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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 4410.t
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4410.t1 | 4410r7 | \([1, -1, 0, -847072809, -9488980043537]\) | \(783736670177727068275201/360150\) | \(30888668478150\) | \([2]\) | \(786432\) | \(3.3167\) | |
| 4410.t2 | 4410r5 | \([1, -1, 0, -52942059, -148255335887]\) | \(191342053882402567201/129708022500\) | \(11124553952405722500\) | \([2, 2]\) | \(393216\) | \(2.9701\) | |
| 4410.t3 | 4410r8 | \([1, -1, 0, -52611309, -150199418237]\) | \(-187778242790732059201/4984939585440150\) | \(-427538931662549743158150\) | \([2]\) | \(786432\) | \(3.3167\) | |
| 4410.t4 | 4410r4 | \([1, -1, 0, -6645879, 6594269485]\) | \(378499465220294881/120530818800\) | \(10337460789429874800\) | \([2]\) | \(196608\) | \(2.6235\) | |
| 4410.t5 | 4410r3 | \([1, -1, 0, -3329559, -2285438387]\) | \(47595748626367201/1215506250000\) | \(104249256113756250000\) | \([2, 2]\) | \(196608\) | \(2.6235\) | |
| 4410.t6 | 4410r2 | \([1, -1, 0, -471879, 73290685]\) | \(135487869158881/51438240000\) | \(4411658315867040000\) | \([2, 2]\) | \(98304\) | \(2.2770\) | |
| 4410.t7 | 4410r1 | \([1, -1, 0, 92601, 8149693]\) | \(1023887723039/928972800\) | \(-79674393570508800\) | \([2]\) | \(49152\) | \(1.9304\) | \(\Gamma_0(N)\)-optimal |
| 4410.t8 | 4410r6 | \([1, -1, 0, 560061, -7308493655]\) | \(226523624554079/269165039062500\) | \(-23085241309204101562500\) | \([2]\) | \(393216\) | \(2.9701\) |
Rank
sage: E.rank()
The elliptic curves in class 4410.t have rank \(0\).
Complex multiplication
The elliptic curves in class 4410.t do not have complex multiplication.Modular form 4410.2.a.t
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
