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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 277725cc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
277725.cc3 | 277725cc1 | \([1, 1, 0, -3703275, -2736585000]\) | \(2428257525121/8150625\) | \(18852890902822265625\) | \([2]\) | \(6488064\) | \(2.5629\) | \(\Gamma_0(N)\)-optimal |
277725.cc2 | 277725cc2 | \([1, 1, 0, -5356400, -53563125]\) | \(7347774183121/4251692025\) | \(9834422010548206640625\) | \([2, 2]\) | \(12976128\) | \(2.9095\) | |
277725.cc1 | 277725cc3 | \([1, 1, 0, -58587025, 172041047500]\) | \(9614816895690721/34652610405\) | \(80153593554294141328125\) | \([2]\) | \(25952256\) | \(3.2561\) | |
277725.cc4 | 277725cc4 | \([1, 1, 0, 21424225, -401711250]\) | \(470166844956479/272118787605\) | \(-629427291198568060078125\) | \([2]\) | \(25952256\) | \(3.2561\) |
Rank
sage: E.rank()
The elliptic curves in class 277725cc have rank \(0\).
Complex multiplication
The elliptic curves in class 277725cc do not have complex multiplication.Modular form 277725.2.a.cc
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.