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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 351120.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 351120.e1 | 351120e7 | \([0, -1, 0, -2409730496, 45531125502720]\) | \(377806291534052689568887263169/100912963819335937500\) | \(413339499804000000000000\) | \([2]\) | \(143327232\) | \(3.9030\) | |
| 351120.e2 | 351120e8 | \([0, -1, 0, -302118976, -937256033024]\) | \(744556702832013561199553089/338208906180283330846500\) | \(1385303679714440523147264000\) | \([2]\) | \(143327232\) | \(3.9030\) | |
| 351120.e3 | 351120e5 | \([0, -1, 0, -254503936, -1562665840640]\) | \(445089424735238304524848129/206488340640267840\) | \(845776243262537072640\) | \([2]\) | \(47775744\) | \(3.3537\) | |
| 351120.e4 | 351120e6 | \([0, -1, 0, -151198976, 705598718976]\) | \(93327647066813251630073089/1506876757438610250000\) | \(6172167198468547584000000\) | \([2, 2]\) | \(71663616\) | \(3.5565\) | |
| 351120.e5 | 351120e4 | \([0, -1, 0, -34374656, 41762451456]\) | \(1096677312076899338462209/450803852032204440000\) | \(1846492577923909386240000\) | \([2]\) | \(47775744\) | \(3.3537\) | |
| 351120.e6 | 351120e2 | \([0, -1, 0, -15988736, -24147394560]\) | \(110358600993178429667329/2339305154932838400\) | \(9581793914604906086400\) | \([2, 2]\) | \(23887872\) | \(3.0072\) | |
| 351120.e7 | 351120e3 | \([0, -1, 0, -608256, 30831820800]\) | \(-6076082794014148609/100253882690711904000\) | \(-410639903501155958784000\) | \([2]\) | \(35831808\) | \(3.2099\) | |
| 351120.e8 | 351120e1 | \([0, -1, 0, 67584, -1141899264]\) | \(8334681620170751/137523678664458240\) | \(-563296987809620951040\) | \([2]\) | \(11943936\) | \(2.6606\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 351120.e have rank \(0\).
Complex multiplication
The elliptic curves in class 351120.e do not have complex multiplication.Modular form 351120.2.a.e
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 4 & 12 \\ 4 & 1 & 3 & 2 & 12 & 6 & 4 & 12 \\ 12 & 3 & 1 & 6 & 4 & 2 & 12 & 4 \\ 2 & 2 & 6 & 1 & 6 & 3 & 2 & 6 \\ 3 & 12 & 4 & 6 & 1 & 2 & 12 & 4 \\ 6 & 6 & 2 & 3 & 2 & 1 & 6 & 2 \\ 4 & 4 & 12 & 2 & 12 & 6 & 1 & 3 \\ 12 & 12 & 4 & 6 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
