Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-2680x-50053\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-2680xz^2-50053z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-42875x-3246250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-31, 65\right) \) | $0.31348842129962495504329837459$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-31:65:1]\) | $0.31348842129962495504329837459$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-125, 400\right) \) | $0.31348842129962495504329837459$ | $\infty$ |
Integral points
\( \left(-31, 65\right) \), \( \left(-31, -35\right) \), \( \left(-25, 41\right) \), \( \left(-25, -17\right) \), \( \left(69, 265\right) \), \( \left(69, -335\right) \)
\([-31:65:1]\), \([-31:-35:1]\), \([-25:41:1]\), \([-25:-17:1]\), \([69:265:1]\), \([69:-335:1]\)
\((-125,\pm 400)\), \((-101,\pm 232)\), \((275,\pm 2400)\)
Invariants
| Conductor: | $N$ | = | \( 2450 \) | = | $2 \cdot 5^{2} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $120050000000$ | = | $2^{7} \cdot 5^{8} \cdot 7^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{2268945}{128} \) | = | $2^{-7} \cdot 3^{3} \cdot 5 \cdot 7^{5}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.88063195564151008172313727645$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.84096336899966126971248652685$ |
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| $abc$ quality: | $Q$ | ≈ | $1.2169692547295907$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.522639428626721$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.31348842129962495504329837459$ |
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| Real period: | $\Omega$ | ≈ | $0.66589236665648031604798968596$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 21 $ = $ 7\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.3837404823508317164903631023 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.383740482 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.665892 \cdot 0.313488 \cdot 21}{1^2} \\ & \approx 4.383740482\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2520 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $7$ | $1$ | $IV$ | additive | 1 | 2 | 4 | 0 |
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 |
|---|---|---|---|
| $2$ | 2G | 8.2.0.2 | $2$ |
| $7$ | 7Ns.2.1 | 7.112.1.2 | $112$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 56.224.5-56.t.1.4, level \( 56 = 2^{3} \cdot 7 \), index $224$, genus $5$, and generators
$\left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right),\left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 43 & 14 \\ 0 & 43 \end{array}\right),\left(\begin{array}{rr} 21 & 36 \\ 20 & 13 \end{array}\right),\left(\begin{array}{rr} 44 & 21 \\ 47 & 47 \end{array}\right),\left(\begin{array}{rr} 49 & 4 \\ 24 & 17 \end{array}\right),\left(\begin{array}{rr} 29 & 14 \\ 0 & 29 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 28 & 1 \end{array}\right),\left(\begin{array}{rr} 35 & 11 \\ 24 & 17 \end{array}\right),\left(\begin{array}{rr} 8 & 21 \\ 35 & 15 \end{array}\right),\left(\begin{array}{rr} 15 & 28 \\ 42 & 15 \end{array}\right),\left(\begin{array}{rr} 21 & 18 \\ 52 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[56])$ is a degree-$13824$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/56\Z)$.
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 | $4$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $14$ | \( 98 = 2 \cdot 7^{2} \) |
| $7$ | additive | $20$ | \( 25 = 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 2450.y consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 2450.i1, its twist by $5$.
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 |
|---|---|---|---|
| $3$ | 3.3.9800.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.768320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.10504375.1 | \(\Z/7\Z\) | not in database |
| $8$ | 8.2.52509870000.1 | \(\Z/3\Z\) | not in database |
| $9$ | 9.3.12867859375.1 | \(\Z/7\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.0.1159072634263427734375.1 | \(\Z/7\Z \oplus \Z/7\Z\) | not in database |
| $18$ | 18.0.303843936636352000000000000.1 | \(\Z/14\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | ss | add | add | ord | ss | ord | ss | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 1,1 | - | - | 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 | 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.