Properties

Label 346560ef
Number of curves $4$
Conductor $346560$
CM no
Rank $1$
Graph

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

Elliptic curves in class 346560ef

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
346560.ef3 346560ef1 \([0, -1, 0, -716705, -231902175]\) \(3301293169/22800\) \(281187735778099200\) \([2]\) \(4423680\) \(2.1817\) \(\Gamma_0(N)\)-optimal
346560.ef2 346560ef2 \([0, -1, 0, -1178785, 104214817]\) \(14688124849/8122500\) \(100173130870947840000\) \([2, 2]\) \(8847360\) \(2.5283\)  
346560.ef1 346560ef3 \([0, -1, 0, -14348065, 20893240225]\) \(26487576322129/44531250\) \(549194796441600000000\) \([2]\) \(17694720\) \(2.8749\)  
346560.ef4 346560ef4 \([0, -1, 0, 4597215, 819283617]\) \(871257511151/527800050\) \(-6509250043994190643200\) \([2]\) \(17694720\) \(2.8749\)  

Rank

sage: E.rank()
 

The elliptic curves in class 346560ef have rank \(1\).

Complex multiplication

The elliptic curves in class 346560ef do not have complex multiplication.

Modular form 346560.2.a.ef

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{9} - 4q^{11} + 2q^{13} - q^{15} + 2q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.