Elliptic curve isogeny class with Cremona label 19a (LMFDB label 19.a)

• Label: 19a
• Number of curves: $3$
• Conductor: $19$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 19a have rank $0$.

L-function data

Bad L-factors:

Prime	L-Factor
$19$	$1 - T$

Good L-factors:

Prime	L-Factor	Isogeny Class over $\mathbb{F}_p$
$2$	$1 + 2 T^{2}$	1.2.a
$3$	$1 + 2 T + 3 T^{2}$	1.3.c
$5$	$1 - 3 T + 5 T^{2}$	1.5.ad
$7$	$1 + T + 7 T^{2}$	1.7.b
$11$	$1 - 3 T + 11 T^{2}$	1.11.ad
$13$	$1 + 4 T + 13 T^{2}$	1.13.e
$17$	$1 + 3 T + 17 T^{2}$	1.17.d
$23$	$1 + 23 T^{2}$	1.23.a
$29$	$1 - 6 T + 29 T^{2}$	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 19a do not have complex multiplication.

Modular form 19.2.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.

$\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)$

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 19a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
19.a2	19a1	$[0, 1, 1, -9, -15]$	$-89915392/6859$	$-6859$	$[3]$	$1$	$-0.51587$	$\Gamma_0(N)$-optimal
19.a1	19a2	$[0, 1, 1, -769, -8470]$	$-50357871050752/19$	$-19$	$[]$	$3$	$0.033439$
19.a3	19a3	$[0, 1, 1, 1, 0]$	$32768/19$	$-19$	$[3]$	$3$	$-1.0652$