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SageMath
E = EllipticCurve("fu1")
E.isogeny_class()
Elliptic curves in class 323400.fu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
323400.fu1 | 323400fu5 | \([0, 1, 0, -2063978408, 36089868272688]\) | \(258286045443018193442/8440380939375\) | \(31776076068368940000000000\) | \([2]\) | \(150994944\) | \(3.9873\) | |
323400.fu2 | 323400fu4 | \([0, 1, 0, -583100408, -5419691553312]\) | \(11647843478225136004/128410942275\) | \(241718703163383600000000\) | \([2]\) | \(75497472\) | \(3.6407\) | |
323400.fu3 | 323400fu3 | \([0, 1, 0, -134603408, 512193272688]\) | \(143279368983686884/22699269140625\) | \(42728741042006250000000000\) | \([2, 2]\) | \(75497472\) | \(3.6407\) | |
323400.fu4 | 323400fu2 | \([0, 1, 0, -37362908, -80195853312]\) | \(12257375872392016/1191317675625\) | \(560629332878422500000000\) | \([2, 2]\) | \(37748736\) | \(3.2942\) | |
323400.fu5 | 323400fu1 | \([0, 1, 0, 2823217, -6012266562]\) | \(84611246065664/580054565475\) | \(-17060709893392068750000\) | \([2]\) | \(18874368\) | \(2.9476\) | \(\Gamma_0(N)\)-optimal |
323400.fu6 | 323400fu6 | \([0, 1, 0, 238923592, 2848978184688]\) | \(400647648358480318/1163177490234375\) | \(-4379093393554687500000000000\) | \([2]\) | \(150994944\) | \(3.9873\) |
Rank
sage: E.rank()
The elliptic curves in class 323400.fu have rank \(1\).
Complex multiplication
The elliptic curves in class 323400.fu do not have complex multiplication.Modular form 323400.2.a.fu
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.