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SageMath
E = EllipticCurve("da1")
E.isogeny_class()
Elliptic curves in class 33600da
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33600.hf7 | 33600da1 | \([0, 1, 0, 335967, 56592063]\) | \(1023887723039/928972800\) | \(-3805072588800000000\) | \([2]\) | \(589824\) | \(2.2526\) | \(\Gamma_0(N)\)-optimal |
| 33600.hf6 | 33600da2 | \([0, 1, 0, -1712033, 505104063]\) | \(135487869158881/51438240000\) | \(210691031040000000000\) | \([2, 2]\) | \(1179648\) | \(2.5991\) | |
| 33600.hf5 | 33600da3 | \([0, 1, 0, -12080033, -15803759937]\) | \(47595748626367201/1215506250000\) | \(4978713600000000000000\) | \([2, 2]\) | \(2359296\) | \(2.9457\) | |
| 33600.hf4 | 33600da4 | \([0, 1, 0, -24112033, 45551504063]\) | \(378499465220294881/120530818800\) | \(493694233804800000000\) | \([2]\) | \(2359296\) | \(2.9457\) | |
| 33600.hf8 | 33600da5 | \([0, 1, 0, 2031967, -50505167937]\) | \(226523624554079/269165039062500\) | \(-1102500000000000000000000\) | \([2]\) | \(4718592\) | \(3.2923\) | |
| 33600.hf2 | 33600da6 | \([0, 1, 0, -192080033, -1024703759937]\) | \(191342053882402567201/129708022500\) | \(531284060160000000000\) | \([2, 2]\) | \(4718592\) | \(3.2923\) | |
| 33600.hf3 | 33600da7 | \([0, 1, 0, -190880033, -1038137759937]\) | \(-187778242790732059201/4984939585440150\) | \(-20418312541962854400000000\) | \([2]\) | \(9437184\) | \(3.6389\) | |
| 33600.hf1 | 33600da8 | \([0, 1, 0, -3073280033, -65577989759937]\) | \(783736670177727068275201/360150\) | \(1475174400000000\) | \([2]\) | \(9437184\) | \(3.6389\) |
Rank
sage: E.rank()
The elliptic curves in class 33600da have rank \(1\).
Complex multiplication
The elliptic curves in class 33600da do not have complex multiplication.Modular form 33600.2.a.da
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
