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Properties

Label 333270.d
Number of curves $4$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 333270.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.d1 333270d4 \([1, -1, 0, -21381750, -38048993780]\) \(10017490085065009/235066440\) \(25367938406790101640\) \([2]\) \(25952256\) \(2.8353\)  
333270.d2 333270d3 \([1, -1, 0, -5765670, 4777229596]\) \(196416765680689/22365315000\) \(2413623711527935515000\) \([2]\) \(25952256\) \(2.8353\)  
333270.d3 333270d2 \([1, -1, 0, -1385550, -548120300]\) \(2725812332209/373262400\) \(40281792555205454400\) \([2, 2]\) \(12976128\) \(2.4888\)  
333270.d4 333270d1 \([1, -1, 0, 137970, -45663404]\) \(2691419471/9891840\) \(-1067509202291159040\) \([2]\) \(6488064\) \(2.1422\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 333270.d have rank \(1\).

Complex multiplication

The elliptic curves in class 333270.d do not have complex multiplication.

Modular form 333270.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - q^{7} - q^{8} + q^{10} - 4 q^{11} - 2 q^{13} + q^{14} + q^{16} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.