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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 333795w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333795.w4 | 333795w1 | \([1, 0, 0, -76591, 8152160]\) | \(2058561081361/12705\) | \(306667814145\) | \([2]\) | \(1179648\) | \(1.3908\) | \(\Gamma_0(N)\)-optimal |
| 333795.w3 | 333795w2 | \([1, 0, 0, -78036, 7828191]\) | \(2177286259681/161417025\) | \(3896214578712225\) | \([2, 2]\) | \(2359296\) | \(1.7373\) | |
| 333795.w5 | 333795w3 | \([1, 0, 0, 73689, 34622826]\) | \(1833318007919/22507682505\) | \(-543280739494530345\) | \([2]\) | \(4718592\) | \(2.0839\) | |
| 333795.w2 | 333795w4 | \([1, 0, 0, -252881, -39694680]\) | \(74093292126001/14707625625\) | \(355006328349605625\) | \([2, 2]\) | \(4718592\) | \(2.0839\) | |
| 333795.w6 | 333795w5 | \([1, 0, 0, 525974, -235810369]\) | \(666688497209279/1381398046875\) | \(-33343590672910546875\) | \([2]\) | \(9437184\) | \(2.4305\) | |
| 333795.w1 | 333795w6 | \([1, 0, 0, -3829256, -2884343355]\) | \(257260669489908001/14267882475\) | \(344391997724203275\) | \([2]\) | \(9437184\) | \(2.4305\) |
Rank
sage: E.rank()
The elliptic curves in class 333795w have rank \(1\).
Complex multiplication
The elliptic curves in class 333795w do not have complex multiplication.Modular form 333795.2.a.w
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
