Elliptic curve isogeny class with Cremona label 411840hc (LMFDB label 411840.hc)

• Label: 411840hc
• Number of curves: $8$
• Conductor: $411840$
• CM: no
• Rank: $1$

Elliptic curves in class 411840hc

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
411840.hc7	411840hc1	\([0, 0, 0, -6577932, 6211625456]\)	\(164711681450297281/8097103872000\)	\(1547380646920323072000\)	\([2]\)	\(21233664\)	\(2.8252\)	\(\Gamma_0(N)\)-optimal^*
411840.hc6	411840hc2	\([0, 0, 0, -18374412, -22293388816]\)	\(3590017885052913601/954068544000000\)	\(182325338066386944000000\)	\([2, 2]\)	\(42467328\)	\(3.1718\)
411840.hc3	411840hc3	\([0, 0, 0, -526360332, 4648071946736]\)	\(84392862605474684114881/11228954880\)	\(2145886694937722880\)	\([2]\)	\(63700992\)	\(3.3745\)	\(\Gamma_0(N)\)-optimal^*
411840.hc8	411840hc4	\([0, 0, 0, 46321908, -144207134224]\)	\(57519563401957999679/80296734375000000\)	\(-15344944902144000000000000\)	\([2]\)	\(84934656\)	\(3.5184\)
411840.hc5	411840hc5	\([0, 0, 0, -271814412, -1724700556816]\)	\(11621808143080380273601/1335706803288000\)	\(255257545171783385088000\)	\([2]\)	\(84934656\)	\(3.5184\)
411840.hc2	411840hc6	\([0, 0, 0, -526406412, 4647217420784]\)	\(84415028961834287121601/30783551683856400\)	\(5882828338634769196646400\)	\([2, 2]\)	\(127401984\)	\(3.7211\)
411840.hc4	411840hc7	\([0, 0, 0, -450478092, 6034853025776]\)	\(-52902632853833942200321/51713453577420277500\)	\(-9882594877882861432995840000\)	\([2]\)	\(254803968\)	\(4.0677\)
411840.hc1	411840hc8	\([0, 0, 0, -603072012, 3204892154864]\)	\(126929854754212758768001/50235797102795981820\)	\(9600210328076490046643896320\)	\([2]\)	\(254803968\)	\(4.0677\)

^*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 411840hc1.

Rank

The elliptic curves in class 411840hc have rank \(1\).

Complex multiplication

The elliptic curves in class 411840hc do not have complex multiplication.

Modular form 411840.2.a.hc

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.