Properties

Label 323400.cz
Number of curves $4$
Conductor $323400$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("cz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 323400.cz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
323400.cz1 323400cz4 \([0, -1, 0, -36476008, -84766747988]\) \(1425631925916578/270703125\) \(1019134462500000000000\) \([2]\) \(18874368\) \(3.0322\)  
323400.cz2 323400cz3 \([0, -1, 0, -15994008, 23846064012]\) \(120186986927618/4332064275\) \(16309216956463200000000\) \([2]\) \(18874368\) \(3.0322\)  
323400.cz3 323400cz2 \([0, -1, 0, -2519008, -1028785988]\) \(939083699236/300155625\) \(565008146010000000000\) \([2, 2]\) \(9437184\) \(2.6856\)  
323400.cz4 323400cz1 \([0, -1, 0, 445492, -109790988]\) \(20777545136/23059575\) \(-10851743756700000000\) \([2]\) \(4718592\) \(2.3391\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 323400.2.a.cz

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