Properties

Label 30345.t
Number of curves $4$
Conductor $30345$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("t1")
 
E.isogeny_class()
 

Elliptic curves in class 30345.t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
30345.t1 30345b4 \([1, 1, 0, -32518, -2270453]\) \(157551496201/13125\) \(316805593125\) \([2]\) \(81920\) \(1.2506\)  
30345.t2 30345b2 \([1, 1, 0, -2173, -30992]\) \(47045881/11025\) \(266116698225\) \([2, 2]\) \(40960\) \(0.90407\)  
30345.t3 30345b1 \([1, 1, 0, -728, 6867]\) \(1771561/105\) \(2534444745\) \([2]\) \(20480\) \(0.55750\) \(\Gamma_0(N)\)-optimal
30345.t4 30345b3 \([1, 1, 0, 5052, -185607]\) \(590589719/972405\) \(-23471492783445\) \([2]\) \(81920\) \(1.2506\)  

Rank

sage: E.rank()
 

The elliptic curves in class 30345.t have rank \(1\).

Complex multiplication

The elliptic curves in class 30345.t do not have complex multiplication.

Modular form 30345.2.a.t

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - q^{7} - 3 q^{8} + q^{9} - q^{10} + q^{12} - 6 q^{13} - q^{14} + q^{15} - q^{16} + q^{18} - 8 q^{19} + O(q^{20})\) 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.