Normalized defining polynomial
Invariants
Degree: |
| ||
Signature: |
| ||
Discriminant: |
|
| |
Root discriminant: |
| ||
Galois root discriminant: | |||
Ramified primes: |
,
|
| |
Discriminant root field: | |||
: |
| ||
This field is not Galois over . | |||
This is not a CM field. | |||
This field has no CM subfields. |
Integral basis (with respect to field generator )
, , , , , ,
Monogenic: | Not computed | |
Index: | ||
Inessential primes: | None |
Class group and class number
Ideal class group: | Trivial group, which has order |
| |
Narrow class group: | Trivial group, which has order |
|
Unit group
Rank: |
| ||
Torsion generator: |
(order )
|
| |
Fundamental units: |
, , ,
|
| |
Regulator: |
|
Class number formula
Galois group
(as 7T5):
A non-solvable group of order 168 |
The 6 conjugacy class representatives for |
Character table for |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and . |
Sibling fields
Degree 8 sibling: | 8.0.251675665475521.3 |
Degree 14 siblings: | deg 14, deg 14 |
Degree 21 sibling: | deg 21 |
Degree 24 sibling: | deg 24 |
Degree 28 sibling: | deg 28 |
Degree 42 siblings: | deg 42, deg 42, deg 42 |
Arithmetically equivalent sibling: | 7.3.15864289.2 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
Cycle type | R |
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
Label | Polynomial | Galois group | Slope content | ||||
---|---|---|---|---|---|---|---|
| Trivial | ||||||
Trivial | |||||||
Trivial | |||||||
7.2.2.2a1.2 | |||||||
| Trivial | ||||||
Deg | |||||||
Deg | |||||||
Deg |
Artin representations
Label | Dimension | Conductor | Artin stem field | Ind | |||
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | ||||||
3.15864289.42t37.a.a | 7.3.15864289.1 | (as 7T5) | |||||
3.15864289.42t37.a.b | 7.3.15864289.1 | (as 7T5) | |||||
* | 6.15864289.7t5.a.a | 7.3.15864289.1 | (as 7T5) | ||||
7.251...521.8t37.a.a | 7.3.15864289.1 | (as 7T5) | |||||
8.251...521.21t14.a.a | 7.3.15864289.1 | (as 7T5) |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.