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SageMath
E = EllipticCurve("ge1")
E.isogeny_class()
Elliptic curves in class 187200ge
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.pp4 | 187200ge1 | \([0, 0, 0, 2934825, -1376507000]\) | \(3834800837445824/3342041015625\) | \(-2436347900390625000000\) | \([2]\) | \(9437184\) | \(2.7912\) | \(\Gamma_0(N)\)-optimal |
187200.pp3 | 187200ge2 | \([0, 0, 0, -14643300, -12204632000]\) | \(7442744143086784/2927948765625\) | \(136606377609000000000000\) | \([2, 2]\) | \(18874368\) | \(3.1378\) | |
187200.pp2 | 187200ge3 | \([0, 0, 0, -105768300, 410068618000]\) | \(350584567631475848/8259273550125\) | \(3082757334037056000000000\) | \([2]\) | \(37748736\) | \(3.4844\) | |
187200.pp1 | 187200ge4 | \([0, 0, 0, -204768300, -1127477882000]\) | \(2543984126301795848/909361981125\) | \(339417540730944000000000\) | \([2]\) | \(37748736\) | \(3.4844\) |
Rank
sage: E.rank()
The elliptic curves in class 187200ge have rank \(0\).
Complex multiplication
The elliptic curves in class 187200ge do not have complex multiplication.Modular form 187200.2.a.ge
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.