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SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 203280.dl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
203280.dl1 | 203280gc6 | \([0, -1, 0, -203870520, -1120317591600]\) | \(258286045443018193442/8440380939375\) | \(30623026580152554240000\) | \([2]\) | \(31457280\) | \(3.4086\) | |
203280.dl2 | 203280gc3 | \([0, -1, 0, -57596040, 168260615712]\) | \(11647843478225136004/128410942275\) | \(232947524923024665600\) | \([4]\) | \(15728640\) | \(3.0620\) | |
203280.dl3 | 203280gc4 | \([0, -1, 0, -13295520, -15897351600]\) | \(143279368983686884/22699269140625\) | \(41178255296547600000000\) | \([2, 2]\) | \(15728640\) | \(3.0620\) | |
203280.dl4 | 203280gc2 | \([0, -1, 0, -3690540, 2490422112]\) | \(12257375872392016/1191317675625\) | \(540285934783462560000\) | \([2, 2]\) | \(7864320\) | \(2.7154\) | |
203280.dl5 | 203280gc1 | \([0, -1, 0, 278865, 186579450]\) | \(84611246065664/580054565475\) | \(-16441632737079303600\) | \([2]\) | \(3932160\) | \(2.3689\) | \(\Gamma_0(N)\)-optimal |
203280.dl6 | 203280gc5 | \([0, -1, 0, 23599800, -88448308848]\) | \(400647648358480318/1163177490234375\) | \(-4220190469687500000000000\) | \([2]\) | \(31457280\) | \(3.4086\) |
Rank
sage: E.rank()
The elliptic curves in class 203280.dl have rank \(0\).
Complex multiplication
The elliptic curves in class 203280.dl do not have complex multiplication.Modular form 203280.2.a.dl
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.