Properties

Label 378.e
Number of curves $3$
Conductor $378$
CM no
Rank $0$
Graph

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

Elliptic curves in class 378.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
378.e1 378a3 \([1, -1, 1, -9560, -357371]\) \(-545407363875/14\) \(-2480058\) \([]\) \(324\) \(0.74319\)  
378.e2 378a2 \([1, -1, 1, -110, -539]\) \(-7414875/2744\) \(-54010152\) \([3]\) \(108\) \(0.19388\)  
378.e3 378a1 \([1, -1, 1, 10, 5]\) \(4492125/3584\) \(-96768\) \([3]\) \(36\) \(-0.35542\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 378.e have rank \(0\).

Complex multiplication

The elliptic curves in class 378.e do not have complex multiplication.

Modular form 378.2.a.e

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.