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SageMath
E = EllipticCurve("ci1")
E.isogeny_class()
Elliptic curves in class 11760.ci
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11760.ci1 | 11760cp7 | \([0, 1, 0, -4181480, -3292509900]\) | \(16778985534208729/81000\) | \(39033114624000\) | \([2]\) | \(165888\) | \(2.2303\) | |
| 11760.ci2 | 11760cp8 | \([0, 1, 0, -355560, -11225292]\) | \(10316097499609/5859375000\) | \(2823576000000000000\) | \([2]\) | \(165888\) | \(2.2303\) | |
| 11760.ci3 | 11760cp6 | \([0, 1, 0, -261480, -51453900]\) | \(4102915888729/9000000\) | \(4337012736000000\) | \([2, 2]\) | \(82944\) | \(1.8837\) | |
| 11760.ci4 | 11760cp5 | \([0, 1, 0, -226200, 41332500]\) | \(2656166199049/33750\) | \(16263797760000\) | \([2]\) | \(55296\) | \(1.6810\) | |
| 11760.ci5 | 11760cp4 | \([0, 1, 0, -53720, -4145772]\) | \(35578826569/5314410\) | \(2560962650480640\) | \([2]\) | \(55296\) | \(1.6810\) | |
| 11760.ci6 | 11760cp2 | \([0, 1, 0, -14520, 605268]\) | \(702595369/72900\) | \(35129803161600\) | \([2, 2]\) | \(27648\) | \(1.3344\) | |
| 11760.ci7 | 11760cp3 | \([0, 1, 0, -10600, -1378252]\) | \(-273359449/1536000\) | \(-740183506944000\) | \([2]\) | \(41472\) | \(1.5371\) | |
| 11760.ci8 | 11760cp1 | \([0, 1, 0, 1160, 47060]\) | \(357911/2160\) | \(-1040883056640\) | \([2]\) | \(13824\) | \(0.98781\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 11760.ci have rank \(1\).
Complex multiplication
The elliptic curves in class 11760.ci do not have complex multiplication.Modular form 11760.2.a.ci
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
