Elliptic curve isogeny class with Cremona label 36a (LMFDB label 36.a)

• Label: 36a
• Number of curves: $4$
• Conductor: $36$
• CM: $\Q(\sqrt{-3})$
• Rank: $0$

Rank

The elliptic curves in class 36a have rank $0$.

L-function data

Bad L-factors:

Prime	L-Factor
$2$	$1$
$3$	$1$

Good L-factors:

Prime	L-Factor	Isogeny Class over $\mathbb{F}_p$
$5$	$1 + 5 T^{2}$	1.5.a
$7$	$1 + 4 T + 7 T^{2}$	1.7.e
$11$	$1 + 11 T^{2}$	1.11.a
$13$	$1 - 2 T + 13 T^{2}$	1.13.ac
$17$	$1 + 17 T^{2}$	1.17.a
$19$	$1 - 8 T + 19 T^{2}$	1.19.ai
$23$	$1 + 23 T^{2}$	1.23.a
$29$	$1 + 29 T^{2}$	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

Each elliptic curve in class 36a has complex multiplication by an order in the imaginary quadratic field $\Q(\sqrt{-3})$.

Modular form 36.2.a.a

Isogeny matrix

The $i,j$ entry is the smallest degree of a cyclic isogeny between the $i$-th and $j$-th curve in the isogeny class, in the Cremona numbering.

$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 36a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
36.a4	36a1	$[0, 0, 0, 0, 1]$	$0$	$-432$	$[6]$	$1$	$-0.81542$	$\Gamma_0(N)$-optimal	$-3$
36.a2	36a2	$[0, 0, 0, -15, 22]$	$54000$	$6912$	$[6]$	$2$	$-0.46884$		$-12$
36.a3	36a3	$[0, 0, 0, 0, -27]$	$0$	$-314928$	$[2]$	$3$	$-0.26611$		$-3$
36.a1	36a4	$[0, 0, 0, -135, -594]$	$54000$	$5038848$	$[2]$	$6$	$0.080464$		$-12$