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SageMath
E = EllipticCurve("lv1")
E.isogeny_class()
Elliptic curves in class 428400lv
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
428400.lv4 | 428400lv1 | \([0, 0, 0, 50325, 7308250]\) | \(302111711/669375\) | \(-31230360000000000\) | \([2]\) | \(3145728\) | \(1.8492\) | \(\Gamma_0(N)\)-optimal* |
428400.lv3 | 428400lv2 | \([0, 0, 0, -399675, 79758250]\) | \(151334226289/28676025\) | \(1337908622400000000\) | \([2, 2]\) | \(6291456\) | \(2.1957\) | \(\Gamma_0(N)\)-optimal* |
428400.lv1 | 428400lv3 | \([0, 0, 0, -6069675, 5755428250]\) | \(530044731605089/26309115\) | \(1227478069440000000\) | \([2]\) | \(12582912\) | \(2.5423\) | \(\Gamma_0(N)\)-optimal* |
428400.lv2 | 428400lv4 | \([0, 0, 0, -1929675, -959111750]\) | \(17032120495489/1339001685\) | \(62472462615360000000\) | \([2]\) | \(12582912\) | \(2.5423\) |
Rank
sage: E.rank()
The elliptic curves in class 428400lv have rank \(0\).
Complex multiplication
The elliptic curves in class 428400lv do not have complex multiplication.Modular form 428400.2.a.lv
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.