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Properties

Label 12615f
Number of curves $8$
Conductor $12615$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 12615f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12615.f7 12615f1 \([1, 0, 1, -18, 4543]\) \(-1/15\) \(-8922349815\) \([2]\) \(6272\) \(0.58822\) \(\Gamma_0(N)\)-optimal
12615.f6 12615f2 \([1, 0, 1, -4223, 103781]\) \(13997521/225\) \(133835247225\) \([2, 2]\) \(12544\) \(0.93480\)  
12615.f5 12615f3 \([1, 0, 1, -8428, -138427]\) \(111284641/50625\) \(30112930625625\) \([2, 2]\) \(25088\) \(1.2814\)  
12615.f4 12615f4 \([1, 0, 1, -67298, 6714041]\) \(56667352321/15\) \(8922349815\) \([2]\) \(25088\) \(1.2814\)  
12615.f2 12615f5 \([1, 0, 1, -113553, -14729777]\) \(272223782641/164025\) \(97565895227025\) \([2, 2]\) \(50176\) \(1.6279\)  
12615.f8 12615f6 \([1, 0, 1, 29417, -1031569]\) \(4733169839/3515625\) \(-2091175737890625\) \([2]\) \(50176\) \(1.6279\)  
12615.f1 12615f7 \([1, 0, 1, -1816578, -942537797]\) \(1114544804970241/405\) \(240903445005\) \([2]\) \(100352\) \(1.9745\)  
12615.f3 12615f8 \([1, 0, 1, -92528, -20347657]\) \(-147281603041/215233605\) \(-128025967716902205\) \([2]\) \(100352\) \(1.9745\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12615f have rank \(1\).

Complex multiplication

The elliptic curves in class 12615f do not have complex multiplication.

Modular form 12615.2.a.f

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.