Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3-x^2-3283x-74657\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3-x^2z-3283xz^2-74657z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-4255200x-3534246000\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(397, 7812\right)\) |
$\hat{h}(P)$ | ≈ | $0.83316000421927884699261405250$ |
Integral points
\( \left(107, 887\right) \), \( \left(107, -888\right) \), \( \left(397, 7812\right) \), \( \left(397, -7813\right) \)
Invariants
Conductor: | \( 175 \) | = | $5^{2} \cdot 7$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-213623046875 $ | = | $-1 \cdot 5^{15} \cdot 7 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{250523582464}{13671875} \) | = | $-1 \cdot 2^{15} \cdot 5^{-9} \cdot 7^{-1} \cdot 197^{3}$ | 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: | $0.93218107984805492160804756825\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.12746212363100473430766790164\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0211201298201755\dots$ | |||
Szpiro ratio: | $6.969032932721161\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.83316000421927884699261405250\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.31435146257085184003506263964\dots$ | 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);
|
Tamagawa product: | $ 4 $ = $ 2^{2}\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)
|
Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L'(E,1) $ ≈ $ 1.0476202635274695823599936515 $ | 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
$\displaystyle 1.047620264 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.314351 \cdot 0.833160 \cdot 4}{1^2} \approx 1.047620264$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 144 | 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 2 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$5$ | $4$ | $I_{9}^{*}$ | additive | 1 | 2 | 15 | 9 |
$7$ | $1$ | $I_{1}$ | 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 |
---|---|---|
$3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 613 & 18 \\ 612 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 281 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 620 & 621 \\ 513 & 8 \end{array}\right),\left(\begin{array}{rr} 286 & 9 \\ 405 & 76 \end{array}\right)$.
The torsion field $K:=\Q(E[630])$ is a degree-$156764160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/630\Z)$.
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 |
---|---|---|---|
$5$ | additive | $18$ | \( 7 \) |
$7$ | nonsplit multiplicative | $8$ | \( 25 = 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 175b
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 35a2, its twist by $5$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-15}) \) | \(\Z/3\Z\) | not in database |
$3$ | 3.1.140.1 | \(\Z/2\Z\) | not in database |
$6$ | 6.0.686000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$6$ | 6.2.218791125.5 | \(\Z/3\Z\) | not in database |
$6$ | 6.0.5907360375.2 | \(\Z/9\Z\) | not in database |
$6$ | 6.0.2646000.1 | \(\Z/6\Z\) | not in database |
$12$ | deg 12 | \(\Z/4\Z\) | not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
$12$ | 12.0.14534321782640625.2 | \(\Z/9\Z\) | not in database |
$12$ | 12.0.343064484000000.3 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$18$ | 18.2.2102060116273937625288000000000.1 | \(\Z/6\Z\) | not in database |
$18$ | 18.0.41374849268619914278543704000000000.1 | \(\Z/18\Z\) | 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 | ss | ord | add | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 1,6 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 0,0 | 2 | - | 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.