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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 7350i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7350.f5 | 7350i1 | \([1, 1, 0, -4925, -295875]\) | \(-7189057/16128\) | \(-29647548000000\) | \([2]\) | \(24576\) | \(1.2736\) | \(\Gamma_0(N)\)-optimal |
7350.f4 | 7350i2 | \([1, 1, 0, -102925, -12741875]\) | \(65597103937/63504\) | \(116737220250000\) | \([2, 2]\) | \(49152\) | \(1.6202\) | |
7350.f1 | 7350i3 | \([1, 1, 0, -1646425, -813818375]\) | \(268498407453697/252\) | \(463242937500\) | \([2]\) | \(98304\) | \(1.9668\) | |
7350.f3 | 7350i4 | \([1, 1, 0, -127425, -6249375]\) | \(124475734657/63011844\) | \(115832506793062500\) | \([2, 2]\) | \(98304\) | \(1.9668\) | |
7350.f2 | 7350i5 | \([1, 1, 0, -1119675, 451177875]\) | \(84448510979617/933897762\) | \(1716752153149031250\) | \([2]\) | \(196608\) | \(2.3134\) | |
7350.f6 | 7350i6 | \([1, 1, 0, 472825, -47666625]\) | \(6359387729183/4218578658\) | \(-7754868133360031250\) | \([2]\) | \(196608\) | \(2.3134\) |
Rank
sage: E.rank()
The elliptic curves in class 7350i have rank \(0\).
Complex multiplication
The elliptic curves in class 7350i do not have complex multiplication.Modular form 7350.2.a.i
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.