Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2+1883755x+2402948582\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z+1883755xz^2+2402948582z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2441346453x+112075348853286\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(6023/4, 738283/8)$ | $4.6656361724839748729981366966$ | $\infty$ |
| $(-3581/4, 3577/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 124215 \) | = | $3 \cdot 5 \cdot 7^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-2920620885885805905075$ | = | $-1 \cdot 3 \cdot 5^{2} \cdot 7^{10} \cdot 13^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{1301812981559}{5143122075} \) | = | $3^{-1} \cdot 5^{-2} \cdot 7^{-4} \cdot 13^{-4} \cdot 61^{3} \cdot 179^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8015534615904290701402381507$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.54612370833200404956081805820$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9306063193741747$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.835765811640873$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.6656361724839748729981366966$ |
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| Real period: | $\Omega$ | ≈ | $0.10180093042514687319172790829$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.7997288270727174587970034243 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.799728827 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.101801 \cdot 4.665636 \cdot 32}{2^2} \\ & \approx 3.799728827\end{aligned}$$
Modular invariants
Modular form 124215.2.a.r
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6193152 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $13$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
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 |
|---|---|---|
| $2$ | 2B | 8.12.0.6 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 935 & 2176 \\ 1556 & 2151 \end{array}\right),\left(\begin{array}{rr} 1460 & 1 \\ 751 & 6 \end{array}\right),\left(\begin{array}{rr} 2177 & 8 \\ 2176 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 167 & 2176 \\ 668 & 2151 \end{array}\right),\left(\begin{array}{rr} 1915 & 1914 \\ 1378 & 283 \end{array}\right),\left(\begin{array}{rr} 1915 & 1912 \\ 842 & 1917 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2178 & 2179 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$81155063808$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 24843 = 3 \cdot 7^{2} \cdot 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 41405 = 5 \cdot 7^{2} \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 24843 = 3 \cdot 7^{2} \cdot 13^{2} \) |
| $7$ | additive | $32$ | \( 2535 = 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 735 = 3 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 124215.r
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1365.f4, its twist by $-91$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{91}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-273}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{91})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.6359808.1 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.364024420171776.68 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | ord | nonsplit | split | add | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 1 | 2 | - | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 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.