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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 30345i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30345.o2 | 30345i1 | \([0, -1, 1, -40845, -4121539]\) | \(-312217698304/125355195\) | \(-3025769668820955\) | \([]\) | \(152064\) | \(1.6792\) | \(\Gamma_0(N)\)-optimal |
30345.o1 | 30345i2 | \([0, -1, 1, -3578205, -2604036922]\) | \(-209906535145406464/6559875\) | \(-158339435443875\) | \([]\) | \(456192\) | \(2.2285\) |
Rank
sage: E.rank()
The elliptic curves in class 30345i have rank \(0\).
Complex multiplication
The elliptic curves in class 30345i do not have complex multiplication.Modular form 30345.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.