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SageMath
E = EllipticCurve("fp1")
E.isogeny_class()
Elliptic curves in class 330330.fp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 330330.fp1 | 330330fp7 | \([1, 0, 0, -274147101, 1746810420081]\) | \(1286229821345376481036009/247265484375000000\) | \(438045888764859375000000\) | \([2]\) | \(119439360\) | \(3.5369\) | |
| 330330.fp2 | 330330fp8 | \([1, 0, 0, -120583581, -493671997455]\) | \(109454124781830273937129/3914078300576808000\) | \(6934028468248150557288000\) | \([2]\) | \(119439360\) | \(3.5369\) | |
| 330330.fp3 | 330330fp5 | \([1, 0, 0, -119529066, -502998862464]\) | \(106607603143751752938169/5290068420\) | \(9371678900203620\) | \([2]\) | \(39813120\) | \(2.9876\) | |
| 330330.fp4 | 330330fp6 | \([1, 0, 0, -18943581, 21175258545]\) | \(424378956393532177129/136231857216000000\) | \(241343045201434176000000\) | \([2, 2]\) | \(59719680\) | \(3.1903\) | |
| 330330.fp5 | 330330fp4 | \([1, 0, 0, -8320386, -5961167640]\) | \(35958207000163259449/12145729518877500\) | \(21516900732192142777500\) | \([2]\) | \(39813120\) | \(2.9876\) | |
| 330330.fp6 | 330330fp2 | \([1, 0, 0, -7470966, -7858941804]\) | \(26031421522845051769/5797789779600\) | \(10271138259737955600\) | \([2, 2]\) | \(19906560\) | \(2.6410\) | |
| 330330.fp7 | 330330fp1 | \([1, 0, 0, -414246, -151592220]\) | \(-4437543642183289/3033210136320\) | \(-5373516782309195520\) | \([2]\) | \(9953280\) | \(2.2944\) | \(\Gamma_0(N)\)-optimal |
| 330330.fp8 | 330330fp3 | \([1, 0, 0, 3359139, 2258091441]\) | \(2366200373628880151/2612420149248000\) | \(-4628061652021936128000\) | \([2]\) | \(29859840\) | \(2.8437\) |
Rank
sage: E.rank()
The elliptic curves in class 330330.fp have rank \(0\).
Complex multiplication
The elliptic curves in class 330330.fp do not have complex multiplication.Modular form 330330.2.a.fp
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
