Properties

Label 311696.i
Number of curves $2$
Conductor $311696$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 311696.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
311696.i1 311696i2 \([0, 1, 0, -1172288, 456981556]\) \(24553362849625/1755162752\) \(12736011796872691712\) \([2]\) \(6881280\) \(2.4128\)  
311696.i2 311696i1 \([0, 1, 0, 66752, 31247412]\) \(4533086375/60669952\) \(-440240213340454912\) \([2]\) \(3440640\) \(2.0663\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 311696.i have rank \(1\).

Complex multiplication

The elliptic curves in class 311696.i do not have complex multiplication.

Modular form 311696.2.a.i

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