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SageMath
E = EllipticCurve("i1")
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)
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.