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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 25410.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25410.w1 | 25410v8 | \([1, 0, 1, -780574, -167063734]\) | \(29689921233686449/10380965400750\) | \(18390513446318070750\) | \([2]\) | \(829440\) | \(2.3980\) | |
25410.w2 | 25410v5 | \([1, 0, 1, -697084, -224072158]\) | \(21145699168383889/2593080\) | \(4593799397880\) | \([2]\) | \(276480\) | \(1.8487\) | |
25410.w3 | 25410v6 | \([1, 0, 1, -326824, 69975266]\) | \(2179252305146449/66177562500\) | \(117237588800062500\) | \([2, 2]\) | \(414720\) | \(2.0514\) | |
25410.w4 | 25410v3 | \([1, 0, 1, -324404, 71090402]\) | \(2131200347946769/2058000\) | \(3645872538000\) | \([2]\) | \(207360\) | \(1.7048\) | |
25410.w5 | 25410v2 | \([1, 0, 1, -43684, -3484318]\) | \(5203798902289/57153600\) | \(101251088769600\) | \([2, 2]\) | \(138240\) | \(1.5021\) | |
25410.w6 | 25410v4 | \([1, 0, 1, -9804, -8742494]\) | \(-58818484369/18600435000\) | \(-32951805229035000\) | \([2]\) | \(276480\) | \(1.8487\) | |
25410.w7 | 25410v1 | \([1, 0, 1, -4964, 46946]\) | \(7633736209/3870720\) | \(6857216593920\) | \([2]\) | \(69120\) | \(1.1555\) | \(\Gamma_0(N)\)-optimal |
25410.w8 | 25410v7 | \([1, 0, 1, 88206, 235655242]\) | \(42841933504271/13565917968750\) | \(-24032851202636718750\) | \([2]\) | \(829440\) | \(2.3980\) |
Rank
sage: E.rank()
The elliptic curves in class 25410.w have rank \(1\).
Complex multiplication
The elliptic curves in class 25410.w do not have complex multiplication.Modular form 25410.2.a.w
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.