Properties

Label 323400.fu
Number of curves $6$
Conductor $323400$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("fu1")
 
E.isogeny_class()
 

Elliptic curves in class 323400.fu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
323400.fu1 323400fu5 \([0, 1, 0, -2063978408, 36089868272688]\) \(258286045443018193442/8440380939375\) \(31776076068368940000000000\) \([2]\) \(150994944\) \(3.9873\)  
323400.fu2 323400fu4 \([0, 1, 0, -583100408, -5419691553312]\) \(11647843478225136004/128410942275\) \(241718703163383600000000\) \([2]\) \(75497472\) \(3.6407\)  
323400.fu3 323400fu3 \([0, 1, 0, -134603408, 512193272688]\) \(143279368983686884/22699269140625\) \(42728741042006250000000000\) \([2, 2]\) \(75497472\) \(3.6407\)  
323400.fu4 323400fu2 \([0, 1, 0, -37362908, -80195853312]\) \(12257375872392016/1191317675625\) \(560629332878422500000000\) \([2, 2]\) \(37748736\) \(3.2942\)  
323400.fu5 323400fu1 \([0, 1, 0, 2823217, -6012266562]\) \(84611246065664/580054565475\) \(-17060709893392068750000\) \([2]\) \(18874368\) \(2.9476\) \(\Gamma_0(N)\)-optimal
323400.fu6 323400fu6 \([0, 1, 0, 238923592, 2848978184688]\) \(400647648358480318/1163177490234375\) \(-4379093393554687500000000000\) \([2]\) \(150994944\) \(3.9873\)  

Rank

sage: E.rank()
 

The elliptic curves in class 323400.fu have rank \(1\).

Complex multiplication

The elliptic curves in class 323400.fu do not have complex multiplication.

Modular form 323400.2.a.fu

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{9} - q^{11} - 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.