Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-312566640x-2126999407968\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-312566640xz^2-2126999407968z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-405086365467x-99236069119058634\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{6180124323592025041003092}{77200269818673191281}, \frac{14924527227080347384808672413972617240}{678310011901066286166592641929}\right)\) |
$\hat{h}(P)$ | ≈ | $55.639205283010446512134074275$ |
Integral points
None
Invariants
Conductor: | \( 122010 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 83$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-5744159273484243360 $ | = | $-1 \cdot 2^{5} \cdot 3^{3} \cdot 5 \cdot 7^{2} \cdot 83^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -\frac{68921624431417353829938395089}{117227740275188640} \) | = | $-1 \cdot 2^{-5} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{4} \cdot 83^{-7} \cdot 127^{3} \cdot 2410927^{3}$ | 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: | $3.2891792701433038950486861181\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $2.9648609119674183441977939942\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.06232460520681\dots$ | |||
Szpiro ratio: | $6.002001225347064\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $55.639205283010446512134074275\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.017952870694978649285634647745\dots$ | 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: | $ 15 $ = $ 5\cdot3\cdot1\cdot1\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) $ ≈ $ 14.983251870258892346805660446 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 14.983251870 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.017953 \cdot 55.639205 \cdot 15}{1^2} \approx 14.983251870$
Modular invariants
Modular form 122010.2.a.dj
For more coefficients, see the Downloads section to the right.
Modular degree: | 20885760 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$3$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$7$ | $1$ | $II$ | Additive | -1 | 2 | 2 | 0 |
$83$ | $1$ | $I_{7}$ | Non-split multiplicative | 1 | 1 | 7 | 7 |
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.1.3 | 7.48.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 69720 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 83 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 69707 & 14 \\ 69706 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 34861 & 14 \\ 34867 & 99 \end{array}\right),\left(\begin{array}{rr} 17433 & 49808 \\ 34846 & 57233 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 17431 & 14 \\ 52297 & 99 \end{array}\right),\left(\begin{array}{rr} 46481 & 14 \\ 46487 & 99 \end{array}\right),\left(\begin{array}{rr} 55777 & 14 \\ 41839 & 99 \end{array}\right),\left(\begin{array}{rr} 21001 & 14 \\ 7567 & 99 \end{array}\right)$.
The torsion field $K:=\Q(E[69720])$ is a degree-$34839993746718720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/69720\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 122010.dj
consists of 2 curves linked by isogenies of
degree 7.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
---|---|---|---|
$3$ | 3.1.488040.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.2372303094336000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | \(\Q(\zeta_{7})\) | \(\Z/7\Z\) | Not in database |
$7$ | 7.1.12252303000000.9 | \(\Z/7\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$18$ | 18.0.4634753389789508219235974934528000000.1 | \(\Z/14\Z\) | Not in database |
$21$ | 21.1.1202090336315934431360526607958288705617920000000000000000000.1 | \(\Z/14\Z\) | 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 | 83 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | split | add | ord | ss | ord | ord | ord | ord | ord | ord | ss | ord | ord | nonsplit |
$\lambda$-invariant(s) | 7 | 2 | 2 | - | 1 | 3,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.