Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-1056556878x+13190899053616\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-1056556878xz^2+13190899053616z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1369297713267x+615438694138659534\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{11172253}{529}, \frac{6503112991}{12167}\right) \) | $14.958977848695477598872621015$ | $\infty$ |
| \( \left(18061, -9031\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([256961819:6503112991:12167]\) | $14.958977848695477598872621015$ | $\infty$ |
| \([18061:-9031:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{402202695}{529}, \frac{1432425596544}{12167}\right) \) | $14.958977848695477598872621015$ | $\infty$ |
| \( \left(650199, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(18061, -9031\right) \)
\([18061:-9031:1]\)
\( \left(650199, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 112530 \) | = | $2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 31$ |
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| Minimal Discriminant: | $\Delta$ | = | $315720421121620031199674640$ | = | $2^{4} \cdot 3 \cdot 5 \cdot 11^{16} \cdot 31^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{73628549562506871957390001}{178215946908754500240} \) | = | $2^{-4} \cdot 3^{-1} \cdot 5^{-1} \cdot 11^{-10} \cdot 31^{-5} \cdot 839^{3} \cdot 499559^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.9635213864605420741183347708$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.7645737500613568020873629818$ |
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| $abc$ quality: | $Q$ | ≈ | $1.011821271387809$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.357888816992408$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $14.958977848695477598872621015$ |
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| Real period: | $\Omega$ | ≈ | $0.054500382982646239381700011980$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2\cdot1\cdot1\cdot2^{2}\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.1527002178282505882576475754 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.152700218 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.054500 \cdot 14.958978 \cdot 40}{2^2} \\ & \approx 8.152700218\end{aligned}$$
Modular invariants
Modular form 112530.2.a.bg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 105600000 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $4$ | $I_{10}^{*}$ | additive | -1 | 2 | 16 | 10 |
| $31$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
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 | $3$ |
| $5$ | 5B.4.2 | 5.12.0.2 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 20460 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 31 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 11 & 16 \\ 20220 & 20111 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6836 & 5 \\ 20415 & 20446 \end{array}\right),\left(\begin{array}{rr} 20441 & 20 \\ 20440 & 21 \end{array}\right),\left(\begin{array}{rr} 18599 & 20440 \\ 1850 & 20259 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 10231 & 20 \\ 10 & 201 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 4636 & 5 \\ 13815 & 20446 \end{array}\right),\left(\begin{array}{rr} 12292 & 5 \\ 20415 & 20446 \end{array}\right)$.
The torsion field $K:=\Q(E[20460])$ is a degree-$90508492800000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20460\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 56265 = 3 \cdot 5 \cdot 11^{2} \cdot 31 \) |
| $3$ | split multiplicative | $4$ | \( 37510 = 2 \cdot 5 \cdot 11^{2} \cdot 31 \) |
| $5$ | split multiplicative | $6$ | \( 726 = 2 \cdot 3 \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| $31$ | split multiplicative | $32$ | \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 112530bf
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 10230bg3, its twist by $-11$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{465}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2 +26 \sqrt{-11}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{110 -22 \sqrt{5}})\) | \(\Z/10\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.684514342400625.8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.8.17112858560015625.7 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $10$ | 10.0.1031890245117187500000000.3 | \(\Z/10\Z\) | 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 |
| $16$ | deg 16 | \(\Z/20\Z\) | not in database |
| $20$ | 20.0.196366079890889247949046790723349785804748535156250000000000000000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $20$ | 20.0.457263716717479671795708718299865722656250000000000000000.1 | \(\Z/20\Z\) | 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 | split | split | ord | add | ord | ord | ss | ord | ord | split | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 2 | 6 | 1 | - | 1 | 1 | 1,1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 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.