Properties

Label 405d
Number of curves $2$
Conductor $405$
CM no
Rank $1$
Graph

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

Elliptic curves in class 405d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405.b2 405d1 \([1, -1, 1, -2, -26]\) \(-9/5\) \(-295245\) \([]\) \(36\) \(-0.27128\) \(\Gamma_0(N)\)-optimal
405.b1 405d2 \([1, -1, 1, -2027, 35776]\) \(-15590912409/78125\) \(-4613203125\) \([]\) \(252\) \(0.70167\)  

Rank

sage: E.rank()
 

The elliptic curves in class 405d have rank \(1\).

Complex multiplication

The elliptic curves in class 405d do not have complex multiplication.

Modular form 405.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + q^{5} - 3 q^{7} + 3 q^{8} - q^{10} + 2 q^{11} - 2 q^{13} + 3 q^{14} - q^{16} - 4 q^{17} - 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.