Properties

Label 3006e
Number of curves $2$
Conductor $3006$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 3006e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3006.d2 3006e1 \([1, -1, 1, -41, 425]\) \(-10218313/96192\) \(-70123968\) \([2]\) \(1152\) \(0.18788\) \(\Gamma_0(N)\)-optimal
3006.d1 3006e2 \([1, -1, 1, -1121, 14681]\) \(213525509833/669336\) \(487945944\) \([2]\) \(2304\) \(0.53445\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3006e have rank \(1\).

Complex multiplication

The elliptic curves in class 3006e do not have complex multiplication.

Modular form 3006.2.a.e

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.