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Properties

Label 350727.d
Number of curves $2$
Conductor $350727$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 350727.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
350727.d1 350727d2 \([1, 1, 1, -1113027, 451383906]\) \(84662348471/25857\) \(46572404367448791\) \([2]\) \(4875264\) \(2.1754\)  
350727.d2 350727d1 \([1, 1, 1, -78832, 5025344]\) \(30080231/11271\) \(20300791647349473\) \([2]\) \(2437632\) \(1.8288\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 350727.d have rank \(1\).

Complex multiplication

The elliptic curves in class 350727.d do not have complex multiplication.

Modular form 350727.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + 2 q^{5} + q^{6} + 3 q^{8} + q^{9} - 2 q^{10} + 2 q^{11} + q^{12} + q^{13} - 2 q^{15} - q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.