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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 163254i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 163254.dj4 | 163254i1 | \([1, 0, 0, 784917, 1885107249]\) | \(11079872671250375/324440155855872\) | \(-1566010664246525672448\) | \([2]\) | \(11059200\) | \(2.7470\) | \(\Gamma_0(N)\)-optimal |
| 163254.dj2 | 163254i2 | \([1, 0, 0, -18927243, 30195711441]\) | \(155355156733986861625/8291568305839392\) | \(40021816522740329860128\) | \([2]\) | \(22118400\) | \(3.0935\) | |
| 163254.dj3 | 163254i3 | \([1, 0, 0, -7086258, -51790953852]\) | \(-8152944444844179625/235342826399858688\) | \(-1135954872552275513966592\) | \([2]\) | \(33177600\) | \(3.2963\) | |
| 163254.dj1 | 163254i4 | \([1, 0, 0, -256286898, -1571565976956]\) | \(385693937170561837203625/2159357734550274048\) | \(10422807347346873727352832\) | \([2]\) | \(66355200\) | \(3.6428\) |
Rank
sage: E.rank()
The elliptic curves in class 163254i have rank \(2\).
Complex multiplication
The elliptic curves in class 163254i do not have complex multiplication.Modular form 163254.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
