Properties

Label 412224.bb
Number of curves $4$
Conductor $412224$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bb1")
 
E.isogeny_class()
 

Elliptic curves in class 412224.bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
412224.bb1 412224bb3 \([0, -1, 0, -274817, 55543233]\) \(70049677573479176/6441\) \(211058688\) \([2]\) \(1130496\) \(1.4789\) \(\Gamma_0(N)\)-optimal*
412224.bb2 412224bb4 \([0, -1, 0, -19457, 628065]\) \(24861294286856/9293699577\) \(304535947739136\) \([2]\) \(1130496\) \(1.4789\)  
412224.bb3 412224bb2 \([0, -1, 0, -17177, 872025]\) \(136845649240768/41486481\) \(169928626176\) \([2, 2]\) \(565248\) \(1.1324\) \(\Gamma_0(N)\)-optimal*
412224.bb4 412224bb1 \([0, -1, 0, -932, 17538]\) \(-1400416996672/1192828113\) \(-76340999232\) \([2]\) \(282624\) \(0.78579\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 412224.bb1.

Rank

sage: E.rank()
 

The elliptic curves in class 412224.bb have rank \(1\).

Complex multiplication

The elliptic curves in class 412224.bb do not have complex multiplication.

Modular form 412224.2.a.bb

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