LMFDB - Galois group: 35T15

Show commands: Magma

magma: G := TransitiveGroup(35, 15);

Group action invariants

Degree $n$:	$35$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$15$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$D_5\times F_7$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$1$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,12,6,27,16,22)(2,11,7,26,17,21)(3,15,8,30,18,25)(4,14,9,29,19,24)(5,13,10,28,20,23)(31,32)(33,35), (1,29,16,9,31,24,11,4,26,19,6,34,21,14)(2,28,17,8,32,23,12,3,27,18,7,33,22,13)(5,30,20,10,35,25,15)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$3$: $C_3$
$4$: $C_2^2$
$6$: $C_6$ x 3
$10$: $D_{5}$
$12$: $C_6\times C_2$
$20$: $D_{10}$
$30$: $D_5\times C_3$
$42$: $F_7$
$60$: 30T5
$84$: $F_7 \times C_2$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_2^2$
$6$:	$C_6$ x 3
$10$:	$D_{5}$
$12$:	$C_6\times C_2$
$20$:	$D_{10}$
$30$:	$D_5\times C_3$
$42$:	$F_7$
$60$:	30T5
$84$:	$F_7 \times C_2$

Subfields

Degree 5: $D_{5}$
Degree 7: $F_7$

Low degree siblings

There are no siblings with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $	$7$	$3$	$( 6,11,21)( 7,12,22)( 8,13,23)( 9,14,24)(10,15,25)(16,31,26)(17,32,27) (18,33,28)(19,34,29)(20,35,30)$
	$ 6, 6, 6, 6, 6, 1, 1, 1, 1, 1 $	$7$	$6$	$( 6,16,11,31,21,26)( 7,17,12,32,22,27)( 8,18,13,33,23,28)( 9,19,14,34,24,29) (10,20,15,35,25,30)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $	$7$	$3$	$( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32) (18,28,33)(19,29,34)(20,30,35)$
	$ 6, 6, 6, 6, 6, 1, 1, 1, 1, 1 $	$7$	$6$	$( 6,26,21,31,11,16)( 7,27,22,32,12,17)( 8,28,23,33,13,18)( 9,29,24,34,14,19) (10,30,25,35,15,20)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $	$7$	$2$	$( 6,31)( 7,32)( 8,33)( 9,34)(10,35)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21) (17,22)(18,23)(19,24)(20,25)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $	$5$	$2$	$( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$
	$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $	$35$	$6$	$( 2, 5)( 3, 4)( 6,11,21)( 7,15,22,10,12,25)( 8,14,23, 9,13,24)(16,31,26) (17,35,27,20,32,30)(18,34,28,19,33,29)$
	$ 6, 6, 6, 6, 6, 2, 2, 1 $	$35$	$6$	$( 2, 5)( 3, 4)( 6,16,11,31,21,26)( 7,20,12,35,22,30)( 8,19,13,34,23,29) ( 9,18,14,33,24,28)(10,17,15,32,25,27)$
	$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $	$35$	$6$	$( 2, 5)( 3, 4)( 6,21,11)( 7,25,12,10,22,15)( 8,24,13, 9,23,14)(16,26,31) (17,30,32,20,27,35)(18,29,33,19,28,34)$
	$ 6, 6, 6, 6, 6, 2, 2, 1 $	$35$	$6$	$( 2, 5)( 3, 4)( 6,26,21,31,11,16)( 7,30,22,35,12,20)( 8,29,23,34,13,19) ( 9,28,24,33,14,18)(10,27,25,32,15,17)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$35$	$2$	$( 2, 5)( 3, 4)( 6,31)( 7,35)( 8,34)( 9,33)(10,32)(11,26)(12,30)(13,29)(14,28) (15,27)(16,21)(17,25)(18,24)(19,23)(20,22)$
	$ 5, 5, 5, 5, 5, 5, 5 $	$2$	$5$	$( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
	$ 15, 15, 5 $	$14$	$15$	$( 1, 2, 3, 4, 5)( 6,12,23, 9,15,21, 7,13,24,10,11,22, 8,14,25)(16,32,28,19,35, 26,17,33,29,20,31,27,18,34,30)$
	$ 30, 5 $	$14$	$30$	$( 1, 2, 3, 4, 5)( 6,17,13,34,25,26, 7,18,14,35,21,27, 8,19,15,31,22,28, 9,20, 11,32,23,29,10,16,12,33,24,30)$
	$ 15, 15, 5 $	$14$	$15$	$( 1, 2, 3, 4, 5)( 6,22,13, 9,25,11, 7,23,14,10,21,12, 8,24,15)(16,27,33,19,30, 31,17,28,34,20,26,32,18,29,35)$
	$ 30, 5 $	$14$	$30$	$( 1, 2, 3, 4, 5)( 6,27,23,34,15,16, 7,28,24,35,11,17, 8,29,25,31,12,18, 9,30, 21,32,13,19,10,26,22,33,14,20)$
	$ 10, 10, 10, 5 $	$14$	$10$	$( 1, 2, 3, 4, 5)( 6,32, 8,34,10,31, 7,33, 9,35)(11,27,13,29,15,26,12,28,14,30) (16,22,18,24,20,21,17,23,19,25)$
	$ 5, 5, 5, 5, 5, 5, 5 $	$2$	$5$	$( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$
	$ 15, 15, 5 $	$14$	$15$	$( 1, 3, 5, 2, 4)( 6,13,25, 7,14,21, 8,15,22, 9,11,23,10,12,24)(16,33,30,17,34, 26,18,35,27,19,31,28,20,32,29)$
	$ 30, 5 $	$14$	$30$	$( 1, 3, 5, 2, 4)( 6,18,15,32,24,26, 8,20,12,34,21,28,10,17,14,31,23,30, 7,19, 11,33,25,27, 9,16,13,35,22,29)$
	$ 15, 15, 5 $	$14$	$15$	$( 1, 3, 5, 2, 4)( 6,23,15, 7,24,11, 8,25,12, 9,21,13,10,22,14)(16,28,35,17,29, 31,18,30,32,19,26,33,20,27,34)$
	$ 30, 5 $	$14$	$30$	$( 1, 3, 5, 2, 4)( 6,28,25,32,14,16, 8,30,22,34,11,18,10,27,24,31,13,20, 7,29, 21,33,15,17, 9,26,23,35,12,19)$
	$ 10, 10, 10, 5 $	$14$	$10$	$( 1, 3, 5, 2, 4)( 6,33,10,32, 9,31, 8,35, 7,34)(11,28,15,27,14,26,13,30,12,29) (16,23,20,22,19,21,18,25,17,24)$
	$ 7, 7, 7, 7, 7 $	$6$	$7$	$( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$
	$ 14, 14, 7 $	$30$	$14$	$( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$
	$ 35 $	$12$	$35$	$( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$
	$ 35 $	$12$	$35$	$( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$420=2^{2} \cdot 3 \cdot 5 \cdot 7$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	420.16	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	3A1	3A-1	5A1	5A2	6A1	6A-1	6B1	6B-1	6C1	6C-1	7A	10A1	10A3	14A	15A1	15A-1	15A2	15A-2	30A1	30A-1	30A7	30A-7	35A1	35A2
	Size	1	5	7	35	7	7	2	2	7	7	35	35	35	35	6	14	14	30	14	14	14	14	14	14	14	14	12	12
	2 P	1A	1A	1A	1A	3A-1	3A1	5A2	5A1	3A1	3A-1	3A1	3A-1	3A1	3A-1	7A	5A1	5A2	7A	15A2	15A-2	15A1	15A-1	15A2	15A-1	15A1	15A-2	35A2	35A1
	3 P	1A	2A	2B	2C	1A	1A	5A2	5A1	2B	2B	2C	2C	2A	2A	7A	10A3	10A1	14A	5A1	5A1	5A2	5A2	10A3	10A1	10A1	10A3	35A2	35A1
	5 P	1A	2A	2B	2C	3A-1	3A1	1A	1A	6A-1	6A1	6B-1	6B1	6C-1	6C1	7A	2B	2B	14A	3A1	3A-1	3A-1	3A1	6A-1	6A-1	6A1	6A1	7A	7A
	7 P	1A	2A	2B	2C	3A1	3A-1	5A2	5A1	6A1	6A-1	6B1	6B-1	6C1	6C-1	1A	10A3	10A1	2A	15A-2	15A2	15A-1	15A1	30A-1	30A-7	30A7	30A1	5A2	5A1
	Type
420.16.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
420.16.1b	R	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
420.16.1c	R	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
420.16.1d	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
420.16.1e1	C	$1$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$
420.16.1e2	C	$1$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$
420.16.1f1	C	$1$	$- 1$	$- 1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$1$	$- 1$	$- 1$	$- 1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$1$	$1$
420.16.1f2	C	$1$	$- 1$	$- 1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$1$	$- 1$	$- 1$	$- 1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$1$	$1$
420.16.1g1	C	$1$	$- 1$	$1$	$- 1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$1$	$1$	$1$	$- 1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$
420.16.1g2	C	$1$	$- 1$	$1$	$- 1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$1$	$1$	$1$	$- 1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$
420.16.1h1	C	$1$	$1$	$- 1$	$- 1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$- 1$	$- 1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$1$	$1$
420.16.1h2	C	$1$	$1$	$- 1$	$- 1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$- 1$	$- 1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$1$	$1$
420.16.2a1	R	$2$	$0$	$2$	$0$	$2$	$2$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$	$2$	$2$	$0$	$0$	$0$	$0$	$2$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$0$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$
420.16.2a2	R	$2$	$0$	$2$	$0$	$2$	$2$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$2$	$2$	$0$	$0$	$0$	$0$	$2$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$	$0$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$
420.16.2b1	R	$2$	$0$	$- 2$	$0$	$2$	$2$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$2$	$- ζ_{5}^{- 1} - ζ_{5}$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$0$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$- ζ_{5}^{- 1} - ζ_{5}$	$- ζ_{5}^{- 1} - ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$
420.16.2b2	R	$2$	$0$	$- 2$	$0$	$2$	$2$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$2$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$- ζ_{5}^{- 1} - ζ_{5}$	$0$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 1} + ζ_{5}$	$- ζ_{5}^{- 1} - ζ_{5}$	$- ζ_{5}^{- 1} - ζ_{5}$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$ζ_{5}^{- 1} + ζ_{5}$	$ζ_{5}^{- 2} + ζ_{5}^{2}$
420.16.2c1	C	$2$	$0$	$2$	$0$	$2 ζ_{15}^{- 5}$	$2 ζ_{15}^{5}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$2 ζ_{15}^{5}$	$2 ζ_{15}^{- 5}$	$0$	$0$	$0$	$0$	$2$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$0$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$ζ_{15} + ζ_{15}^{4}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$ζ_{15} + ζ_{15}^{4}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$
420.16.2c2	C	$2$	$0$	$2$	$0$	$2 ζ_{15}^{5}$	$2 ζ_{15}^{- 5}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$2 ζ_{15}^{- 5}$	$2 ζ_{15}^{5}$	$0$	$0$	$0$	$0$	$2$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$0$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$ζ_{15} + ζ_{15}^{4}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$ζ_{15} + ζ_{15}^{4}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$
420.16.2c3	C	$2$	$0$	$2$	$0$	$2 ζ_{15}^{- 5}$	$2 ζ_{15}^{5}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$2 ζ_{15}^{5}$	$2 ζ_{15}^{- 5}$	$0$	$0$	$0$	$0$	$2$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$0$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$ζ_{15} + ζ_{15}^{4}$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$ζ_{15} + ζ_{15}^{4}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$
420.16.2c4	C	$2$	$0$	$2$	$0$	$2 ζ_{15}^{5}$	$2 ζ_{15}^{- 5}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$2 ζ_{15}^{- 5}$	$2 ζ_{15}^{5}$	$0$	$0$	$0$	$0$	$2$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$0$	$ζ_{15} + ζ_{15}^{4}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$ζ_{15} + ζ_{15}^{4}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$
420.16.2d1	C	$2$	$0$	$- 2$	$0$	$2 ζ_{15}^{- 5}$	$2 ζ_{15}^{5}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$- 2 ζ_{15}^{5}$	$- 2 ζ_{15}^{- 5}$	$0$	$0$	$0$	$0$	$2$	$- ζ_{15}^{- 3} - ζ_{15}^{3}$	$- ζ_{15}^{- 6} - ζ_{15}^{6}$	$0$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$ζ_{15} + ζ_{15}^{4}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$- ζ_{15} - ζ_{15}^{4}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} + ζ_{15}^{7}$	$- 1 + ζ_{15} + ζ_{15}^{4} - ζ_{15}^{5}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$
420.16.2d2	C	$2$	$0$	$- 2$	$0$	$2 ζ_{15}^{5}$	$2 ζ_{15}^{- 5}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$- 2 ζ_{15}^{- 5}$	$- 2 ζ_{15}^{5}$	$0$	$0$	$0$	$0$	$2$	$- ζ_{15}^{- 3} - ζ_{15}^{3}$	$- ζ_{15}^{- 6} - ζ_{15}^{6}$	$0$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$ζ_{15} + ζ_{15}^{4}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} + ζ_{15}^{7}$	$- ζ_{15} - ζ_{15}^{4}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- 1 + ζ_{15} + ζ_{15}^{4} - ζ_{15}^{5}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$
420.16.2d3	C	$2$	$0$	$- 2$	$0$	$2 ζ_{15}^{- 5}$	$2 ζ_{15}^{5}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$- 2 ζ_{15}^{5}$	$- 2 ζ_{15}^{- 5}$	$0$	$0$	$0$	$0$	$2$	$- ζ_{15}^{- 6} - ζ_{15}^{6}$	$- ζ_{15}^{- 3} - ζ_{15}^{3}$	$0$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$ζ_{15} + ζ_{15}^{4}$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$- 1 + ζ_{15} + ζ_{15}^{4} - ζ_{15}^{5}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- ζ_{15} - ζ_{15}^{4}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} + ζ_{15}^{7}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$
420.16.2d4	C	$2$	$0$	$- 2$	$0$	$2 ζ_{15}^{5}$	$2 ζ_{15}^{- 5}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$	$- 2 ζ_{15}^{- 5}$	$- 2 ζ_{15}^{5}$	$0$	$0$	$0$	$0$	$2$	$- ζ_{15}^{- 6} - ζ_{15}^{6}$	$- ζ_{15}^{- 3} - ζ_{15}^{3}$	$0$	$ζ_{15} + ζ_{15}^{4}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} - ζ_{15}^{7}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$1 - ζ_{15} - ζ_{15}^{4} + ζ_{15}^{5}$	$1 - ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- 1 + ζ_{15} + ζ_{15}^{4} - ζ_{15}^{5}$	$- 1 + ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + ζ_{15}^{4} + ζ_{15}^{7}$	$- ζ_{15} - ζ_{15}^{4}$	$ζ_{15}^{- 3} + ζ_{15}^{3}$	$ζ_{15}^{- 6} + ζ_{15}^{6}$
420.16.6a	R	$6$	$6$	$0$	$0$	$0$	$0$	$6$	$6$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$
420.16.6b	R	$6$	$- 6$	$0$	$0$	$0$	$0$	$6$	$6$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$0$	$0$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$
420.16.12a1	R	$12$	$0$	$0$	$0$	$0$	$0$	$6 ζ_{5}^{- 2} + 6 ζ_{5}^{2}$	$6 ζ_{5}^{- 1} + 6 ζ_{5}$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$	$- ζ_{5}^{- 1} - ζ_{5}$
420.16.12a2	R	$12$	$0$	$0$	$0$	$0$	$0$	$6 ζ_{5}^{- 1} + 6 ζ_{5}$	$6 ζ_{5}^{- 2} + 6 ζ_{5}^{2}$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{5}^{- 1} - ζ_{5}$	$- ζ_{5}^{- 2} - ζ_{5}^{2}$

magma: CharacterTable(G);

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	35T15
Degree	$35$
Order	$420$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$D_5\times F_7$