Elliptic curve isogeny class with LMFDB label 448448.by (Cremona label 448448by)

• Label: 448448.by
• Number of curves: $4$
• Conductor: $448448$
• CM: no
• Rank: $2$

Elliptic curves in class 448448.by

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
448448.by1	448448by3	\([0, 0, 0, -31095596, 66741571120]\)	\(107818231938348177/4463459\)	\(137657447321698304\)	\([2]\)	\(14942208\)	\(2.7749\)	\(\Gamma_0(N)\)-optimal^*
448448.by2	448448by4	\([0, 0, 0, -3153836, -404064976]\)	\(112489728522417/62811265517\)	\(1937160949415158218752\)	\([2]\)	\(14942208\)	\(2.7749\)
448448.by3	448448by2	\([0, 0, 0, -1946476, 1039454640]\)	\(26444947540257/169338169\)	\(5222554991245656064\)	\([2, 2]\)	\(7471104\)	\(2.4283\)	\(\Gamma_0(N)\)-optimal^*
448448.by4	448448by1	\([0, 0, 0, -49196, 35414064]\)	\(-426957777/17320303\)	\(-534175108994695168\)	\([2]\)	\(3735552\)	\(2.0818\)	\(\Gamma_0(N)\)-optimal^*

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 448448.by1.

Rank

The elliptic curves in class 448448.by have rank \(2\).

Complex multiplication

The elliptic curves in class 448448.by do not have complex multiplication.

Modular form 448448.2.a.by

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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.