Properties

Label 12870.w
Number of curves $4$
Conductor $12870$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 12870.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12870.w1 12870x4 \([1, -1, 0, -1443834, 668110788]\) \(456612868287073618849/12544848030000\) \(9145194213870000\) \([4]\) \(196608\) \(2.1660\)  
12870.w2 12870x3 \([1, -1, 0, -402714, -88828380]\) \(9908022260084596129/1047363281250000\) \(763527832031250000\) \([2]\) \(196608\) \(2.1660\)  
12870.w3 12870x2 \([1, -1, 0, -93834, 9580788]\) \(125337052492018849/18404100000000\) \(13416588900000000\) \([2, 2]\) \(98304\) \(1.8195\)  
12870.w4 12870x1 \([1, -1, 0, 9846, 809460]\) \(144794100308831/474439680000\) \(-345866526720000\) \([2]\) \(49152\) \(1.4729\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 12870.w have rank \(1\).

Complex multiplication

The elliptic curves in class 12870.w do not have complex multiplication.

Modular form 12870.2.a.w

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