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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 290145i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
290145.i4 | 290145i1 | \([1, 0, 0, 35725, 4837560]\) | \(8477185319/21880935\) | \(-13015290423285135\) | \([4]\) | \(1612800\) | \(1.7758\) | \(\Gamma_0(N)\)-optimal |
290145.i3 | 290145i2 | \([1, 0, 0, -304880, 54361527]\) | \(5268932332201/900900225\) | \(535876463724147225\) | \([2, 2]\) | \(3225600\) | \(2.1224\) | |
290145.i1 | 290145i3 | \([1, 0, 0, -4657055, 3867737262]\) | \(18778886261717401/732035835\) | \(435431986465708035\) | \([2]\) | \(6451200\) | \(2.4690\) | |
290145.i2 | 290145i4 | \([1, 0, 0, -1402385, -588117900]\) | \(512787603508921/45649063125\) | \(27153127328551138125\) | \([2]\) | \(6451200\) | \(2.4690\) |
Rank
sage: E.rank()
The elliptic curves in class 290145i have rank \(1\).
Complex multiplication
The elliptic curves in class 290145i do not have complex multiplication.Modular form 290145.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.