_{7}^{2}$ | |