Properties

Label 369600.pv
Number of curves $8$
Conductor $369600$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("pv1")
 
E.isogeny_class()
 

Elliptic curves in class 369600.pv

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
369600.pv1 369600pv7 \([0, 1, 0, -410284033, 3001022612063]\) \(1864737106103260904761/129177711985836360\) \(529111908293985730560000000\) \([2]\) \(127401984\) \(3.8757\)  
369600.pv2 369600pv4 \([0, 1, 0, -403204033, 3116138012063]\) \(1769857772964702379561/691787250\) \(2833560576000000000\) \([2]\) \(42467328\) \(3.3264\)  
369600.pv3 369600pv6 \([0, 1, 0, -81004033, -224274987937]\) \(14351050585434661561/3001282273281600\) \(12293252191361433600000000\) \([2, 2]\) \(63700992\) \(3.5291\)  
369600.pv4 369600pv3 \([0, 1, 0, -76396033, -257024043937]\) \(12038605770121350841/757333463040\) \(3102037864611840000000\) \([2]\) \(31850496\) \(3.1826\)  
369600.pv5 369600pv2 \([0, 1, 0, -25204033, 48668012063]\) \(432288716775559561/270140062500\) \(1106493696000000000000\) \([2, 2]\) \(21233664\) \(2.9798\)  
369600.pv6 369600pv5 \([0, 1, 0, -20452033, 67585724063]\) \(-230979395175477481/348191894531250\) \(-1426194000000000000000000\) \([2]\) \(42467328\) \(3.3264\)  
369600.pv7 369600pv1 \([0, 1, 0, -1876033, 449036063]\) \(178272935636041/81841914000\) \(335224479744000000000\) \([2]\) \(10616832\) \(2.6333\) \(\Gamma_0(N)\)-optimal
369600.pv8 369600pv8 \([0, 1, 0, 174547967, -1353559275937]\) \(143584693754978072519/276341298967965000\) \(-1131893960572784640000000000\) \([2]\) \(127401984\) \(3.8757\)  

Rank

sage: E.rank()
 

The elliptic curves in class 369600.pv have rank \(1\).

Complex multiplication

The elliptic curves in class 369600.pv do not have complex multiplication.

Modular form 369600.2.a.pv

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{7} + q^{9} + q^{11} + 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.