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SageMath
E = EllipticCurve("if1")
E.isogeny_class()
Elliptic curves in class 129360.if
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.if1 | 129360if4 | \([0, 1, 0, -76260, -4428600]\) | \(1628514404944/664335375\) | \(20008548488544000\) | \([2]\) | \(995328\) | \(1.8256\) | |
| 129360.if2 | 129360if2 | \([0, 1, 0, -35100, 2519208]\) | \(158792223184/16335\) | \(491979882240\) | \([2]\) | \(331776\) | \(1.2763\) | |
| 129360.if3 | 129360if1 | \([0, 1, 0, -2025, 45198]\) | \(-488095744/200475\) | \(-377370932400\) | \([2]\) | \(165888\) | \(0.92968\) | \(\Gamma_0(N)\)-optimal |
| 129360.if4 | 129360if3 | \([0, 1, 0, 15615, -496350]\) | \(223673040896/187171875\) | \(-352329342750000\) | \([2]\) | \(497664\) | \(1.4790\) |
Rank
sage: E.rank()
The elliptic curves in class 129360.if have rank \(0\).
Complex multiplication
The elliptic curves in class 129360.if do not have complex multiplication.Modular form 129360.2.a.if
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.
