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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 18480.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18480.k1 | 18480bn7 | \([0, -1, 0, -407487816, 3166191056880]\) | \(1826870018430810435423307849/7641104625000000000\) | \(31297964544000000000000\) | \([2]\) | \(3981312\) | \(3.5254\) | |
| 18480.k2 | 18480bn6 | \([0, -1, 0, -25863496, 47862413296]\) | \(467116778179943012100169/28800309694464000000\) | \(117966068508524544000000\) | \([2, 2]\) | \(1990656\) | \(3.1788\) | |
| 18480.k3 | 18480bn4 | \([0, -1, 0, -7004376, 628859376]\) | \(9278380528613437145689/5328033205714065000\) | \(21823624010604810240000\) | \([2]\) | \(1327104\) | \(2.9761\) | |
| 18480.k4 | 18480bn3 | \([0, -1, 0, -4891976, -3240986640]\) | \(3160944030998056790089/720291785342976000\) | \(2950315152764829696000\) | \([2]\) | \(995328\) | \(2.8322\) | |
| 18480.k5 | 18480bn2 | \([0, -1, 0, -4589656, -3767862800]\) | \(2610383204210122997209/12104550027662400\) | \(49580236913305190400\) | \([2, 2]\) | \(663552\) | \(2.6295\) | |
| 18480.k6 | 18480bn1 | \([0, -1, 0, -4584536, -3776726544]\) | \(2601656892010848045529/56330588160\) | \(230730089103360\) | \([2]\) | \(331776\) | \(2.2829\) | \(\Gamma_0(N)\)-optimal |
| 18480.k7 | 18480bn5 | \([0, -1, 0, -2256856, -7597387280]\) | \(-310366976336070130009/5909282337130963560\) | \(-24204420452888426741760\) | \([2]\) | \(1327104\) | \(2.9761\) | |
| 18480.k8 | 18480bn8 | \([0, -1, 0, 20216504, 199815821296]\) | \(223090928422700449019831/4340371122724101696000\) | \(-17778160118677920546816000\) | \([2]\) | \(3981312\) | \(3.5254\) |
Rank
sage: E.rank()
The elliptic curves in class 18480.k have rank \(1\).
Complex multiplication
The elliptic curves in class 18480.k do not have complex multiplication.Modular form 18480.2.a.k
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
