Properties

Label 42978.c
Number of curves $4$
Conductor $42978$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 42978.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42978.c1 42978b4 \([1, 1, 0, -16059278499, -783322981022457]\) \(458038307459437803276572539343003833/86000447460798\) \(86000447460798\) \([2]\) \(40919040\) \(3.9675\)  
42978.c2 42978b2 \([1, 1, 0, -1003704909, -12239735151735]\) \(111825759760338976846738658338393/1532291201797601099556\) \(1532291201797601099556\) \([2, 2]\) \(20459520\) \(3.6209\)  
42978.c3 42978b3 \([1, 1, 0, -1002813239, -12262566362085]\) \(-111527993597885114164012178708473/413980765601504764798430334\) \(-413980765601504764798430334\) \([2]\) \(40919040\) \(3.9675\)  
42978.c4 42978b1 \([1, 1, 0, -62787289, -190908660587]\) \(27374041292637614212993237273/101051566314387812377488\) \(101051566314387812377488\) \([2]\) \(10229760\) \(3.2743\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 42978.c have rank \(1\).

Complex multiplication

The elliptic curves in class 42978.c do not have complex multiplication.

Modular form 42978.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} + q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - q^{19} + O(q^{20})\) Copy content 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.