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Properties

Label 1078f
Number of curves $2$
Conductor $1078$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 1078f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1078.b2 1078f1 \([1, 0, 1, -712, 7494]\) \(-338608873/13552\) \(-1594379248\) \([2]\) \(768\) \(0.53407\) \(\Gamma_0(N)\)-optimal
1078.b1 1078f2 \([1, 0, 1, -11492, 473190]\) \(1426487591593/2156\) \(253651244\) \([2]\) \(1536\) \(0.88064\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1078f have rank \(1\).

Complex multiplication

The elliptic curves in class 1078f do not have complex multiplication.

Modular form 1078.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{2} - 2 q^{3} + q^{4} - 2 q^{5} + 2 q^{6} - q^{8} + q^{9} + 2 q^{10} + q^{11} - 2 q^{12} + 4 q^{13} + 4 q^{15} + q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) 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.