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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 10830.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10830.e1 | 10830e4 | \([1, 1, 0, -167226398, -531563735292]\) | \(10993009831928446009969/3767761230468750000\) | \(177257646485046386718750000\) | \([2]\) | \(6220800\) | \(3.7386\) | |
10830.e2 | 10830e2 | \([1, 1, 0, -149811758, -705838483788]\) | \(7903870428425797297009/886464000000\) | \(41704479854784000000\) | \([2]\) | \(2073600\) | \(3.1893\) | |
10830.e3 | 10830e1 | \([1, 1, 0, -9339438, -11090483532]\) | \(-1914980734749238129/20440940544000\) | \(-961662056361099264000\) | \([2]\) | \(1036800\) | \(2.8427\) | \(\Gamma_0(N)\)-optimal |
10830.e4 | 10830e3 | \([1, 1, 0, 30861522, -57697813068]\) | \(69096190760262356111/70568821500000000\) | \(-3319972378599241500000000\) | \([2]\) | \(3110400\) | \(3.3920\) |
Rank
sage: E.rank()
The elliptic curves in class 10830.e have rank \(0\).
Complex multiplication
The elliptic curves in class 10830.e do not have complex multiplication.Modular form 10830.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.