Properties

Label 1305.b
Number of curves $4$
Conductor $1305$
CM no
Rank $0$
Graph

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

Elliptic curves in class 1305.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1305.b1 1305d3 \([1, -1, 1, -2318, 42896]\) \(1888690601881/31827645\) \(23202353205\) \([2]\) \(1536\) \(0.78696\)  
1305.b2 1305d2 \([1, -1, 1, -293, -844]\) \(3803721481/1703025\) \(1241505225\) \([2, 2]\) \(768\) \(0.44038\)  
1305.b3 1305d1 \([1, -1, 1, -248, -1438]\) \(2305199161/1305\) \(951345\) \([2]\) \(384\) \(0.093810\) \(\Gamma_0(N)\)-optimal
1305.b4 1305d4 \([1, -1, 1, 1012, -7108]\) \(157376536199/118918125\) \(-86691313125\) \([2]\) \(1536\) \(0.78696\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1305.b have rank \(0\).

Complex multiplication

The elliptic curves in class 1305.b do not have complex multiplication.

Modular form 1305.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - q^{5} + 4 q^{7} + 3 q^{8} + q^{10} + 4 q^{11} + 6 q^{13} - 4 q^{14} - q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.