Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-6953828x-9139360633\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-6953828xz^2-9139360633z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-111261243x-585030341738\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(8241, 698149\right) \) | $2.1081510362071145333073631321$ | $\infty$ |
| \( \left(3141, -1571\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([8241:698149:1]\) | $2.1081510362071145333073631321$ | $\infty$ |
| \([3141:-1571:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(32963, 5618160\right) \) | $2.1081510362071145333073631321$ | $\infty$ |
| \( \left(12563, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(3141, -1571\right) \), \( \left(8241, 698149\right) \), \( \left(8241, -706391\right) \), \( \left(3933541, 7799484189\right) \), \( \left(3933541, -7803417731\right) \)
\([3141:-1571:1]\), \([8241:698149:1]\), \([8241:-706391:1]\), \([3933541:7799484189:1]\), \([3933541:-7803417731:1]\)
\( \left(12563, 0\right) \), \((32963,\pm 5618160)\), \((15734163,\pm 62411607680)\)
Invariants
| Conductor: | $N$ | = | \( 26010 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 17^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-14577317354677787688960$ | = | $-1 \cdot 2^{18} \cdot 3^{13} \cdot 5 \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{2113364608155289}{828431400960} \) | = | $-1 \cdot 2^{-18} \cdot 3^{-7} \cdot 5^{-1} \cdot 17^{-2} \cdot 181^{3} \cdot 709^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9619404220872619711813316982$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.99602760572509908535894177080$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9973642386708145$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.842401885498882$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.1081510362071145333073631321$ |
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| Real period: | $\Omega$ | ≈ | $0.045600718789507656869821316038$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 288 $ = $ ( 2 \cdot 3^{2} )\cdot2^{2}\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.9215905848880659132443165721 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.921590585 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.045601 \cdot 2.108151 \cdot 288}{2^2} \\ & \approx 6.921590585\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2322432 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $18$ | $I_{18}$ | split multiplicative | -1 | 1 | 18 | 18 |
| $3$ | $4$ | $I_{7}^{*}$ | additive | -1 | 2 | 13 | 7 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $17$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
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 $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 257 & 1786 \\ 1784 & 255 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1021 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 241 & 4 \\ 482 & 9 \end{array}\right),\left(\begin{array}{rr} 682 & 1 \\ 679 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2037 & 4 \\ 2036 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1634 & 1 \\ 1223 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$231022264320$ 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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 13005 = 3^{2} \cdot 5 \cdot 17^{2} \) |
| $3$ | additive | $8$ | \( 1445 = 5 \cdot 17^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 26010bk
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 510a1, its twist by $-51$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.277440.3 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.17318914560000.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.974188944000000.51 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | split | add | nonsplit | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 9 | - | 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 |
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.