Properties

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

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Show commands: SageMath
E = EllipticCurve("qf1")
 
E.isogeny_class()
 

Elliptic curves in class 369600qf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
369600.qf6 369600qf1 \([0, 1, 0, -458453633, 3778101904863]\) \(2601656892010848045529/56330588160\) \(230730089103360000000\) \([2]\) \(63700992\) \(3.4342\) \(\Gamma_0(N)\)-optimal
369600.qf5 369600qf2 \([0, 1, 0, -458965633, 3769239696863]\) \(2610383204210122997209/12104550027662400\) \(49580236913305190400000000\) \([2, 2]\) \(127401984\) \(3.7808\)  
369600.qf4 369600qf3 \([0, 1, 0, -489197633, 3242454232863]\) \(3160944030998056790089/720291785342976000\) \(2950315152764829696000000000\) \([2]\) \(191102976\) \(3.9835\)  
369600.qf7 369600qf4 \([0, 1, 0, -225685633, 7598064336863]\) \(-310366976336070130009/5909282337130963560\) \(-24204420452888426741760000000\) \([2]\) \(254803968\) \(4.1274\)  
369600.qf3 369600qf5 \([0, 1, 0, -700437633, -626758063137]\) \(9278380528613437145689/5328033205714065000\) \(21823624010604810240000000000\) \([2]\) \(254803968\) \(4.1274\)  
369600.qf2 369600qf6 \([0, 1, 0, -2586349633, -47854654247137]\) \(467116778179943012100169/28800309694464000000\) \(117966068508524544000000000000\) \([2, 2]\) \(382205952\) \(4.3301\)  
369600.qf8 369600qf7 \([0, 1, 0, 2021650367, -199821886247137]\) \(223090928422700449019831/4340371122724101696000\) \(-17778160118677920546816000000000\) \([2]\) \(764411904\) \(4.6767\)  
369600.qf1 369600qf8 \([0, 1, 0, -40748781633, -3166068810535137]\) \(1826870018430810435423307849/7641104625000000000\) \(31297964544000000000000000000\) \([2]\) \(764411904\) \(4.6767\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 369600.2.a.qf

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.