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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 30030a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30030.a3 | 30030a1 | \([1, 1, 0, -37128, 2737728]\) | \(5660393911359932809/1087710624000\) | \(1087710624000\) | \([2]\) | \(98304\) | \(1.3101\) | \(\Gamma_0(N)\)-optimal |
30030.a2 | 30030a2 | \([1, 1, 0, -41048, 2119152]\) | \(7649204088039921289/2455152950250000\) | \(2455152950250000\) | \([2, 2]\) | \(196608\) | \(1.6567\) | |
30030.a4 | 30030a3 | \([1, 1, 0, 116452, 14624652]\) | \(174646038940465958711/192852576007591500\) | \(-192852576007591500\) | \([2]\) | \(393216\) | \(2.0033\) | |
30030.a1 | 30030a4 | \([1, 1, 0, -261268, -49896812]\) | \(1972354483749778615369/70549971679687500\) | \(70549971679687500\) | \([2]\) | \(393216\) | \(2.0033\) |
Rank
sage: E.rank()
The elliptic curves in class 30030a have rank \(1\).
Complex multiplication
The elliptic curves in class 30030a do not have complex multiplication.Modular form 30030.2.a.a
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.