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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 74970q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 74970.m7 | 74970q1 | \([1, -1, 0, -14255775, 24697778125]\) | \(-3735772816268612449/909650165760000\) | \(-78017166184242216960000\) | \([2]\) | \(7077888\) | \(3.1110\) | \(\Gamma_0(N)\)-optimal |
| 74970.m6 | 74970q2 | \([1, -1, 0, -240047775, 1431517413325]\) | \(17836145204788591940449/770635366502400\) | \(66094406090324185190400\) | \([2, 2]\) | \(14155776\) | \(3.4576\) | |
| 74970.m8 | 74970q3 | \([1, -1, 0, 102591585, -168020096819]\) | \(1392333139184610040991/947901937500000000\) | \(-81297872267759437500000000\) | \([2]\) | \(21233664\) | \(3.6603\) | |
| 74970.m5 | 74970q4 | \([1, -1, 0, -252042975, 1280553023245]\) | \(20645800966247918737249/3688936444974392640\) | \(316385769500983601063749440\) | \([2]\) | \(28311552\) | \(3.8042\) | |
| 74970.m3 | 74970q5 | \([1, -1, 0, -3840724575, 91616228952205]\) | \(73054578035931991395831649/136386452160\) | \(11697336958715271360\) | \([2]\) | \(28311552\) | \(3.8042\) | |
| 74970.m4 | 74970q6 | \([1, -1, 0, -448658415, -1400945846819]\) | \(116454264690812369959009/57505157319440250000\) | \(4931994280783148133770250000\) | \([2, 2]\) | \(42467328\) | \(4.0069\) | |
| 74970.m1 | 74970q7 | \([1, -1, 0, -5865240915, -172763199766319]\) | \(260174968233082037895439009/223081361502731896500\) | \(19132823043488045665778476500\) | \([2]\) | \(84934656\) | \(4.3535\) | |
| 74970.m2 | 74970q8 | \([1, -1, 0, -3852075915, 91047445072681]\) | \(73704237235978088924479009/899277423164136103500\) | \(77127536287663499913249523500\) | \([2]\) | \(84934656\) | \(4.3535\) |
Rank
sage: E.rank()
The elliptic curves in class 74970q have rank \(1\).
Complex multiplication
The elliptic curves in class 74970q do not have complex multiplication.Modular form 74970.2.a.q
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
