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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 82110.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82110.z1 | 82110y4 | \([1, 0, 1, -3088994359, -62459382879718]\) | \(3259680673368195763530660699476329/201095781785032168161288000000\) | \(201095781785032168161288000000\) | \([2]\) | \(173187072\) | \(4.3745\) | |
| 82110.z2 | 82110y2 | \([1, 0, 1, -548298544, 4921801751126]\) | \(18229509733721637057438776494969/84465268068629449376512200\) | \(84465268068629449376512200\) | \([6]\) | \(57729024\) | \(3.8252\) | |
| 82110.z3 | 82110y1 | \([1, 0, 1, -16857544, 154988557526]\) | \(-529793445280877735265310969/10070862368645180735640000\) | \(-10070862368645180735640000\) | \([6]\) | \(28864512\) | \(3.4787\) | \(\Gamma_0(N)\)-optimal |
| 82110.z4 | 82110y3 | \([1, 0, 1, 151005641, -4075878879718]\) | \(380805515386703785466660523671/7397142570224064000000000000\) | \(-7397142570224064000000000000\) | \([2]\) | \(86593536\) | \(4.0280\) |
Rank
sage: E.rank()
The elliptic curves in class 82110.z have rank \(0\).
Complex multiplication
The elliptic curves in class 82110.z do not have complex multiplication.Modular form 82110.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
