Elliptic curve isogeny class with Cremona label 42978r (LMFDB label 42978.u)

• Label: 42978r
• Number of curves: $2$
• Conductor: $42978$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 42978r have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(3\)	\(1 - T\)
\(13\)	\(1 - T\)
\(19\)	\(1 + T\)
\(29\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 4 T + 5 T^{2}\)	1.5.e
\(7\)	\( 1 + 2 T + 7 T^{2}\)	1.7.c
\(11\)	\( 1 + 3 T + 11 T^{2}\)	1.11.d
\(17\)	\( 1 + 7 T + 17 T^{2}\)	1.17.h
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 42978r do not have complex multiplication.

Modular form 42978.2.a.r

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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 42978r

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
42978.u1	42978r1	\([1, 0, 0, -459769306, 3796241996132]\)	\(-10748395438529140294639078020769/5737242625602477531070464\)	\(-5737242625602477531070464\)	\([7]\)	\(11063808\)	\(3.7003\)	\(\Gamma_0(N)\)-optimal
42978.u2	42978r2	\([1, 0, 0, 2127889334, -194909741524108]\)	\(1065542619208351347902742829533791/17028260093251190608019507801424\)	\(-17028260093251190608019507801424\)	\([]\)	\(77446656\)	\(4.6733\)