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Properties

Label 12495.i
Number of curves $2$
Conductor $12495$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 12495.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12495.i1 12495c2 \([1, 1, 0, -61295938, 163572097417]\) \(216486375407331255135001/27004994294227023375\) \(3177110573721515073045375\) \([2]\) \(1935360\) \(3.4312\)  
12495.i2 12495c1 \([1, 1, 0, 5680937, 13235803792]\) \(172343644217341694999/742780064187984375\) \(-87387331771652173734375\) \([2]\) \(967680\) \(3.0846\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 12495.i have rank \(1\).

Complex multiplication

The elliptic curves in class 12495.i do not have complex multiplication.

Modular form 12495.2.a.i

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.