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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 2070i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2070.h4 | 2070i1 | \([1, -1, 0, -40419, 5567125]\) | \(-10017490085065009/12502381363200\) | \(-9114236013772800\) | \([2]\) | \(14336\) | \(1.7558\) | \(\Gamma_0(N)\)-optimal |
2070.h3 | 2070i2 | \([1, -1, 0, -777699, 264057493]\) | \(71356102305927901489/35540674560000\) | \(25909151754240000\) | \([2, 2]\) | \(28672\) | \(2.1024\) | |
2070.h2 | 2070i3 | \([1, -1, 0, -910179, 168062485]\) | \(114387056741228939569/49503729150000000\) | \(36088218550350000000\) | \([2]\) | \(57344\) | \(2.4489\) | |
2070.h1 | 2070i4 | \([1, -1, 0, -12441699, 16894588693]\) | \(292169767125103365085489/72534787200\) | \(52877859868800\) | \([2]\) | \(57344\) | \(2.4489\) |
Rank
sage: E.rank()
The elliptic curves in class 2070i have rank \(0\).
Complex multiplication
The elliptic curves in class 2070i do not have complex multiplication.Modular form 2070.2.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.