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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 39270o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
39270.l4 | 39270o1 | \([1, 1, 0, 93013, -160290819]\) | \(88991263500189913799/11156306844639559680\) | \(-11156306844639559680\) | \([2]\) | \(1376256\) | \(2.3342\) | \(\Gamma_0(N)\)-optimal |
39270.l3 | 39270o2 | \([1, 1, 0, -3921067, -2895484931]\) | \(6667095482374842146270521/240543066599132774400\) | \(240543066599132774400\) | \([2, 2]\) | \(2752512\) | \(2.6808\) | |
39270.l2 | 39270o3 | \([1, 1, 0, -9883947, 8001081981]\) | \(106786168592175009377891641/34113595164994089360000\) | \(34113595164994089360000\) | \([2]\) | \(5505024\) | \(3.0274\) | |
39270.l1 | 39270o4 | \([1, 1, 0, -62183467, -188764193411]\) | \(26591847997770272732680824121/36570473891173918080\) | \(36570473891173918080\) | \([2]\) | \(5505024\) | \(3.0274\) |
Rank
sage: E.rank()
The elliptic curves in class 39270o have rank \(0\).
Complex multiplication
The elliptic curves in class 39270o do not have complex multiplication.Modular form 39270.2.a.o
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.