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Properties

Label 456304bd
Number of curves $3$
Conductor $456304$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve([0, 1, 0, -268704, 53520884]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 456304bd have rank \(0\).

Complex multiplication

The elliptic curves in class 456304bd do not have complex multiplication.

Modular form 456304.2.a.bd

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 456304bd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
456304.bd3 456304bd1 \([0, 1, 0, -268704, 53520884]\) \(11134383337/316\) \(60893177430016\) \([]\) \(2280960\) \(1.7466\) \(\Gamma_0(N)\)-optimal
456304.bd2 456304bd2 \([0, 1, 0, -470864, -37451116]\) \(59914169497/31554496\) \(6080549125451677696\) \([]\) \(6842880\) \(2.2959\)  
456304.bd1 456304bd3 \([0, 1, 0, -30130624, -63669043916]\) \(15698803397448457/20709376\) \(3990695276053528576\) \([]\) \(20528640\) \(2.8452\)