Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-10273526088x-402773373059778\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-10273526088xz^2-402773373059778z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-13314489810723x-18791594776129844898\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{1279551750691779579415715031882914110972967205926582270686748112770138458693108143888868013347829759394800330069}{4756057210795169514131373942600130233629110895651100986650909991012383368896278537089705635716106789786896}, \frac{41839950465672503700751292552942321639992308375986642461584444190514025825998998785474604732337102198388167154959097408338405150066084665202179169822233693111987562955}{327997554118354972384485858361987741026020797074397520964495115329576903957917708063435409449752272784007040546036833436043010334134494324276350277643435370944}\right)\) |
$\hat{h}(P)$ | ≈ | $251.61378178035375794193578451$ |
Integral points
None
Invariants
Conductor: | \( 75810 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-683652529355875573834020235590 $ | = | $-1 \cdot 2 \cdot 3^{31} \cdot 5 \cdot 7^{3} \cdot 19^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -\frac{371620692159996346278931}{2118619749253938210} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-31} \cdot 5^{-1} \cdot 7^{-3} \cdot 727^{3} \cdot 98893^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
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: | $4.5666736424771438979159420661\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $2.3583444081023135529091714922\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.0428605295052948\dots$ | |||
Szpiro ratio: | $7.189570928189096\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $251.61378178035375794193578451\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.0074953454679563649408743849130\dots$ | 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: | $ 2 $ = $ 1\cdot1\cdot1\cdot1\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)
|
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) $ ≈ $ 3.7718644378854726575641040824 $ | 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
$\displaystyle 3.771864438 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.007495 \cdot 251.613782 \cdot 2}{1^2} \approx 3.771864438$
Modular invariants
Modular form 75810.2.a.c
For more coefficients, see the Downloads section to the right.
Modular degree: | 217694400 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$3$ | $1$ | $I_{31}$ | Non-split multiplicative | 1 | 1 | 31 | 31 |
$5$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$7$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$19$ | $2$ | $III^{*}$ | Additive | 1 | 2 | 9 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15960 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 9577 & 2 \\ 9577 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 15959 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3991 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15959 & 2 \\ 15958 & 3 \end{array}\right),\left(\begin{array}{rr} 13681 & 2 \\ 13681 & 3 \end{array}\right),\left(\begin{array}{rr} 5321 & 2 \\ 5321 & 3 \end{array}\right),\left(\begin{array}{rr} 4201 & 2 \\ 4201 & 3 \end{array}\right),\left(\begin{array}{rr} 7981 & 2 \\ 7981 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[15960])$ is a degree-$4392005035622400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15960\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 75810a consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 75810cr1, its twist by $-19$.
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 |
---|---|---|---|
$3$ | 3.1.15960.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.4065356736000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | 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 | nonsplit | nonsplit | nonsplit | nonsplit | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 2 | 1 | 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 |
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.