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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 6384.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6384.i1 | 6384q6 | \([0, -1, 0, -156587368, 754246418800]\) | \(103665426767620308239307625/5961940992\) | \(24420110303232\) | \([2]\) | \(497664\) | \(2.9554\) | |
| 6384.i2 | 6384q5 | \([0, -1, 0, -9786728, 11787501936]\) | \(25309080274342544331625/191933498523648\) | \(786159609952862208\) | \([2]\) | \(248832\) | \(2.6088\) | |
| 6384.i3 | 6384q4 | \([0, -1, 0, -1934968, 1033192048]\) | \(195607431345044517625/752875610010048\) | \(3083778498601156608\) | \([2]\) | \(165888\) | \(2.4061\) | |
| 6384.i4 | 6384q3 | \([0, -1, 0, -178808, -834960]\) | \(154357248921765625/89242711068672\) | \(365538144537280512\) | \([2]\) | \(82944\) | \(2.0595\) | |
| 6384.i5 | 6384q2 | \([0, -1, 0, -127048, -16288016]\) | \(55369510069623625/3916046302812\) | \(16040125656317952\) | \([2]\) | \(55296\) | \(1.8568\) | |
| 6384.i6 | 6384q1 | \([0, -1, 0, -124808, -16929552]\) | \(52492168638015625/293197968\) | \(1200938876928\) | \([2]\) | \(27648\) | \(1.5102\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6384.i have rank \(0\).
Complex multiplication
The elliptic curves in class 6384.i do not have complex multiplication.Modular form 6384.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
