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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 30345.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30345.f1 | 30345m4 | \([1, 1, 1, -487260, 130706322]\) | \(530044731605089/26309115\) | \(635038078641435\) | \([2]\) | \(294912\) | \(1.9118\) | |
30345.f2 | 30345m3 | \([1, 1, 1, -154910, -21879898]\) | \(17032120495489/1339001685\) | \(32320245562803765\) | \([2]\) | \(294912\) | \(1.9118\) | |
30345.f3 | 30345m2 | \([1, 1, 1, -32085, 1800762]\) | \(151334226289/28676025\) | \(692169532083225\) | \([2, 2]\) | \(147456\) | \(1.5652\) | |
30345.f4 | 30345m1 | \([1, 1, 1, 4040, 167912]\) | \(302111711/669375\) | \(-16157085249375\) | \([4]\) | \(73728\) | \(1.2186\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 30345.f have rank \(0\).
Complex multiplication
The elliptic curves in class 30345.f do not have complex multiplication.Modular form 30345.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.