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

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

Rank

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

Modular form 19.2.a.a

Isogeny matrix

The \((i,j)\)-th entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

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

Isogeny graph

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 19.a

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