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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 12100c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12100.c2 | 12100c1 | \([0, -1, 0, 11092, -3121688]\) | \(176/5\) | \(-4287177620000000\) | \([]\) | \(38016\) | \(1.6796\) | \(\Gamma_0(N)\)-optimal |
12100.c1 | 12100c2 | \([0, -1, 0, -1319908, -583437688]\) | \(-296587984/125\) | \(-107179440500000000\) | \([]\) | \(114048\) | \(2.2289\) |
Rank
sage: E.rank()
The elliptic curves in class 12100c have rank \(1\).
Complex multiplication
The elliptic curves in class 12100c do not have complex multiplication.Modular form 12100.2.a.c
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.