Properties

Label 68970.e
Number of curves $4$
Conductor $68970$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 68970.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68970.e1 68970e4 \([1, 1, 0, -75143, 7885563]\) \(26487576322129/44531250\) \(78889825781250\) \([2]\) \(327680\) \(1.5619\)  
68970.e2 68970e2 \([1, 1, 0, -6173, 36777]\) \(14688124849/8122500\) \(14389504222500\) \([2, 2]\) \(163840\) \(1.2153\)  
68970.e3 68970e1 \([1, 1, 0, -3753, -89547]\) \(3301293169/22800\) \(40391590800\) \([2]\) \(81920\) \(0.86872\) \(\Gamma_0(N)\)-optimal
68970.e4 68970e3 \([1, 1, 0, 24077, 321127]\) \(871257511151/527800050\) \(-935029984378050\) \([2]\) \(327680\) \(1.5619\)  

Rank

sage: E.rank()
 

The elliptic curves in class 68970.e have rank \(1\).

Complex multiplication

The elliptic curves in class 68970.e do not have complex multiplication.

Modular form 68970.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} - q^{12} - 2q^{13} + q^{15} + q^{16} - 2q^{17} - q^{18} + q^{19} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.