Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+552278x+68146637\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+552278xz^2+68146637z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+715752261x+3177302239062\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(1061, 42467\right)\) |
$\hat{h}(P)$ | ≈ | $2.0642626470206162648123084785$ |
Torsion generators
\( \left(-\frac{481}{4}, \frac{481}{8}\right) \)
Integral points
\( \left(1061, 42467\right) \), \( \left(1061, -43528\right) \)
Invariants
Conductor: | \( 32487 \) | = | $3 \cdot 7^{2} \cdot 13 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-12784337402104656297 $ | = | $-1 \cdot 3^{8} \cdot 7^{14} \cdot 13^{2} \cdot 17 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{158346567380527343}{108665074944153} \) | = | $3^{-8} \cdot 7^{-8} \cdot 13^{-2} \cdot 17^{-1} \cdot 541007^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.3548417573043155541679270421\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.3818866827766589016152506704\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9631127158592017\dots$ | |||
Szpiro ratio: | $4.936088273872958\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.0642626470206162648123084785\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.14169147262354586618715938905\dots$ | 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: | $ 64 $ = $ 2^{3}\cdot2^{2}\cdot2\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: | $2$ | 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) $ ≈ $ 4.6798146294100795709226626654 $ | 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
$\displaystyle 4.679814629 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.141691 \cdot 2.064263 \cdot 64}{2^2} \approx 4.679814629$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 589824 | 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
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
$7$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
$13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
$17$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 |
---|---|---|
$2$ | 2B | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 74256 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 8752 & 5 \\ 21795 & 74242 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 18308 & 18249 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 49505 & 16 \\ 24760 & 129 \end{array}\right),\left(\begin{array}{rr} 42431 & 74240 \\ 42424 & 74127 \end{array}\right),\left(\begin{array}{rr} 74241 & 16 \\ 74240 & 17 \end{array}\right),\left(\begin{array}{rr} 64982 & 18569 \\ 9285 & 37130 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 74158 & 74243 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 34016 & 73941 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 74252 & 74253 \end{array}\right)$.
The torsion field $K:=\Q(E[74256])$ is a degree-$25429452313853952$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/74256\Z)$.
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 |
---|---|---|---|
$2$ | good | $2$ | \( 833 = 7^{2} \cdot 17 \) |
$3$ | split multiplicative | $4$ | \( 10829 = 7^{2} \cdot 13 \cdot 17 \) |
$7$ | additive | $32$ | \( 663 = 3 \cdot 13 \cdot 17 \) |
$13$ | nonsplit multiplicative | $14$ | \( 2499 = 3 \cdot 7^{2} \cdot 17 \) |
$17$ | nonsplit multiplicative | $18$ | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 32487.e
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 4641.b6, its twist by $-7$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$2$ | \(\Q(\sqrt{7}) \) | \(\Z/4\Z\) | not in database |
$2$ | \(\Q(\sqrt{-119}) \) | \(\Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{7}, \sqrt{-17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{7}, \sqrt{442})\) | \(\Z/8\Z\) | not in database |
$4$ | \(\Q(\sqrt{7}, \sqrt{-26})\) | \(\Z/8\Z\) | not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.927268850704.3 | \(\Z/8\Z\) | not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | deg 8 | \(\Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$16$ | deg 16 | \(\Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/12\Z\) | not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | ord | split | ord | add | ord | nonsplit | nonsplit | ord | ss | ord | ss | ord | ord | ord | ss |
$\lambda$-invariant(s) | 8 | 6 | 3 | - | 1 | 1 | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | 1 | 0 | 0 | - | 0 | 0 | 0 | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.