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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 141120.ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141120.ef1 | 141120dz7 | \([0, 0, 0, -9913186668, 379898903267408]\) | \(4791901410190533590281/41160000\) | \(925403326804131840000\) | \([2]\) | \(84934656\) | \(4.0642\) | |
| 141120.ef2 | 141120dz6 | \([0, 0, 0, -619587948, 5935643093072]\) | \(1169975873419524361/108425318400\) | \(2437734459600516258201600\) | \([2, 2]\) | \(42467328\) | \(3.7177\) | |
| 141120.ef3 | 141120dz8 | \([0, 0, 0, -574429548, 6837564721232]\) | \(-932348627918877961/358766164249920\) | \(-8066166227930844485470126080\) | \([2]\) | \(84934656\) | \(4.0642\) | |
| 141120.ef4 | 141120dz4 | \([0, 0, 0, -122986668, 515740787408]\) | \(9150443179640281/184570312500\) | \(4149707998464000000000000\) | \([2]\) | \(28311552\) | \(3.5149\) | |
| 141120.ef5 | 141120dz3 | \([0, 0, 0, -41560428, 78374627408]\) | \(353108405631241/86318776320\) | \(1940711437613938696519680\) | \([2]\) | \(21233664\) | \(3.3711\) | |
| 141120.ef6 | 141120dz2 | \([0, 0, 0, -16299948, -13297319728]\) | \(21302308926361/8930250000\) | \(200779471797682176000000\) | \([2, 2]\) | \(14155776\) | \(3.1684\) | |
| 141120.ef7 | 141120dz1 | \([0, 0, 0, -14042028, -20245391152]\) | \(13619385906841/6048000\) | \(135977631693668352000\) | \([2]\) | \(7077888\) | \(2.8218\) | \(\Gamma_0(N)\)-optimal |
| 141120.ef8 | 141120dz5 | \([0, 0, 0, 54260052, -97658855728]\) | \(785793873833639/637994920500\) | \(-14344087024170010017792000\) | \([2]\) | \(28311552\) | \(3.5149\) |
Rank
sage: E.rank()
The elliptic curves in class 141120.ef have rank \(0\).
Complex multiplication
The elliptic curves in class 141120.ef do not have complex multiplication.Modular form 141120.2.a.ef
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.
