Properties

Label 2070.b
Number of curves $6$
Conductor $2070$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 2070.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2070.b1 2070f3 \([1, -1, 0, -993600, -380962764]\) \(148809678420065817601/20700\) \(15090300\) \([2]\) \(12288\) \(1.7023\)  
2070.b2 2070f5 \([1, -1, 0, -232470, 37012950]\) \(1905890658841300321/293666194803750\) \(214082656011933750\) \([2]\) \(24576\) \(2.0489\)  
2070.b3 2070f4 \([1, -1, 0, -63720, -5613300]\) \(39248884582600321/3935264062500\) \(2868807501562500\) \([2, 2]\) \(12288\) \(1.7023\)  
2070.b4 2070f2 \([1, -1, 0, -62100, -5940864]\) \(36330796409313601/428490000\) \(312369210000\) \([2, 2]\) \(6144\) \(1.3557\)  
2070.b5 2070f1 \([1, -1, 0, -3780, -97200]\) \(-8194759433281/965779200\) \(-704053036800\) \([2]\) \(3072\) \(1.0092\) \(\Gamma_0(N)\)-optimal
2070.b6 2070f6 \([1, -1, 0, 79110, -27294894]\) \(75108181893694559/484313964843750\) \(-353064880371093750\) \([2]\) \(24576\) \(2.0489\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2070.b have rank \(1\).

Complex multiplication

The elliptic curves in class 2070.b do not have complex multiplication.

Modular form 2070.2.a.b

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.