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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 227430.ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 227430.ba1 | 227430er8 | \([1, -1, 0, -1141154460, -14837352383984]\) | \(4791901410190533590281/41160000\) | \(1411641768768840000\) | \([2]\) | \(55738368\) | \(3.5238\) | |
| 227430.ba2 | 227430er6 | \([1, -1, 0, -71323740, -231809428400]\) | \(1169975873419524361/108425318400\) | \(3718603212961629081600\) | \([2, 2]\) | \(27869184\) | \(3.1772\) | |
| 227430.ba3 | 227430er7 | \([1, -1, 0, -66125340, -267036905840]\) | \(-932348627918877961/358766164249920\) | \(-12304404826923450932470080\) | \([2]\) | \(55738368\) | \(3.5238\) | |
| 227430.ba4 | 227430er5 | \([1, -1, 0, -14157585, -20139648359]\) | \(9150443179640281/184570312500\) | \(6330105986387695312500\) | \([2]\) | \(18579456\) | \(2.9745\) | |
| 227430.ba5 | 227430er3 | \([1, -1, 0, -4784220, -3059866544]\) | \(353108405631241/86318776320\) | \(2960427358657110343680\) | \([2]\) | \(13934592\) | \(2.8306\) | |
| 227430.ba6 | 227430er2 | \([1, -1, 0, -1876365, 519819925]\) | \(21302308926361/8930250000\) | \(306275848045382250000\) | \([2, 2]\) | \(9289728\) | \(2.6279\) | |
| 227430.ba7 | 227430er1 | \([1, -1, 0, -1616445, 791124421]\) | \(13619385906841/6048000\) | \(207424912961952000\) | \([2]\) | \(4644864\) | \(2.2813\) | \(\Gamma_0(N)\)-optimal |
| 227430.ba8 | 227430er4 | \([1, -1, 0, 6246135, 3812681425]\) | \(785793873833639/637994920500\) | \(-21880959136058198704500\) | \([2]\) | \(18579456\) | \(2.9745\) |
Rank
sage: E.rank()
The elliptic curves in class 227430.ba have rank \(0\).
Complex multiplication
The elliptic curves in class 227430.ba do not have complex multiplication.Modular form 227430.2.a.ba
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
