Properties

Label 162624he
Number of curves $4$
Conductor $162624$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("he1")
 
E.isogeny_class()
 

Elliptic curves in class 162624he

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162624.bd4 162624he1 \([0, -1, 0, -751329, -916386975]\) \(-100999381393/723148272\) \(-335833012867587637248\) \([2]\) \(4423680\) \(2.6217\) \(\Gamma_0(N)\)-optimal
162624.bd3 162624he2 \([0, -1, 0, -19491809, -33041317791]\) \(1763535241378513/4612311396\) \(2141976261822816190464\) \([2, 2]\) \(8847360\) \(2.9683\)  
162624.bd2 162624he3 \([0, -1, 0, -27158369, -4624446495]\) \(4770223741048753/2740574865798\) \(1272734167814103545413632\) \([4]\) \(17694720\) \(3.3149\)  
162624.bd1 162624he4 \([0, -1, 0, -311672929, -2117753608991]\) \(7209828390823479793/49509306\) \(22992324040239611904\) \([2]\) \(17694720\) \(3.3149\)  

Rank

sage: E.rank()
 

The elliptic curves in class 162624he have rank \(1\).

Complex multiplication

The elliptic curves in class 162624he do not have complex multiplication.

Modular form 162624.2.a.he

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{7} + q^{9} - 2 q^{13} + 2 q^{15} + 2 q^{17} + O(q^{20})\) 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.