Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-6355692x-6193904976\)
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(homogenize, simplify) |
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\(y^2z=x^3-6355692xz^2-6193904976z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6355692x-6193904976\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
| Conductor: | \( 1270080 \) | = | $2^{6} \cdot 3^{4} \cdot 5 \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-142275702804480000000 $ | = | $-1 \cdot 2^{18} \cdot 3^{10} \cdot 5^{7} \cdot 7^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{15590912409}{78125} \) | = | $-1 \cdot 3^{2} \cdot 5^{-7} \cdot 1201^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\Z$ | |||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $\mathrm{SU}(2)$ | |||
| Faltings height: | $2.7143472059508806243626390714\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-0.21383887997345206847858984661\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0070302306950922\dots$ | |||
| Szpiro ratio: | $4.170652208347135\dots$ | |||
BSD invariants
| Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.047527906750374259648726028398\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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| Tamagawa product: | $ 42 $ = $ 2\cdot3\cdot7\cdot1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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| Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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| Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L(E,1) $ ≈ $ 1.9961720835157189052464931927 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 1.996172084 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.047528 \cdot 1.000000 \cdot 42}{1^2} \approx 1.996172084$
Modular invariants
Modular form 1270080.2.a.pn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 42577920 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}^{*}$ | Additive | 1 | 6 | 18 | 0 |
| $3$ | $3$ | $IV^{*}$ | Additive | -1 | 4 | 10 | 0 |
| $5$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
| $7$ | $1$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 2507 & 14 \\ 2506 & 15 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1887 & 1432 \\ 1274 & 1567 \end{array}\right),\left(\begin{array}{rr} 1273 & 2506 \\ 1274 & 2505 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1763 & 2506 \\ 2261 & 2421 \end{array}\right),\left(\begin{array}{rr} 1259 & 0 \\ 0 & 2519 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 2511 & 2510 \\ 1442 & 2411 \end{array}\right),\left(\begin{array}{rr} 631 & 14 \\ 637 & 99 \end{array}\right)$.
The torsion field $K:=\Q(E[2520])$ is a degree-$60197437440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2520\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 1270080.pn
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 405.b1, its twist by $-56$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.