Properties

Label 30345.f
Number of curves $4$
Conductor $30345$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 30345.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
30345.f1 30345m4 \([1, 1, 1, -487260, 130706322]\) \(530044731605089/26309115\) \(635038078641435\) \([2]\) \(294912\) \(1.9118\)  
30345.f2 30345m3 \([1, 1, 1, -154910, -21879898]\) \(17032120495489/1339001685\) \(32320245562803765\) \([2]\) \(294912\) \(1.9118\)  
30345.f3 30345m2 \([1, 1, 1, -32085, 1800762]\) \(151334226289/28676025\) \(692169532083225\) \([2, 2]\) \(147456\) \(1.5652\)  
30345.f4 30345m1 \([1, 1, 1, 4040, 167912]\) \(302111711/669375\) \(-16157085249375\) \([4]\) \(73728\) \(1.2186\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 30345.f have rank \(0\).

Complex multiplication

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

Modular form 30345.2.a.f

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