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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 12100g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12100.i4 | 12100g1 | \([0, -1, 0, -137133, -19099738]\) | \(643956736/15125\) | \(6698715031250000\) | \([2]\) | \(103680\) | \(1.8222\) | \(\Gamma_0(N)\)-optimal |
12100.i3 | 12100g2 | \([0, -1, 0, -303508, 36469512]\) | \(436334416/171875\) | \(1217948187500000000\) | \([2]\) | \(207360\) | \(2.1688\) | |
12100.i2 | 12100g3 | \([0, -1, 0, -1347133, 594672762]\) | \(610462990336/8857805\) | \(3923035470901250000\) | \([2]\) | \(311040\) | \(2.3715\) | |
12100.i1 | 12100g4 | \([0, -1, 0, -21478508, 38320869512]\) | \(154639330142416/33275\) | \(235794769100000000\) | \([2]\) | \(622080\) | \(2.7181\) |
Rank
sage: E.rank()
The elliptic curves in class 12100g have rank \(1\).
Complex multiplication
The elliptic curves in class 12100g do not have complex multiplication.Modular form 12100.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.