Properties

Label 13680c
Number of curves $2$
Conductor $13680$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 13680c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13680.d2 13680c1 \([0, 0, 0, -483, 4082]\) \(450714348/475\) \(13132800\) \([2]\) \(5120\) \(0.28284\) \(\Gamma_0(N)\)-optimal
13680.d1 13680c2 \([0, 0, 0, -603, 1898]\) \(438512454/225625\) \(12476160000\) \([2]\) \(10240\) \(0.62941\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13680c have rank \(2\).

Complex multiplication

The elliptic curves in class 13680c do not have complex multiplication.

Modular form 13680.2.a.c

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