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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 5070v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5070.w8 | 5070v1 | \([1, 0, 0, 250, 4692]\) | \(357911/2160\) | \(-10425907440\) | \([2]\) | \(4608\) | \(0.60419\) | \(\Gamma_0(N)\)-optimal |
5070.w6 | 5070v2 | \([1, 0, 0, -3130, 60800]\) | \(702595369/72900\) | \(351874376100\) | \([2, 2]\) | \(9216\) | \(0.95076\) | |
5070.w7 | 5070v3 | \([1, 0, 0, -2285, -137775]\) | \(-273359449/1536000\) | \(-7413978624000\) | \([2]\) | \(13824\) | \(1.1535\) | |
5070.w5 | 5070v4 | \([1, 0, 0, -11580, -414090]\) | \(35578826569/5314410\) | \(25651642017690\) | \([2]\) | \(18432\) | \(1.2973\) | |
5070.w4 | 5070v5 | \([1, 0, 0, -48760, 4140122]\) | \(2656166199049/33750\) | \(162904803750\) | \([2]\) | \(18432\) | \(1.2973\) | |
5070.w3 | 5070v6 | \([1, 0, 0, -56365, -5145583]\) | \(4102915888729/9000000\) | \(43441281000000\) | \([2, 2]\) | \(27648\) | \(1.5001\) | |
5070.w1 | 5070v7 | \([1, 0, 0, -901365, -329456583]\) | \(16778985534208729/81000\) | \(390971529000\) | \([2]\) | \(55296\) | \(1.8466\) | |
5070.w2 | 5070v8 | \([1, 0, 0, -76645, -1117975]\) | \(10316097499609/5859375000\) | \(28282083984375000\) | \([2]\) | \(55296\) | \(1.8466\) |
Rank
sage: E.rank()
The elliptic curves in class 5070v have rank \(0\).
Complex multiplication
The elliptic curves in class 5070v do not have complex multiplication.Modular form 5070.2.a.v
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.