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Properties

Label 162435l
Number of curves $4$
Conductor $162435$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("l1")
 
E.isogeny_class()
 

Elliptic curves in class 162435l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162435.ba3 162435l1 \([1, 0, 0, -1275, -16920]\) \(1948441249/89505\) \(10530173745\) \([2]\) \(129024\) \(0.68476\) \(\Gamma_0(N)\)-optimal
162435.ba2 162435l2 \([1, 0, 0, -3480, 56727]\) \(39616946929/10989225\) \(1292871332025\) \([2, 2]\) \(258048\) \(1.0313\)  
162435.ba1 162435l3 \([1, 0, 0, -51255, 4461582]\) \(126574061279329/16286595\) \(1916101615155\) \([2]\) \(516096\) \(1.3779\)  
162435.ba4 162435l4 \([1, 0, 0, 9015, 374100]\) \(688699320191/910381875\) \(-107105517211875\) \([2]\) \(516096\) \(1.3779\)  

Rank

sage: E.rank()
 

The elliptic curves in class 162435l have rank \(1\).

Complex multiplication

The elliptic curves in class 162435l do not have complex multiplication.

Modular form 162435.2.a.l

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.