Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-488599680x+4113962955900\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-488599680xz^2+4113962955900z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-39576574107x+2998960265128806\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(11685, 0)$ | $0$ | $2$ |
| $(13810, 0)$ | $0$ | $2$ |
Integral points
\( \left(-25494, 0\right) \), \( \left(11685, 0\right) \), \( \left(13810, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 34680 \) | = | $2^{3} \cdot 3 \cdot 5 \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $\Delta$ | = | $154278981280586449296000000$ | = | $2^{10} \cdot 3^{14} \cdot 5^{6} \cdot 17^{10} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | $j$ | = | \( \frac{521902963282042184836}{6241849278890625} \) | = | $2^{2} \cdot 3^{-14} \cdot 5^{-6} \cdot 7^{3} \cdot 17^{-4} \cdot 724567^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8363960888003584697357738314$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8421667663056293384299797546$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $Q$ | ≈ | $1.0333686215042301$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.852432003738785$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Mordell-Weil rank: | $r$ | = | $ 0$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $\Omega$ | ≈ | $0.057920646001713508937630204603$ | 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: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2\cdot2\cdot( 2 \cdot 3 )\cdot2^{2} $ | 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: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ | 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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| Special value: | $ L(E,1)$ | ≈ | $3.1277148840925294826320310486 $ | 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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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 3.127714884 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{9 \cdot 0.057921 \cdot 1.000000 \cdot 96}{4^2} \approx 3.127714884$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15482880 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | 1 | 3 | 10 | 0 |
| $3$ | $2$ | $I_{14}$ | nonsplit multiplicative | 1 | 1 | 14 | 14 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $17$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 171 & 1802 \\ 646 & 239 \end{array}\right),\left(\begin{array}{rr} 1361 & 1564 \\ 442 & 1089 \end{array}\right),\left(\begin{array}{rr} 1021 & 1802 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 511 & 1564 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2037 & 4 \\ 2036 & 5 \end{array}\right),\left(\begin{array}{rr} 1799 & 0 \\ 0 & 2039 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$57755566080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | additive | $2$ | \( 289 = 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 2312 = 2^{3} \cdot 17^{2} \) |
| $5$ | split multiplicative | $6$ | \( 6936 = 2^{3} \cdot 3 \cdot 17^{2} \) |
| $7$ | good | $2$ | \( 11560 = 2^{3} \cdot 5 \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 120 = 2^{3} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 34680l
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2040f2, its twist by $17$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-15}, \sqrt{51})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{10}, \sqrt{34})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{-34})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.76785869193364478361600000000.4 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 17 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | split | add |
| $\lambda$-invariant(s) | - | 2 | 1 | - |
| $\mu$-invariant(s) | - | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.