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Properties

Label 487305dc
Number of curves $4$
Conductor $487305$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dc1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 487305dc have rank \(1\).

Complex multiplication

The elliptic curves in class 487305dc do not have complex multiplication.

Modular form 487305.2.a.dc

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{4} - q^{5} - 3 q^{8} - q^{10} + 4 q^{11} - q^{13} - q^{16} - q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 487305dc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
487305.dc3 487305dc1 \([1, -1, 0, -11475, 456840]\) \(1948441249/89505\) \(7676496660105\) \([2]\) \(1032192\) \(1.2341\) \(\Gamma_0(N)\)-optimal
487305.dc2 487305dc2 \([1, -1, 0, -31320, -1531629]\) \(39616946929/10989225\) \(942503201046225\) \([2, 2]\) \(2064384\) \(1.5806\)  
487305.dc4 487305dc3 \([1, -1, 0, 81135, -10100700]\) \(688699320191/910381875\) \(-78079922047456875\) \([2]\) \(4128768\) \(1.9272\)  
487305.dc1 487305dc4 \([1, -1, 0, -461295, -120462714]\) \(126574061279329/16286595\) \(1396838077447995\) \([2]\) \(4128768\) \(1.9272\)