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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 7605.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7605.g1 | 7605q7 | \([1, -1, 1, -3285392, 2292896454]\) | \(1114544804970241/405\) | \(1425091223205\) | \([2]\) | \(73728\) | \(2.1227\) | |
7605.g2 | 7605q5 | \([1, -1, 1, -205367, 35854134]\) | \(272223782641/164025\) | \(577161945398025\) | \([2, 2]\) | \(36864\) | \(1.7761\) | |
7605.g3 | 7605q8 | \([1, -1, 1, -167342, 49512714]\) | \(-147281603041/215233605\) | \(-757351904751288405\) | \([2]\) | \(73728\) | \(2.1227\) | |
7605.g4 | 7605q3 | \([1, -1, 1, -121712, -16313124]\) | \(56667352321/15\) | \(52781156415\) | \([2]\) | \(18432\) | \(1.4295\) | |
7605.g5 | 7605q4 | \([1, -1, 1, -15242, 338784]\) | \(111284641/50625\) | \(178136402900625\) | \([2, 2]\) | \(18432\) | \(1.4295\) | |
7605.g6 | 7605q2 | \([1, -1, 1, -7637, -251364]\) | \(13997521/225\) | \(791717346225\) | \([2, 2]\) | \(9216\) | \(1.0829\) | |
7605.g7 | 7605q1 | \([1, -1, 1, -32, -11046]\) | \(-1/15\) | \(-52781156415\) | \([2]\) | \(4608\) | \(0.73636\) | \(\Gamma_0(N)\)-optimal |
7605.g8 | 7605q6 | \([1, -1, 1, 53203, 2501646]\) | \(4733169839/3515625\) | \(-12370583534765625\) | \([2]\) | \(36864\) | \(1.7761\) |
Rank
sage: E.rank()
The elliptic curves in class 7605.g have rank \(1\).
Complex multiplication
The elliptic curves in class 7605.g do not have complex multiplication.Modular form 7605.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.