Properties

Label 14400ef
Number of curves $8$
Conductor $14400$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("ef1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 14400ef

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
14400.o8 14400ef1 \([0, 0, 0, 21300, 3674000]\) \(357911/2160\) \(-6449725440000000\) \([2]\) \(73728\) \(1.7155\) \(\Gamma_0(N)\)-optimal
14400.o6 14400ef2 \([0, 0, 0, -266700, 48026000]\) \(702595369/72900\) \(217678233600000000\) \([2, 2]\) \(147456\) \(2.0620\)  
14400.o7 14400ef3 \([0, 0, 0, -194700, -108214000]\) \(-273359449/1536000\) \(-4586471424000000000\) \([2]\) \(221184\) \(2.2648\)  
14400.o5 14400ef4 \([0, 0, 0, -986700, -324934000]\) \(35578826569/5314410\) \(15868743229440000000\) \([2]\) \(294912\) \(2.4086\)  
14400.o4 14400ef5 \([0, 0, 0, -4154700, 3259514000]\) \(2656166199049/33750\) \(100776960000000000\) \([2]\) \(294912\) \(2.4086\)  
14400.o3 14400ef6 \([0, 0, 0, -4802700, -4043446000]\) \(4102915888729/9000000\) \(26873856000000000000\) \([2, 2]\) \(442368\) \(2.6113\)  
14400.o1 14400ef7 \([0, 0, 0, -76802700, -259067446000]\) \(16778985534208729/81000\) \(241864704000000000\) \([2]\) \(884736\) \(2.9579\)  
14400.o2 14400ef8 \([0, 0, 0, -6530700, -874294000]\) \(10316097499609/5859375000\) \(17496000000000000000000\) \([2]\) \(884736\) \(2.9579\)  

Rank

sage: E.rank()
 

The elliptic curves in class 14400ef have rank \(1\).

Complex multiplication

The elliptic curves in class 14400ef do not have complex multiplication.

Modular form 14400.2.a.ef

sage: E.q_eigenform(10)
 
\(q - 4q^{7} + 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().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

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.