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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 27720.u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27720.u1 | 27720k6 | \([0, 0, 0, -15163923, 22727580622]\) | \(258286045443018193442/8440380939375\) | \(12601421219439360000\) | \([2]\) | \(1048576\) | \(2.7590\) | |
| 27720.u2 | 27720k4 | \([0, 0, 0, -4284003, -3412861202]\) | \(11647843478225136004/128410942275\) | \(95858254764518400\) | \([2]\) | \(524288\) | \(2.4124\) | |
| 27720.u3 | 27720k3 | \([0, 0, 0, -988923, 322575622]\) | \(143279368983686884/22699269140625\) | \(16944913616400000000\) | \([2, 2]\) | \(524288\) | \(2.4124\) | |
| 27720.u4 | 27720k2 | \([0, 0, 0, -274503, -50494502]\) | \(12257375872392016/1191317675625\) | \(222328469895840000\) | \([2, 2]\) | \(262144\) | \(2.0658\) | |
| 27720.u5 | 27720k1 | \([0, 0, 0, 20742, -3786743]\) | \(84611246065664/580054565475\) | \(-6765756451700400\) | \([2]\) | \(131072\) | \(1.7192\) | \(\Gamma_0(N)\)-optimal |
| 27720.u6 | 27720k5 | \([0, 0, 0, 1755357, 1794058558]\) | \(400647648358480318/1163177490234375\) | \(-1736614687500000000000\) | \([2]\) | \(1048576\) | \(2.7590\) |
Rank
sage: E.rank()
The elliptic curves in class 27720.u have rank \(0\).
Complex multiplication
The elliptic curves in class 27720.u do not have complex multiplication.Modular form 27720.2.a.u
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
