Properties

Label 312.d
Number of curves $2$
Conductor $312$
CM no
Rank $1$
Graph

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

Elliptic curves in class 312.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
312.d1 312f2 \([0, 1, 0, -60, 144]\) \(94875856/9477\) \(2426112\) \([2]\) \(96\) \(-0.038467\)  
312.d2 312f1 \([0, 1, 0, 5, 14]\) \(702464/4563\) \(-73008\) \([2]\) \(48\) \(-0.38504\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 312.d have rank \(1\).

Complex multiplication

The elliptic curves in class 312.d do not have complex multiplication.

Modular form 312.2.a.d

sage: E.q_eigenform(10)
 
\(q + q^{3} - 4q^{5} - 4q^{7} + q^{9} - 2q^{11} - q^{13} - 4q^{15} - 6q^{17} + 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.