Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3-377686875x+14420422656\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3-377686875xz^2+14420422656z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-6042990000x+922907050000\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(20519, 950665\right)\) |
$\hat{h}(P)$ | ≈ | $4.9120215501585368855402623796$ |
Integral points
\( \left(20519, 950665\right) \), \( \left(20519, -950666\right) \)
Invariants
Conductor: | \( 455175 \) | = | $3^{2} \cdot 5^{2} \cdot 7 \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $3447976822042782125126953125 $ | = | $3^{11} \cdot 5^{10} \cdot 7^{5} \cdot 17^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{7057510400}{4084101} \) | = | $2^{12} \cdot 3^{-5} \cdot 5^{2} \cdot 7^{-5} \cdot 41^{3}$ | 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: | $3.9734330910708953502123009414\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.041981321667071867839772084826\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.2807716361702774\dots$ | |||
Szpiro ratio: | $5.439050735141807\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $4.9120215501585368855402623796\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.037642836512316638468110851792\dots$ | 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: | $ 20 $ = $ 2\cdot1\cdot5\cdot2 $ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 3.6980484831518789328496688844 $ | 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
$\displaystyle 3.698048483 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.037643 \cdot 4.912022 \cdot 20}{1^2} \approx 3.698048483$
Modular invariants
Modular form 455175.2.a.o
For more coefficients, see the Downloads section to the right.
Modular degree: | 404736000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{5}^{*}$ | Additive | -1 | 2 | 11 | 5 |
$5$ | $1$ | $II^{*}$ | Additive | 1 | 2 | 10 | 0 |
$7$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$17$ | $2$ | $III^{*}$ | Additive | 1 | 2 | 9 | 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 |
---|---|---|
$5$ | 5B | 5.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3570 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 1189 & 3560 \\ 2375 & 3519 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 2551 & 10 \\ 2045 & 51 \end{array}\right),\left(\begin{array}{rr} 3567 & 3560 \\ 3550 & 1361 \end{array}\right),\left(\begin{array}{rr} 3561 & 10 \\ 3560 & 11 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 3515 & 3451 \end{array}\right),\left(\begin{array}{rr} 3357 & 3560 \\ 2965 & 83 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3570])$ is a degree-$454825082880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3570\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 455175.o
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 151725.l2, its twist by $-255$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.