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SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 43120cv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43120.q3 | 43120cv1 | \([0, 1, 0, -43920, 68929300]\) | \(-19443408769/4249907200\) | \(-2047989072579788800\) | \([2]\) | \(663552\) | \(2.1929\) | \(\Gamma_0(N)\)-optimal |
43120.q2 | 43120cv2 | \([0, 1, 0, -2803600, 1789865748]\) | \(5057359576472449/51765560000\) | \(24945321445130240000\) | \([2]\) | \(1327104\) | \(2.5395\) | |
43120.q4 | 43120cv3 | \([0, 1, 0, 395120, -1857051372]\) | \(14156681599871/3100231750000\) | \(-1493971620477952000000\) | \([2]\) | \(1990656\) | \(2.7422\) | |
43120.q1 | 43120cv4 | \([0, 1, 0, -20474960, -34656469100]\) | \(1969902499564819009/63690429687500\) | \(30691800524000000000000\) | \([2]\) | \(3981312\) | \(3.0888\) |
Rank
sage: E.rank()
The elliptic curves in class 43120cv have rank \(1\).
Complex multiplication
The elliptic curves in class 43120cv do not have complex multiplication.Modular form 43120.2.a.cv
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.